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<section class="ltx_section ltx_leqno">
<h1 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">§1.6 </span>Vectors and Vector-Valued Functions</h1>
<div id="info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for </dd>
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<h6>Contents</h6>
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<li class="ltx_tocentry"></li>
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<li class="ltx_tocentry"></li>
</ul>
<section id="i" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.6(i) </span>Vectors</h2>
<div id="SS1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Marsden and Tromba (, Chapter 1)</cite>.
For (, pp. 82–84)</cite>.</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="25px" altimg-valign="-7px" altimg-width="124px" alttext="\displaystyle=(a_{1},a_{2},a_{3})," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
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<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="19px" altimg-valign="-2px" altimg-width="19px" alttext="\displaystyle\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="118px" alttext="\displaystyle=(b_{1},b_{2},b_{3})." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><msub><mi>b</mi><mn>1</mn></msub><mo>,</mo><msub><mi>b</mi><mn>2</mn></msub><mo>,</mo><msub><mi>b</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Dot Product (or Scalar Product)</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="22px" altimg-valign="-5px" altimg-width="236px" alttext="\mathbf{a}\cdot\mathbf{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}." display="block"><mrow><mrow><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow><mo>=</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>3</mn></msub><mo></mo><msub><mi>b</mi><mn>3</mn></msub></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Magnitude and Angle of Vector <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math>
</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="27px" altimg-valign="-7px" altimg-width="122px" alttext="\|\mathbf{a}\|=\sqrt{\mathbf{a}\cdot\mathbf{a}}," display="block"><mrow><mrow><mrow><mo>∥</mo><mi mathvariant="bold">a</mi><mo>∥</mo></mrow><mo>=</mo><msqrt><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">a</mi></mrow></msqrt></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="51px" altimg-valign="-21px" altimg-width="152px" alttext="\cos\theta=\frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\;\|\mathbf{b}\|};" display="block"><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.6#Px2.p1">θ</mi></mrow><mo>=</mo><mfrac><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow><mrow><mrow><mo>∥</mo><mi mathvariant="bold">a</mi><mo rspace="5.3pt">∥</mo></mrow><mo></mo><mrow><mo>∥</mo><mi mathvariant="bold">b</mi><mo>∥</mo></mrow></mrow></mfrac></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and
<a href="./1.6#Px2.p1" title="Magnitude and Angle of Vector a ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.6#Px2.p1">θ</mi></math>: angle between <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.6#Px2.p1">θ</mi></math> is the angle between <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math>.</p>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Unit Vectors</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="E5" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="19px" altimg-valign="-2px" altimg-width="12px" alttext="\displaystyle\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(1,0,0)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-6px" altimg-width="13px" alttext="\displaystyle\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(0,1,0)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="19px" altimg-valign="-2px" altimg-width="18px" alttext="\displaystyle\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(0,0,1)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector (locally)</span>,
<span class="ltx_text"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector (locally)</span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector (locally)</span>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="23px" altimg-valign="-6px" altimg-width="182px" alttext="\mathbf{a}=a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}." display="block"><mrow><mrow><mi mathvariant="bold">a</mi><mo>=</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>3</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cross Product (or Vector Product)</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-6px" altimg-width="44px" alttext="\displaystyle\mathbf{i}\times\mathbf{j}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">i</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-6px" altimg-width="45px" alttext="\displaystyle=\mathbf{k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-6px" altimg-width="49px" alttext="\displaystyle\mathbf{j}\times\mathbf{k}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">j</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-6px" altimg-width="39px" alttext="\displaystyle=\mathbf{i}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="21px" altimg-valign="-4px" altimg-width="49px" alttext="\displaystyle\mathbf{k}\times\mathbf{i}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">k</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-6px" altimg-width="40px" alttext="\displaystyle=\mathbf{j}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E8" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-6px" altimg-width="44px" alttext="\displaystyle\mathbf{j}\times\mathbf{i}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">j</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-6px" altimg-width="60px" alttext="\displaystyle=-\mathbf{k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-6px" altimg-width="49px" alttext="\displaystyle\mathbf{k}\times\mathbf{j}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">k</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-6px" altimg-width="54px" alttext="\displaystyle=-\mathbf{i}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="21px" altimg-valign="-4px" altimg-width="49px" alttext="\displaystyle\mathbf{i}\times\mathbf{k}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">i</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-6px" altimg-width="55px" alttext="\displaystyle=-\mathbf{j}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\mathbf{a}\times\mathbf{b}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\
a_{1}&a_{2}&a_{3}\\
b_{1}&b_{2}&b_{3}\end{vmatrix}\\
=(a_{2}b_{3}-a_{3}b_{2})\mathbf{i}+(a_{3}b_{1}-a_{1}b_{3})\mathbf{j}+(a_{1}b_{%
2}-a_{2}b_{1})\mathbf{k}\\
=\|\mathbf{a}\|\|\mathbf{b}\|(\sin\theta)\mathbf{n}," display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><mi mathvariant="bold">a</mi><mo>×</mo><mi mathvariant="bold">b</mi></mrow></mtd><mtd><mo>=</mo><mrow><mo href="./1.3#i">|</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">i</mi></mtd><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">j</mi></mtd><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">k</mi></mtd></mtr><mtr><mtd columnalign="center"><msub><mi>a</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>2</mn></msub></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd columnalign="center"><msub><mi>b</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><msub><mi>b</mi><mn>2</mn></msub></mtd><mtd columnalign="center"><msub><mi>b</mi><mn>3</mn></msub></mtd></mtr></mtable><mo href="./1.3#i">|</mo></mrow><mo>=</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>3</mn></msub></mrow><mo>-</mo><mrow><msub><mi>a</mi><mn>3</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>3</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo>-</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>3</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo>-</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mn>1</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><mrow><mo>∥</mo><mi mathvariant="bold">a</mi><mo>∥</mo></mrow><mo></mo><mrow><mo>∥</mo><mi mathvariant="bold">b</mi><mo>∥</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.6#Px2.p1">θ</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi mathvariant="bold">n</mi></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.3#i" title="§1.3(i) Definitions and Elementary Properties ‣ §1.3 Determinants ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\det" display="inline"><mo href="./1.3#i">det</mo></math>: determinant</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.6#Px2.p1" title="Magnitude and Angle of Vector a ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.6#Px2.p1">θ</mi></math>: angle between <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math></a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{n}" display="inline"><mi mathvariant="bold">n</mi></math> is the unit vector normal to <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math>
whose direction is determined by the right-hand rule; see
Figure .</p>
</div>
<figure id="F1" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1.6.1: </span>Vector notation. Right-hand rule for cross products.
</figcaption>
<div id="F1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</figure>
<div id="Px4.p2" class="ltx_para">
<p class="ltx_p">Area of parallelogram with vectors <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math> as sides
<math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="=\|\mathbf{a}\times\mathbf{b}\|" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>∥</mo><mrow><mi mathvariant="bold">a</mi><mo>×</mo><mi mathvariant="bold">b</mi></mrow><mo>∥</mo></mrow></mrow></math>.
</p>
</div>
<div id="Px4.p3" class="ltx_para">
<p class="ltx_p">Volume of a parallelepiped with vectors <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math>, <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math>, and
<math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math> as edges
<math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="125px" alttext="=\left|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})\right|" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>|</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">b</mi><mo>×</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>|</mo></mrow></mrow></math>.
</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E10">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="25px" altimg-valign="-7px" altimg-width="104px" alttext="\displaystyle\mathbf{a}\times(\mathbf{b}\times\mathbf{c})" display="inline"><mrow><mi mathvariant="bold">a</mi><mo>×</mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">b</mi><mo>×</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="25px" altimg-valign="-7px" altimg-width="185px" alttext="\displaystyle=\mathbf{b}(\mathbf{a}\cdot\mathbf{c})-\mathbf{c}(\mathbf{a}\cdot%
\mathbf{b})," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi mathvariant="bold">b</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="25px" altimg-valign="-7px" altimg-width="104px" alttext="\displaystyle(\mathbf{a}\times\mathbf{b})\times\mathbf{c}" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">a</mi><mo>×</mo><mi mathvariant="bold">b</mi></mrow><mo stretchy="false">)</mo></mrow><mo>×</mo><mi mathvariant="bold">c</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="185px" alttext="\displaystyle=\mathbf{b}(\mathbf{a}\cdot\mathbf{c})-\mathbf{a}(\mathbf{b}\cdot%
\mathbf{c})." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi mathvariant="bold">b</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi mathvariant="bold">a</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">b</mi><mo>⋅</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.6(ii) </span>Vectors: Alternative Notations</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">The following notations are often used in the physics literature; see for
example <cite class="ltx_cite ltx_citemacro_citet">Lorentz<span class="ltx_text ltx_bib_etal"> et al.</span> (, pp. 122–123)</cite>.</p>
</div>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Einstein Summation Convention</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<p class="ltx_p">Much vector algebra involves summation over suffices of products of vector
components. In almost all cases of repeated suffices, we can suppress the
summation notation entirely, if it is understood that an implicit sum is to be
taken over any repeated suffix. Thus pairs of indefinite suffices in an
expression are resolved by being summed over (or “traced” over).</p>
</div>
</section>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Example</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="70px" altimg-valign="-30px" altimg-width="207px" alttext="a_{j}b_{j}=\sum_{j=1}^{3}a_{j}b_{j}=\mathbf{a}\cdot\mathbf{b}." display="block"><mrow><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>=</mo><mn>1</mn></mrow><mn>3</mn></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow></mrow><mo>=</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px6.p2" class="ltx_para">
<p class="ltx_p">Next,</p>
<table id="E13" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-5px" altimg-width="25px" alttext="\displaystyle\mathbf{e}_{1}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(1,0,0)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-5px" altimg-width="25px" alttext="\displaystyle\mathbf{e}_{2}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(0,1,0)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="17px" altimg-valign="-5px" altimg-width="25px" alttext="\displaystyle\mathbf{e}_{3}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mn>3</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(0,0,1);" display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m84.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="\mathbf{e}_{j}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: unit vectors (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare (). Thus <math class="ltx_Math" altimg="m108.png" altimg-height="18px" altimg-valign="-8px" altimg-width="80px" alttext="a_{j}\mathbf{e}_{j}=\mathbf{a}" display="inline"><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow><mo>=</mo><mi mathvariant="bold">a</mi></mrow></math>.</p>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Levi-Civita Symbol</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="92px" altimg-valign="-40px" altimg-width="461px" alttext="\epsilon_{jk\ell}=\begin{cases}+1,&\text{if }j,k,\ell\text{ is even %
permutation of }1,2,3,\\
-1,&\text{if }j,k,\ell\text{ is odd permutation of }1,2,3,\\
\phantom{-}0,&\text{otherwise}.\end{cases}" display="block"><mrow><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mo>+</mo><mn>1</mn></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mrow><mtext>if </mtext><mo></mo><mi href="./1.1#p2.t1.r4">j</mi></mrow><mo>,</mo><mi href="./1.1#p2.t1.r4">k</mi><mo>,</mo><mrow><mi mathvariant="normal">ℓ</mi><mo></mo><mtext> is even permutation of </mtext><mo></mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow><mo>-</mo><mn>1</mn></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mrow><mtext>if </mtext><mo></mo><mi href="./1.1#p2.t1.r4">j</mi></mrow><mo>,</mo><mi href="./1.1#p2.t1.r4">k</mi><mo>,</mo><mrow><mi mathvariant="normal">ℓ</mi><mo></mo><mtext> is odd permutation of </mtext><mo></mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mpadded lspace="7.5pt" width="+7.5pt"><mn>0</mn></mpadded><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>otherwise</mtext><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-8px" altimg-width="36px" alttext="\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi class="ltx_nvar" href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" mathvariant="normal">ℓ</mi></mrow></msub></math>: Levi-Civita symbol</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Examples</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<table id="E15" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="17px" altimg-valign="-5px" altimg-width="39px" alttext="\displaystyle\epsilon_{123}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mn>1</mn><mo href="./1.6#E14"></mo><mn>2</mn><mo href="./1.6#E14"></mo><mn>3</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="22px" altimg-valign="-6px" altimg-width="102px" alttext="\displaystyle=\epsilon_{312}=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi href="./1.6#E14">ϵ</mi><mrow><mn>3</mn><mo href="./1.6#E14"></mo><mn>1</mn><mo href="./1.6#E14"></mo><mn>2</mn></mrow></msub><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="17px" altimg-valign="-5px" altimg-width="39px" alttext="\displaystyle\epsilon_{213}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mn>2</mn><mo href="./1.6#E14"></mo><mn>1</mn><mo href="./1.6#E14"></mo><mn>3</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="22px" altimg-valign="-6px" altimg-width="118px" alttext="\displaystyle=\epsilon_{321}=-1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi href="./1.6#E14">ϵ</mi><mrow><mn>3</mn><mo href="./1.6#E14"></mo><mn>2</mn><mo href="./1.6#E14"></mo><mn>1</mn></mrow></msub><mo>=</mo><mrow><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="17px" altimg-valign="-5px" altimg-width="39px" alttext="\displaystyle\epsilon_{221}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mn>2</mn><mo href="./1.6#E14"></mo><mn>2</mn><mo href="./1.6#E14"></mo><mn>1</mn></mrow></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-2px" altimg-width="43px" alttext="\displaystyle=0." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.6#E14" title="(1.6.14) ‣ Levi-Civita Symbol ‣ §1.6(ii) Vectors: Alternative Notations ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-8px" altimg-width="36px" alttext="\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi class="ltx_nvar" href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" mathvariant="normal">ℓ</mi></mrow></msub></math>: Levi-Civita symbol</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="25px" altimg-valign="-8px" altimg-width="271px" alttext="\epsilon_{jk\ell}\epsilon_{\ell mn}=\delta_{j,m}\delta_{k,n}-\delta_{j,n}%
\delta_{k,m}," display="block"><mrow><mrow><mrow><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo></mo><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi mathvariant="normal">ℓ</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r5">m</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow><mo>=</mo><mrow><mrow><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./1.1#p2.t1.r5">m</mi></mrow></msub><mo></mo><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow><mo>-</mo><mrow><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub><mo></mo><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./1.1#p2.t1.r5">m</mi></mrow></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r4" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m61.png" altimg-height="23px" altimg-valign="-8px" altimg-width="35px" alttext="\delta_{\NVar{j},\NVar{k}}" display="inline"><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi class="ltx_nvar">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi class="ltx_nvar">k</mi></mrow></msub></math>: Kronecker delta</a>,
<a href="./1.6#E14" title="(1.6.14) ‣ Levi-Civita Symbol ‣ §1.6(ii) Vectors: Alternative Notations ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-8px" altimg-width="36px" alttext="\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi class="ltx_nvar" href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" mathvariant="normal">ℓ</mi></mrow></msub></math>: Levi-Civita symbol</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m62.png" altimg-height="23px" altimg-valign="-8px" altimg-width="34px" alttext="\delta_{j,k}" display="inline"><msub><mi href="./front/introduction#Sx4.p1.t1.r4">δ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./front/introduction#Sx4.p1.t1.r4">,</mo><mi href="./1.1#p2.t1.r4">k</mi></mrow></msub></math> is the Kronecker delta.</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="23px" altimg-valign="-8px" altimg-width="152px" alttext="\mathbf{e}_{j}\times\mathbf{e}_{k}=\epsilon_{jk\ell}\mathbf{e}_{\ell};" display="block"><mrow><mrow><mrow><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo>×</mo><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow><mo>=</mo><mrow><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo></mo><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi mathvariant="normal">ℓ</mi></msub></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E14" title="(1.6.14) ‣ Levi-Civita Symbol ‣ §1.6(ii) Vectors: Alternative Notations ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-8px" altimg-width="36px" alttext="\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi class="ltx_nvar" href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" mathvariant="normal">ℓ</mi></mrow></msub></math>: Levi-Civita symbol</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.6#E13" title="(1.6.13) ‣ Example ‣ §1.6(ii) Vectors: Alternative Notations ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="\mathbf{e}_{j}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: unit vectors</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare ().
</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="25px" altimg-valign="-8px" altimg-width="226px" alttext="a_{j}\mathbf{e}_{j}\times b_{k}\mathbf{e}_{k}=\epsilon_{jk\ell}a_{j}b_{k}%
\mathbf{e}_{\ell};" display="block"><mrow><mrow><mrow><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow><mo>×</mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow><mo></mo><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow><mo>=</mo><mrow><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo></mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi mathvariant="normal">ℓ</mi></msub></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E14" title="(1.6.14) ‣ Levi-Civita Symbol ‣ §1.6(ii) Vectors: Alternative Notations ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-8px" altimg-width="36px" alttext="\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}" display="inline"><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi class="ltx_nvar" href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi class="ltx_nvar" mathvariant="normal">ℓ</mi></mrow></msub></math>: Levi-Civita symbol</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.6#E13" title="(1.6.13) ‣ Example ‣ §1.6(ii) Vectors: Alternative Notations ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m84.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="\mathbf{e}_{j}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: unit vectors</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">compare ().</p>
</div>
<div id="Px8.p2" class="ltx_para">
<p class="ltx_p">Lastly, the volume of a parallelepiped with vectors <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math>, <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math>,
and <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math> as edges is <math class="ltx_Math" altimg="m136.png" altimg-height="24px" altimg-valign="-8px" altimg-width="101px" alttext="|\epsilon_{jk\ell}a_{j}b_{k}c_{\ell}|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14"></mo><mi href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14"></mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo></mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><msub><mi>c</mi><mi mathvariant="normal">ℓ</mi></msub></mrow><mo stretchy="false">|</mo></mrow></math>.
</p>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.6(iii) </span>Vector-Valued Functions</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Marsden and Tromba (, pp. 144–147, 273–283)</cite>.</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<section id="Px9" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Del Operator</h3>
<div id="Px9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px9.p1" class="ltx_para">
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="50px" altimg-valign="-20px" altimg-width="210px" alttext="\nabla=\mathbf{i}\frac{\partial}{\partial x}+\mathbf{j}\frac{\partial}{%
\partial y}+\mathbf{k}\frac{\partial}{\partial z}." display="block"><mrow><mrow><mo>∇</mo><mo>=</mo><mrow><mrow><mi href="./1.6#E5" mathvariant="bold">i</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mi href="./1.6#E5" mathvariant="bold">j</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mi href="./1.6#E5" mathvariant="bold">k</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px9.p2" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">gradient</em>
of a differentiable scalar function <math class="ltx_Math" altimg="m109.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="f(x,y,z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="50px" altimg-valign="-20px" altimg-width="306px" alttext="\operatorname{grad}f=\nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{%
\partial f}{\partial y}\mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}." display="block"><mrow><mrow><mrow><mo href="./1.6#E20">grad</mo><mi>f</mi></mrow><mo>=</mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo>=</mo><mrow><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m97.png" altimg-height="21px" altimg-valign="-6px" altimg-width="43px" alttext="\operatorname{grad}" display="inline"><mo href="./1.6#E20">grad</mo></math>: gradient of differentiable scalar function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px9.p3" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">divergence</em>
of a differentiable vector-valued function
<math class="ltx_Math" altimg="m70.png" altimg-height="21px" altimg-valign="-6px" altimg-width="185px" alttext="\mathbf{F}=F_{1}\mathbf{i}+F_{2}\mathbf{j}+F_{3}\mathbf{k}" display="inline"><mrow><mi mathvariant="bold">F</mi><mo>=</mo><mrow><mrow><msub><mi href="./1.6#Px9.p3">F</mi><mn>1</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><msub><mi href="./1.6#Px9.p3">F</mi><mn>2</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><msub><mi href="./1.6#Px9.p3">F</mi><mn>3</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow></math> is
</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="50px" altimg-valign="-20px" altimg-width="319px" alttext="\operatorname{div}\mathbf{F}=\nabla\cdot\mathbf{F}=\frac{\partial F_{1}}{%
\partial x}+\frac{\partial F_{2}}{\partial y}+\frac{\partial F_{3}}{\partial z}." display="block"><mrow><mrow><mrow><mo href="./1.6#E21">div</mo><mi mathvariant="bold">F</mi></mrow><mo>=</mo><mrow><mo>∇</mo><mo>⋅</mo><mi mathvariant="bold">F</mi></mrow><mo>=</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>1</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>2</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>3</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m96.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\operatorname{div}" display="inline"><mo href="./1.6#E21">div</mo></math>: divergence of vector-valued function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.6#Px9.p3" title="Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="F_{j}" display="inline"><msub><mi href="./1.6#Px9.p3">F</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: vector function components</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px9.p4" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">curl</em> of
<math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi mathvariant="bold">F</mi></math> is
</p>
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="152px" altimg-valign="-43px" altimg-width="564px" alttext="\operatorname{curl}\mathbf{F}=\nabla\times\mathbf{F}=\begin{vmatrix}\mathbf{i}%
&\mathbf{j}&\mathbf{k}\\
\displaystyle{\frac{\partial}{\partial x}}&\displaystyle{\frac{\partial}{%
\partial y}}&\displaystyle{\frac{\partial}{\partial z}}\\
F_{1}&F_{2}&F_{3}\end{vmatrix}\\
=\left(\frac{\partial F_{3}}{\partial y}-\frac{\partial F_{2}}{\partial z}%
\right)\mathbf{i}+\left(\frac{\partial F_{1}}{\partial z}-\frac{\partial F_{3}%
}{\partial x}\right)\mathbf{j}+\left(\frac{\partial F_{2}}{\partial x}-\frac{%
\partial F_{1}}{\partial y}\right)\mathbf{k}." display="block"><mrow><mrow><mrow><mo href="./1.6#E22">curl</mo><mi mathvariant="bold">F</mi></mrow><mo>=</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">F</mi></mrow><mo>=</mo><mrow><mo href="./1.3#i">|</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">i</mi></mtd><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">j</mi></mtd><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">k</mi></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mtd><mtd columnalign="center"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mtd><mtd columnalign="center"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="center"><msub><mi href="./1.6#Px9.p3">F</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><msub><mi href="./1.6#Px9.p3">F</mi><mn>2</mn></msub></mtd><mtd columnalign="center"><msub><mi href="./1.6#Px9.p3">F</mi><mn>3</mn></msub></mtd></mtr></mtable><mo href="./1.3#i">|</mo></mrow><mo>=</mo><mrow><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>3</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>2</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>1</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>3</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>2</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#Px9.p3">F</mi><mn>1</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-2px" altimg-width="38px" alttext="\operatorname{curl}" display="inline"><mo href="./1.6#E22">curl</mo></math>: of vector-valued function</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.3#i" title="§1.3(i) Definitions and Elementary Properties ‣ §1.3 Determinants ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\det" display="inline"><mo href="./1.3#i">det</mo></math>: determinant</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a> and
<a href="./1.6#Px9.p3" title="Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="F_{j}" display="inline"><msub><mi href="./1.6#Px9.p3">F</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: vector function components</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="25px" altimg-valign="-7px" altimg-width="193px" alttext="\nabla(fg)=f\nabla g+g\nabla f," display="block"><mrow><mrow><mrow><mo>∇</mo><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo></mo><mi>g</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow></mrow><mo>+</mo><mrow><mi>g</mi><mo></mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="28px" altimg-valign="-7px" altimg-width="249px" alttext="\nabla(f/g)=(g\nabla f-f\nabla g)/g^{2}," display="block"><mrow><mrow><mrow><mo>∇</mo><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo>/</mo><mi>g</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>g</mi><mo></mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><msup><mi>g</mi><mn>2</mn></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E25" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E25.png" altimg-height="25px" altimg-valign="-7px" altimg-width="264px" alttext="\nabla\cdot(f\mathbf{F})=f(\nabla\cdot\mathbf{F})+\mathbf{F}\cdot\nabla f," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo></mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>⋅</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="25px" altimg-valign="-7px" altimg-width="372px" alttext="\nabla\cdot(\mathbf{F}\times\mathbf{G})=\mathbf{G}\cdot(\nabla\times\mathbf{F}%
)-\mathbf{F}\cdot(\nabla\times\mathbf{G})," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">F</mi><mo>×</mo><mi mathvariant="bold">G</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi mathvariant="bold">G</mi><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">G</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="258px" alttext="\nabla\cdot(\nabla\times\mathbf{F})=\operatorname{div}\operatorname{curl}%
\mathbf{F}=0," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo href="./1.6#E21">div</mo><mrow><mo href="./1.6#E22">curl</mo><mi mathvariant="bold">F</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E22" title="(1.6.22) ‣ Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-2px" altimg-width="38px" alttext="\operatorname{curl}" display="inline"><mo href="./1.6#E22">curl</mo></math>: of vector-valued function</a> and
<a href="./1.6#E21" title="(1.6.21) ‣ Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\operatorname{div}" display="inline"><mo href="./1.6#E21">div</mo></math>: divergence of vector-valued function</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="25px" altimg-valign="-7px" altimg-width="311px" alttext="\nabla\times(f\mathbf{F})=f(\nabla\times\mathbf{F})+(\nabla f)\times\mathbf{F}," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>×</mo><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo></mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo stretchy="false">)</mo></mrow><mo>×</mo><mi mathvariant="bold">F</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="25px" altimg-valign="-7px" altimg-width="250px" alttext="\nabla\times(\nabla f)=\operatorname{curl}\operatorname{grad}f=0," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>×</mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo href="./1.6#E22">curl</mo><mrow><mo href="./1.6#E20">grad</mo><mi>f</mi></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.6#E22" title="(1.6.22) ‣ Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-2px" altimg-width="38px" alttext="\operatorname{curl}" display="inline"><mo href="./1.6#E22">curl</mo></math>: of vector-valued function</a> and
<a href="./1.6#E20" title="(1.6.20) ‣ Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="21px" altimg-valign="-6px" altimg-width="43px" alttext="\operatorname{grad}" display="inline"><mo href="./1.6#E20">grad</mo></math>: gradient of differentiable scalar function</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="28px" altimg-valign="-7px" altimg-width="151px" alttext="\nabla^{2}f=\nabla\cdot(\nabla f)," display="block"><mrow><mrow><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow><mo>=</mo><mrow><mo>∇</mo><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="28px" altimg-valign="-7px" altimg-width="341px" alttext="\nabla^{2}(fg)=f\nabla^{2}g+g\nabla^{2}f+2(\nabla f\cdot\nabla g)," display="block"><mrow><mrow><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo></mo><mi>g</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>g</mi></mrow></mrow><mo>+</mo><mrow><mi>g</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo>⋅</mo><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="25px" altimg-valign="-7px" altimg-width="174px" alttext="\nabla\cdot(\nabla f\times\nabla g)=0," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo>×</mo><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="28px" altimg-valign="-7px" altimg-width="305px" alttext="\nabla\cdot(f\nabla g-g\nabla f)=f\nabla^{2}g-g\nabla^{2}f," display="block"><mrow><mrow><mrow><mo>∇</mo><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow></mrow><mo>-</mo><mrow><mi>g</mi><mo></mo><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>g</mi></mrow></mrow><mo>-</mo><mrow><mi>g</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="28px" altimg-valign="-7px" altimg-width="406px" alttext="\nabla\times(\nabla\times\mathbf{F})=\operatorname{curl}\operatorname{curl}%
\mathbf{F}=\nabla(\nabla\cdot\mathbf{F})-\nabla^{2}\mathbf{F}." display="block"><mrow><mrow><mrow><mo>∇</mo><mo>×</mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo href="./1.6#E22">curl</mo><mrow><mo href="./1.6#E22">curl</mo><mi mathvariant="bold">F</mi></mrow></mrow><mo>=</mo><mrow><mrow><mo>∇</mo><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>⋅</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi mathvariant="bold">F</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.6#E22" title="(1.6.22) ‣ Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="18px" altimg-valign="-2px" altimg-width="38px" alttext="\operatorname{curl}" display="inline"><mo href="./1.6#E22">curl</mo></math>: of vector-valued function</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.6(iv) </span>Path and Line Integrals</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Marsden and Tromba (, pp. 396–417, 470)</cite>.</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">Note: The terminology <em class="ltx_emph ltx_font_italic">open</em> and <em class="ltx_emph ltx_font_italic">closed sets</em> and <em class="ltx_emph ltx_font_italic">boundary
points</em> in the <math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(x,y)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> plane that is used in this subsection and
§ is analogous to that introduced for the complex plane in
§.
</p>
</div>
<div id="SS4.p2" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m79.png" altimg-height="23px" altimg-valign="-7px" altimg-width="197px" alttext="\mathbf{c}(t)=(x(t),y(t),z(t))" display="inline"><mrow><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, with <math class="ltx_Math" altimg="m121.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math> ranging
over an interval and <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="122px" alttext="x(t),y(t),z(t)" display="inline"><mrow><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> differentiable, defines a <em class="ltx_emph ltx_font_italic">path</em>.</p>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="26px" altimg-valign="-7px" altimg-width="227px" alttext="\mathbf{c}^{\prime}(t)=(x^{\prime}(t),y^{\prime}(t),z^{\prime}(t))." display="block"><mrow><mrow><mrow><msup><mi mathvariant="bold">c</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>x</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msup><mi>y</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msup><mi>z</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">length</em> of a path for <math class="ltx_Math" altimg="m107.png" altimg-height="20px" altimg-valign="-5px" altimg-width="84px" alttext="a\leq t\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>b</mi></mrow></math> is
</p>
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="56px" altimg-valign="-20px" altimg-width="119px" alttext="\int_{a}^{b}\|\mathbf{c}^{\prime}(t)\|\mathrm{d}t." display="block"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mo>∥</mo><mrow><msup><mi mathvariant="bold">c</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>∥</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">path integral</em> of a continuous function <math class="ltx_Math" altimg="m109.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="f(x,y,z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is
</p>
<table id="E37" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="56px" altimg-valign="-20px" altimg-width="347px" alttext="\int_{\mathbf{c}}f\mathrm{d}s=\int^{b}_{a}f(x(t),y(t),z(t))\|\mathbf{c}^{%
\prime}(t)\|\mathrm{d}t." display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><mi>f</mi><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>s</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo>∥</mo><mrow><msup><mi mathvariant="bold">c</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>∥</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">line integral</em> of a vector-valued function <math class="ltx_Math" altimg="m70.png" altimg-height="21px" altimg-valign="-6px" altimg-width="185px" alttext="\mathbf{F}=F_{1}\mathbf{i}+F_{2}\mathbf{j}+F_{3}\mathbf{k}" display="inline"><mrow><mi mathvariant="bold">F</mi><mo>=</mo><mrow><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>3</mn></msub><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow></math> along <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math> is given
by
</p>
<table id="E38" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E38.png" altimg-height="56px" altimg-valign="-21px" altimg-width="807px" alttext="\int_{\mathbf{c}}\mathbf{F}\cdot\mathrm{d}\mathbf{s}=\int^{b}_{a}\mathbf{F}(%
\mathbf{c}(t))\cdot\mathbf{c}^{\prime}(t)\mathrm{d}t=\int^{b}_{a}\left(F_{1}%
\frac{\mathrm{d}x}{\mathrm{d}t}+F_{2}\frac{\mathrm{d}y}{\mathrm{d}t}+F_{3}%
\frac{\mathrm{d}z}{\mathrm{d}t}\right)\mathrm{d}t=\int_{\mathbf{c}}F_{1}%
\mathrm{d}x+F_{2}\mathrm{d}y+F_{3}\mathrm{d}z." display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mrow><mi mathvariant="bold">F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>⋅</mo><mrow><msup><mi mathvariant="bold">c</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mo>(</mo><mrow><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>x</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>y</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>3</mn></msub><mo></mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>z</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>3</mn></msub><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.6#SS4.p2" title="§1.6(iv) Path and Line Integrals ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="F_{j}" display="inline"><msub><mi href="./1.6#SS4.p2">F</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: vector function components</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">A path <math class="ltx_Math" altimg="m83.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\mathbf{c}_{1}(t)" display="inline"><mrow><msub><mi mathvariant="bold">c</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m122.png" altimg-height="23px" altimg-valign="-7px" altimg-width="75px" alttext="t\in[a,b]" display="inline"><mrow><mi>t</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>, is a <em class="ltx_emph ltx_font_italic">reparametrization</em> of
<math class="ltx_Math" altimg="m80.png" altimg-height="24px" altimg-valign="-7px" altimg-width="43px" alttext="\mathbf{c}(t^{\prime})" display="inline"><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m124.png" altimg-height="24px" altimg-valign="-7px" altimg-width="92px" alttext="t^{\prime}\in[a^{\prime},b^{\prime}]" display="inline"><mrow><msup><mi>t</mi><mo>′</mo></msup><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><msup><mi>a</mi><mo>′</mo></msup><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><msup><mi>b</mi><mo>′</mo></msup><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>, if <math class="ltx_Math" altimg="m82.png" altimg-height="24px" altimg-valign="-7px" altimg-width="111px" alttext="\mathbf{c}_{1}(t)=\mathbf{c}(t^{\prime})" display="inline"><mrow><mrow><msub><mi mathvariant="bold">c</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo></mrow></mrow></mrow></math> and
<math class="ltx_Math" altimg="m123.png" altimg-height="24px" altimg-valign="-7px" altimg-width="78px" alttext="t^{\prime}=h(t)" display="inline"><mrow><msup><mi>t</mi><mo>′</mo></msup><mo>=</mo><mrow><mi href="./1.6#SS4.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> with <math class="ltx_Math" altimg="m116.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="h(t)" display="inline"><mrow><mi href="./1.6#SS4.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> differentiable and monotonic. If <math class="ltx_Math" altimg="m112.png" altimg-height="24px" altimg-valign="-7px" altimg-width="84px" alttext="h(a)=a^{\prime}" display="inline"><mrow><mrow><mi href="./1.6#SS4.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mi>a</mi><mo>′</mo></msup></mrow></math> and
<math class="ltx_Math" altimg="m115.png" altimg-height="24px" altimg-valign="-7px" altimg-width="81px" alttext="h(b)=b^{\prime}" display="inline"><mrow><mrow><mi href="./1.6#SS4.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mi>b</mi><mo>′</mo></msup></mrow></math>, then the
reparametrization is called <em class="ltx_emph ltx_font_italic">orientation-preserving</em>, and
</p>
<table id="E39" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E39.png" altimg-height="54px" altimg-valign="-22px" altimg-width="191px" alttext="\int_{\mathbf{c}}\mathbf{F}\cdot\mathrm{d}\mathbf{s}=\int_{\mathbf{c}_{1}}%
\mathbf{F}\cdot\mathrm{d}\mathbf{s}." display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi mathvariant="bold">c</mi><mn>1</mn></msub></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If <math class="ltx_Math" altimg="m113.png" altimg-height="24px" altimg-valign="-7px" altimg-width="83px" alttext="h(a)=b^{\prime}" display="inline"><mrow><mrow><mi href="./1.6#SS4.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mi>b</mi><mo>′</mo></msup></mrow></math> and <math class="ltx_Math" altimg="m114.png" altimg-height="24px" altimg-valign="-7px" altimg-width="83px" alttext="h(b)=a^{\prime}" display="inline"><mrow><mrow><mi href="./1.6#SS4.p2">h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msup><mi>a</mi><mo>′</mo></msup></mrow></math>, then the reparametrization is
<em class="ltx_emph ltx_font_italic">orientation-reversing</em> and
</p>
<table id="E40" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E40.png" altimg-height="54px" altimg-valign="-22px" altimg-width="209px" alttext="\int_{\mathbf{c}}\mathbf{F}\cdot\mathrm{d}\mathbf{s}=-\int_{\mathbf{c}_{1}}%
\mathbf{F}\cdot\mathrm{d}\mathbf{s}." display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi mathvariant="bold">c</mi><mn>1</mn></msub></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In either case</p>
<table id="E41" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E41.png" altimg-height="54px" altimg-valign="-22px" altimg-width="157px" alttext="\int_{\mathbf{c}}f\mathrm{d}s=\int_{\mathbf{c}_{1}}f\mathrm{d}s," display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><mi>f</mi><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>s</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi mathvariant="bold">c</mi><mn>1</mn></msub></msub><mrow><mi>f</mi><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>s</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> is continuous, and</p>
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E42.png" altimg-height="52px" altimg-valign="-20px" altimg-width="274px" alttext="\int_{\mathbf{c}}\nabla f\cdot\mathrm{d}\mathbf{s}=f(\mathbf{c}(b))-f(\mathbf{%
c}(a))," display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi mathvariant="bold">c</mi></msub><mrow><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> is continuously differentiable.</p>
</div>
<div id="SS4.p3" class="ltx_para">
<p class="ltx_p">The geometrical image <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math> of a path <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math> is called a <em class="ltx_emph ltx_font_italic">simple
closed curve</em>
if <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math> is one-to-one, with the exception <math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="\mathbf{c}(a)=\mathbf{c}(b)" display="inline"><mrow><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi mathvariant="bold">c</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. The curve <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math> is <em class="ltx_emph ltx_font_italic">piecewise differentiable</em>
if <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math> is piecewise differentiable. Note that <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math> can be given an
orientation by means of <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math>.
</p>
</div>
<section id="Px10" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Green’s Theorem</h3>
<div id="Px10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px10.p1" class="ltx_para">
<p class="ltx_p">Let</p>
<table id="E43" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E43.png" altimg-height="25px" altimg-valign="-7px" altimg-width="267px" alttext="\mathbf{F}(x,y)=F_{1}(x,y)\mathbf{i}+F_{2}(x,y)\mathbf{j}" display="block"><mrow><mrow><mi mathvariant="bold">F</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E43.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#SS4.p2" title="§1.6(iv) Path and Line Integrals ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="F_{j}" display="inline"><msub><mi href="./1.6#SS4.p2">F</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: vector function components</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#Px10.p1">S</mi></math> be the closed and bounded point set in the <math class="ltx_Math" altimg="m40.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(x,y)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> plane having a
simple closed curve <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math>
as boundary. If <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math> is oriented in the positive (anticlockwise) sense, then</p>
<table id="E44" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.44</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E44.png" altimg-height="53px" altimg-valign="-21px" altimg-width="480px" alttext="\iint_{S}\left(\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{%
\partial y}\right)\mathrm{d}A=\int_{C}\mathbf{F}\cdot\mathrm{d}\mathbf{s}=\int%
_{C}F_{1}\mathrm{d}x+F_{2}\mathrm{d}y." display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#Px10.p1">S</mi></msub><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>A</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.6#Px10.p1">C</mi></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.6#Px10.p1">C</mi></msub><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow><mo>+</mo><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E44.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#Px10.p1" title="Green’s Theorem ‣ §1.6(iv) Path and Line Integrals ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#Px10.p1">S</mi></math>: closed region</a>,
<a href="./1.6#Px10.p1" title="Green’s Theorem ‣ §1.6(iv) Path and Line Integrals ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math>: closed curve</a> and
<a href="./1.6#SS4.p2" title="§1.6(iv) Path and Line Integrals ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-8px" altimg-width="26px" alttext="F_{j}" display="inline"><msub><mi href="./1.6#SS4.p2">F</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: vector function components</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Sufficient conditions for this result to hold are that <math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="F_{1}(x,y)" display="inline"><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> and
<math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="F_{2}(x,y)" display="inline"><mrow><msub><mi href="./1.6#SS4.p2">F</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> are continuously differentiable on <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#Px10.p1">S</mi></math>, and <math class="ltx_Math" altimg="m45.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.6#Px10.p1">C</mi></math> is piecewise
differentiable.</p>
</div>
<div id="Px10.p2" class="ltx_para">
<p class="ltx_p">The area of <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#Px10.p1">S</mi></math> can be found from () by taking
<math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="\mathbf{F}(x,y)=-y\mathbf{i}" display="inline"><mrow><mrow><mi mathvariant="bold">F</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><mi>y</mi><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m129.png" altimg-height="21px" altimg-valign="-6px" altimg-width="23px" alttext="x\mathbf{j}" display="inline"><mrow><mi>x</mi><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></math>, or
<math class="ltx_Math" altimg="m41.png" altimg-height="27px" altimg-valign="-9px" altimg-width="105px" alttext="-\frac{1}{2}y\mathbf{i}+\frac{1}{2}x\mathbf{j}" display="inline"><mrow><mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>y</mi><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>x</mi><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></mrow></math>.</p>
</div>
</section>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.6(v) </span>Surfaces and Integrals over Surfaces</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Marsden and Tromba (, pp. 421–459, 485, 506)</cite>.</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<p class="ltx_p">A <em class="ltx_emph ltx_font_italic">parametrized surface</em> <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> is defined by
</p>
<table id="E45" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.45</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E45.png" altimg-height="25px" altimg-valign="-7px" altimg-width="299px" alttext="\boldsymbol{{\Phi}}(u,v)=(x(u,v),y(u,v),z(u,v))" display="block"><mrow><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E45.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="\boldsymbol{{\Phi}}(x,y,z)" display="inline"><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: parameterization (locally)</span></dd>
<dt>Symbols:</dt>
<dd><a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m38.png" altimg-height="23px" altimg-valign="-7px" altimg-width="92px" alttext="(u,v)\in D" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.6#SS5.p1">D</mi></mrow></math>, an open set in the plane.</p>
</div>
<div id="SS5.p2" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math>, <math class="ltx_Math" altimg="m132.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi>y</mi></math>, and <math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi>z</mi></math> continuously differentiable, the vectors</p>
<table id="E46" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.46</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E46.png" altimg-height="47px" altimg-valign="-16px" altimg-width="407px" alttext="\mathbf{T}_{u}=\frac{\partial x}{\partial u}(u_{0},v_{0})\mathbf{i}+\frac{%
\partial y}{\partial u}(u_{0},v_{0})\mathbf{j}+\frac{\partial z}{\partial u}(u%
_{0},v_{0})\mathbf{k}" display="block"><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>=</mo><mrow><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>u</mi></mrow></mfrac><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>u</mi></mrow></mfrac><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>u</mi></mrow></mfrac><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E46.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E47" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.47</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E47.png" altimg-height="47px" altimg-valign="-16px" altimg-width="404px" alttext="\mathbf{T}_{v}=\frac{\partial x}{\partial v}(u_{0},v_{0})\mathbf{i}+\frac{%
\partial y}{\partial v}(u_{0},v_{0})\mathbf{j}+\frac{\partial z}{\partial v}(u%
_{0},v_{0})\mathbf{k}" display="block"><mrow><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub><mo>=</mo><mrow><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac><mo></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo></mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E47.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>,
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and
<a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">are tangent to the surface at <math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="\boldsymbol{{\Phi}}(u_{0},v_{0})" display="inline"><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo>,</mo><msub><mi>v</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math>. The surface is
<em class="ltx_emph ltx_font_italic">smooth</em>
at this point if <math class="ltx_Math" altimg="m73.png" altimg-height="21px" altimg-valign="-6px" altimg-width="117px" alttext="\mathbf{T}_{u}\times\mathbf{T}_{v}\not=0" display="inline"><mrow><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow><mo>≠</mo><mn>0</mn></mrow></math>. A surface is
<em class="ltx_emph ltx_font_italic">smooth</em> if it is smooth at every point. The vector
<math class="ltx_Math" altimg="m72.png" altimg-height="21px" altimg-valign="-5px" altimg-width="80px" alttext="\mathbf{T}_{u}\times\mathbf{T}_{v}" display="inline"><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow></math> at <math class="ltx_Math" altimg="m39.png" altimg-height="23px" altimg-valign="-7px" altimg-width="68px" alttext="(u_{0},v_{0})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>v</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> is normal to the surface at
<math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="84px" alttext="\boldsymbol{{\Phi}}(u_{0},v_{0})" display="inline"><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo>,</mo><msub><mi>v</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math>.
</p>
</div>
<div id="SS5.p3" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">area</em> <math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math> of a parametrized smooth surface is given by
</p>
<table id="E48" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.48</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E48.png" altimg-height="52px" altimg-valign="-20px" altimg-width="264px" alttext="A(S)=\iint_{D}\|\mathbf{T}_{u}\times\mathbf{T}_{v}\|\mathrm{d}u\mathrm{d}v," display="block"><mrow><mrow><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">D</mi></msub><mrow><mrow><mo>∥</mo><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow><mo>∥</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>v</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E48.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math>: area of a parameterized smooth surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a> and
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.6#SS5.p1">D</mi></math>: open set in the plane</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E49" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.49</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E49.png" altimg-height="66px" altimg-valign="-23px" altimg-width="506px" alttext="\|\mathbf{T}_{u}\times\mathbf{T}_{v}\|=\sqrt{\left(\frac{\partial(x,y)}{%
\partial(u,v)}\right)^{2}+\left(\frac{\partial(y,z)}{\partial(u,v)}\right)^{2}%
+\left(\frac{\partial(x,z)}{\partial(u,v)}\right)^{2}}." display="block"><mrow><mrow><mrow><mo>∥</mo><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow><mo>∥</mo></mrow><mo>=</mo><msqrt><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>z</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>z</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E49.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The area is independent of the parametrizations.</p>
</div>
<div id="SS5.p4" class="ltx_para">
<p class="ltx_p">For a sphere <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="139px" alttext="x=\rho\sin\theta\cos\phi" display="inline"><mrow><mi>x</mi><mo>=</mo><mrow><mi href="./1.6#SS5.p4">ρ</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.6#SS5.p4">θ</mi></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.6#SS5.p4">ϕ</mi></mrow></mrow></mrow></math>,
<math class="ltx_Math" altimg="m130.png" altimg-height="21px" altimg-valign="-6px" altimg-width="136px" alttext="y=\rho\sin\theta\sin\phi" display="inline"><mrow><mi>y</mi><mo>=</mo><mrow><mi href="./1.6#SS5.p4">ρ</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.6#SS5.p4">θ</mi></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.6#SS5.p4">ϕ</mi></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m133.png" altimg-height="21px" altimg-valign="-6px" altimg-width="95px" alttext="z=\rho\cos\theta" display="inline"><mrow><mi>z</mi><mo>=</mo><mrow><mi href="./1.6#SS5.p4">ρ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.6#SS5.p4">θ</mi></mrow></mrow></mrow></math>,
</p>
<table id="E50" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.50</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E50.png" altimg-height="29px" altimg-valign="-8px" altimg-width="209px" alttext="\|\mathbf{T}_{\theta}\times\mathbf{T}_{\phi}\|=\rho^{2}\left|\sin\theta\right|." display="block"><mrow><mrow><mrow><mo>∥</mo><mrow><msub><mi mathvariant="bold">T</mi><mi href="./1.6#SS5.p4">θ</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi href="./1.6#SS5.p4">ϕ</mi></msub></mrow><mo>∥</mo></mrow><mo>=</mo><mrow><msup><mi href="./1.6#SS5.p4">ρ</mi><mn>2</mn></msup><mo></mo><mrow><mo>|</mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.6#SS5.p4">θ</mi></mrow><mo>|</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E50.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.6#SS5.p4" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m102.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./1.6#SS5.p4">ρ</mi></math>: radius</a>,
<a href="./1.6#SS5.p4" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.6#SS5.p4">θ</mi></math>: angle</a> and
<a href="./1.6#SS5.p4" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m100.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.6#SS5.p4">ϕ</mi></math>: angle</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS5.p5" class="ltx_para">
<p class="ltx_p">For a surface <math class="ltx_Math" altimg="m134.png" altimg-height="23px" altimg-valign="-7px" altimg-width="99px" alttext="z=f(x,y)" display="inline"><mrow><mi>z</mi><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>,</p>
<table id="E51" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.51</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E51.png" altimg-height="66px" altimg-valign="-23px" altimg-width="362px" alttext="A(S)=\iint_{D}\sqrt{1+\left(\frac{\partial f}{\partial x}\right)^{2}+\left(%
\frac{\partial f}{\partial y}\right)^{2}}\mathrm{d}A." display="block"><mrow><mrow><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">D</mi></msub><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#E48">A</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E51.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.6#SS5.p1">D</mi></math>: open set in the plane</a> and
<a href="./1.6#E48" title="(1.6.48) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math>: area of a parameterized smooth surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS5.p6" class="ltx_para">
<p class="ltx_p">For a surface of revolution, <math class="ltx_Math" altimg="m131.png" altimg-height="23px" altimg-valign="-7px" altimg-width="80px" alttext="y=f(x)" display="inline"><mrow><mi>y</mi><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m128.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="x\in[a,b]" display="inline"><mrow><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>, about the <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math>-axis,
</p>
<table id="E52" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.52</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E52.png" altimg-height="56px" altimg-valign="-20px" altimg-width="334px" alttext="A(S)=2\pi\int^{b}_{a}|f(x)|\sqrt{1+(f^{\prime}(x))^{2}}\mathrm{d}x," display="block"><mrow><mrow><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E52.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a> and
<a href="./1.6#E48" title="(1.6.48) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math>: area of a parameterized smooth surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and about the <math class="ltx_Math" altimg="m132.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi>y</mi></math>-axis,</p>
<table id="E53" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.53</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E53.png" altimg-height="56px" altimg-valign="-20px" altimg-width="307px" alttext="A(S)=2\pi\int^{b}_{a}|x|\sqrt{1+(f^{\prime}(x))^{2}}\mathrm{d}x." display="block"><mrow><mrow><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow><mo></mo><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E53.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a> and
<a href="./1.6#E48" title="(1.6.48) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math>: area of a parameterized smooth surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS5.p7" class="ltx_para">
<p class="ltx_p">The integral of a continuous function <math class="ltx_Math" altimg="m109.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="f(x,y,z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> over a surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> is
</p>
<table id="E54" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.54</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E54.png" altimg-height="52px" altimg-valign="-20px" altimg-width="451px" alttext="\iint_{S}f(x,y,z)\mathrm{d}S=\iint_{D}f(\boldsymbol{{\Phi}}(u,v))\|\mathbf{T}_%
{u}\times\mathbf{T}_{v}\|\mathrm{d}u\mathrm{d}v." display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">S</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#SS5.p1">S</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">D</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo>∥</mo><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow><mo>∥</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>v</mi></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E54.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a>,
<a href="./1.6#E45" title="(1.6.45) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="\boldsymbol{{\Phi}}(x,y,z)" display="inline"><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: parameterization</a> and
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.6#SS5.p1">D</mi></math>: open set in the plane</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For a vector-valued function <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi mathvariant="bold">F</mi></math>,</p>
<table id="E55" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.55</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E55.png" altimg-height="52px" altimg-valign="-20px" altimg-width="336px" alttext="\iint_{S}\mathbf{F}\cdot\mathrm{d}\mathbf{S}=\iint_{D}\mathbf{F}\cdot(\mathbf{%
T}_{u}\times\mathbf{T}_{v})\mathrm{d}u\mathrm{d}v," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">S</mi></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">S</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">D</mi></msub><mrow><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>v</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E55.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a> and
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.6#SS5.p1">D</mi></math>: open set in the plane</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m90.png" altimg-height="18px" altimg-valign="-2px" altimg-width="28px" alttext="\mathrm{d}\mathbf{S}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">S</mi></mrow></math> is the surface element with an attached normal
direction <math class="ltx_Math" altimg="m72.png" altimg-height="21px" altimg-valign="-5px" altimg-width="80px" alttext="\mathbf{T}_{u}\times\mathbf{T}_{v}" display="inline"><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow></math>.</p>
</div>
<div id="SS5.p8" class="ltx_para">
<p class="ltx_p">A surface is <em class="ltx_emph ltx_font_italic">orientable</em>
if a continuously varying normal can be defined at all points of the surface.
An orientable surface is <em class="ltx_emph ltx_font_italic">oriented</em> if suitable normals have been chosen.
A parametrization <math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\boldsymbol{{\Phi}}(u,v)" display="inline"><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow></math> of an oriented surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> is
<em class="ltx_emph ltx_font_italic">orientation preserving</em> if <math class="ltx_Math" altimg="m72.png" altimg-height="21px" altimg-valign="-5px" altimg-width="80px" alttext="\mathbf{T}_{u}\times\mathbf{T}_{v}" display="inline"><mrow><msub><mi mathvariant="bold">T</mi><mi>u</mi></msub><mo>×</mo><msub><mi mathvariant="bold">T</mi><mi>v</mi></msub></mrow></math> has the
same direction as the chosen normal at each point of <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>, otherwise it is
<em class="ltx_emph ltx_font_italic">orientation reversing</em>.</p>
</div>
<div id="SS5.p9" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m57.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="\boldsymbol{{\Phi}}_{1}" display="inline"><msub><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mn>1</mn></msub></math> and <math class="ltx_Math" altimg="m58.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="\boldsymbol{{\Phi}}_{2}" display="inline"><msub><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mn>2</mn></msub></math> are both orientation preserving or
both orientation reversing parametrizations of <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> defined on open sets <math class="ltx_Math" altimg="m47.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="D_{1}" display="inline"><msub><mi href="./1.6#SS5.p1">D</mi><mn>1</mn></msub></math>
and <math class="ltx_Math" altimg="m48.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="D_{2}" display="inline"><msub><mi href="./1.6#SS5.p1">D</mi><mn>2</mn></msub></math> respectively, then</p>
<table id="E56" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.56</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E56.png" altimg-height="54px" altimg-valign="-24px" altimg-width="309px" alttext="\iint_{\boldsymbol{{\Phi}}_{1}(D_{1})}\mathbf{F}\cdot\mathrm{d}\mathbf{S}=%
\iint_{\boldsymbol{{\Phi}}_{2}(D_{2})}\mathbf{F}\cdot\mathrm{d}\mathbf{S};" display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mrow><msub><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.6#SS5.p1">D</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mrow></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">S</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mrow><msub><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.6#SS5.p1">D</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mrow></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">S</mi></mrow></mrow></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E56.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#E45" title="(1.6.45) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m56.png" altimg-height="23px" altimg-valign="-7px" altimg-width="86px" alttext="\boldsymbol{{\Phi}}(x,y,z)" display="inline"><mrow><mi href="./1.6#E45" mathvariant="bold">Φ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: parameterization</a> and
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.6#SS5.p1">D</mi></math>: open set in the plane</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">otherwise, one is the negative of the other.</p>
</div>
<section id="Px11" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Stokes’s Theorem</h3>
<div id="Px11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px11.p1" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> is an oriented surface with boundary <math class="ltx_Math" altimg="m98.png" altimg-height="18px" altimg-valign="-2px" altimg-width="29px" alttext="\partial S" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.6#SS5.p1">S</mi></mrow></math> which is oriented
so that its direction is clockwise relative to the normals of <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>. Then</p>
<table id="E57" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.57</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E57.png" altimg-height="52px" altimg-valign="-20px" altimg-width="267px" alttext="\iint_{S}(\nabla\times\mathbf{F})\cdot\mathrm{d}\mathbf{S}=\int_{\partial S}%
\mathbf{F}\cdot\mathrm{d}\mathbf{s}," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">S</mi></msub><mrow><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>×</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">S</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.6#SS5.p1">S</mi></mrow></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">s</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E57.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m68.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi mathvariant="bold">F</mi></math> is a continuously differentiable vector-valued function.</p>
</div>
</section>
<section id="Px12" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Gauss’s (or Divergence) Theorem</h3>
<div id="Px12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px12.p1" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> is a piecewise smooth surface which forms the complete boundary of
a bounded closed point set <math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./1.6#Px12.p1">V</mi></math>, and <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math> is oriented by its normal being
outwards
from <math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./1.6#Px12.p1">V</mi></math>. Then</p>
<table id="E58" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.58</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E58.png" altimg-height="52px" altimg-valign="-20px" altimg-width="269px" alttext="\iiint_{V}(\nabla\cdot\mathbf{F})\mathrm{d}V=\iint_{S}\mathbf{F}\cdot\mathrm{d%
}\mathbf{S}," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∭</mo><mi href="./1.6#Px12.p1">V</mi></msub><mrow><mrow><mo stretchy="false">(</mo><mrow><mo>∇</mo><mo>⋅</mo><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#Px12.p1">V</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">S</mi></msub><mrow><mi mathvariant="bold">F</mi><mo>⋅</mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi mathvariant="bold">S</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E58.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a> and
<a href="./1.6#Px12.p1" title="Gauss’s (or Divergence) Theorem ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./1.6#Px12.p1">V</mi></math>: closed region</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">when <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi mathvariant="bold">F</mi></math> is a continuously differentiable vector-valued function.</p>
</div>
</section>
<section id="Px13" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Green’s Theorem (for Volume)</h3>
<div id="Px13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px13.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> and <math class="ltx_Math" altimg="m111.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi>g</mi></math> twice-continuously differentiable functions
</p>
<table id="E59" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.59</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E59.png" altimg-height="51px" altimg-valign="-20px" altimg-width="380px" alttext="\iiint_{V}(f\nabla^{2}g+\nabla f\cdot\nabla g)\mathrm{d}V=\iint_{S}f\frac{%
\partial g}{\partial n}\mathrm{d}A," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∭</mo><mi href="./1.6#Px12.p1">V</mi></msub><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>g</mi></mrow></mrow><mo>+</mo><mrow><mrow><mo>∇</mo><mo></mo><mi>f</mi></mrow><mo>⋅</mo><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#Px12.p1">V</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">S</mi></msub><mrow><mi>f</mi><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#E48">A</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E59.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a>,
<a href="./1.6#E48" title="(1.6.48) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math>: area of a parameterized smooth surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math></a> and
<a href="./1.6#Px12.p1" title="Gauss’s (or Divergence) Theorem ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./1.6#Px12.p1">V</mi></math>: closed region</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E60" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.60</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E60.png" altimg-height="53px" altimg-valign="-21px" altimg-width="453px" alttext="\iiint_{V}(f\nabla^{2}g-g\nabla^{2}f)\mathrm{d}V=\iint_{S}\left(f\frac{%
\partial g}{\partial n}-g\frac{\partial f}{\partial n}\right)\mathrm{d}A," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∭</mo><mi href="./1.6#Px12.p1">V</mi></msub><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>g</mi></mrow></mrow><mo>-</mo><mrow><mi>g</mi><mo></mo><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#Px12.p1">V</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.6#SS5.p1">S</mi></msub><mrow><mrow><mo>(</mo><mrow><mrow><mi>f</mi><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mfrac></mrow><mo>-</mo><mrow><mi>g</mi><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mfrac></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.6#E48">A</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E60.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m66.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m110.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m99.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.6#SS5.p1" title="§1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math>: parameterized surface</a>,
<a href="./1.6#E48" title="(1.6.48) ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m44.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="A(S)" display="inline"><mrow><mi href="./1.6#E48">A</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.6#SS5.p1">S</mi><mo stretchy="false">)</mo></mrow></mrow></math>: area of a parameterized smooth surface <math class="ltx_Math" altimg="m52.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.6#SS5.p1">S</mi></math></a> and
<a href="./1.6#Px12.p1" title="Gauss’s (or Divergence) Theorem ‣ §1.6(v) Surfaces and Integrals over Surfaces ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./1.6#Px12.p1">V</mi></math>: closed region</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="145px" alttext="\ifrac{\partial g}{\partial n}=\nabla g\cdot\mathbf{n}" display="inline"><mrow><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mrow><mo>=</mo><mrow><mrow><mo>∇</mo><mo></mo><mi>g</mi></mrow><mo>⋅</mo><mi mathvariant="bold">n</mi></mrow></mrow></math> is the derivative of <math class="ltx_Math" altimg="m111.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="g" display="inline"><mi>g</mi></math>
normal to the surface outwards from <math class="ltx_Math" altimg="m53.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="V" display="inline"><mi href="./1.6#Px12.p1">V</mi></math> and <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{n}" display="inline"><mi mathvariant="bold">n</mi></math> is the unit outer
normal vector.
</p>
</div>
</section>
</section>
</section>
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<h1 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">§1.5 </span>Calculus of Two or More Variables</h1>
<div id="info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>Referenced by:</dt>
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<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for </dd>
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<h6>Contents</h6>
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<section id="i" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.5(i) </span>Partial Derivatives</h2>
<div id="SS1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Marsden and Tromba (, Chapters 2, 3)</cite>.</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<p class="ltx_p">A function <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">continuous at a point</em> <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> if</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="40px" altimg-valign="-21px" altimg-width="243px" alttext="\lim_{(x,y)\to(a,b)}f(x,y)=f(a,b)," display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo>→</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></munder><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a></dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">that is, for every arbitrarily small positive constant <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi>ϵ</mi></math> there exists
<math class="ltx_Math" altimg="m86.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi>δ</mi></math> (<math class="ltx_Math" altimg="m62.png" altimg-height="17px" altimg-valign="-3px" altimg-width="35px" alttext=">0" display="inline"><mrow><mi></mi><mo>></mo><mn>0</mn></mrow></math>) such that</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="266px" alttext="|f(a+\alpha,b+\beta)-f(a,b)|<\epsilon," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>+</mo><mi>α</mi></mrow><mo>,</mo><mrow><mi>b</mi><mo>+</mo><mi>β</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi>ϵ</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for all <math class="ltx_Math" altimg="m82.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\alpha" display="inline"><mi>α</mi></math> and <math class="ltx_Math" altimg="m84.png" altimg-height="21px" altimg-valign="-6px" altimg-width="17px" alttext="\beta" display="inline"><mi>β</mi></math> that satisfy <math class="ltx_Math" altimg="m147.png" altimg-height="23px" altimg-valign="-7px" altimg-width="97px" alttext="|\alpha|,|\beta|<\delta" display="inline"><mrow><mrow><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow><mo>,</mo><mrow><mo stretchy="false">|</mo><mi>β</mi><mo stretchy="false">|</mo></mrow></mrow><mo><</mo><mi>δ</mi></mrow></math>.</p>
</div>
<div id="SS1.p2" class="ltx_para">
<p class="ltx_p">A function is <em class="ltx_emph ltx_font_italic">continuous on a point set</em> <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#SS1.p2">D</mi></math> if it is continuous at all
points of <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#SS1.p2">D</mi></math>. A function <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">piecewise continuous</em> on
<math class="ltx_Math" altimg="m71.png" altimg-height="21px" altimg-valign="-5px" altimg-width="64px" alttext="I_{1}\times I_{2}" display="inline"><mrow><msub><mi href="./1.5#SS1.p2">I</mi><mn>1</mn></msub><mo>×</mo><msub><mi href="./1.5#SS1.p2">I</mi><mn>2</mn></msub></mrow></math>, where <math class="ltx_Math" altimg="m70.png" altimg-height="21px" altimg-valign="-5px" altimg-width="22px" alttext="I_{1}" display="inline"><msub><mi href="./1.5#SS1.p2">I</mi><mn>1</mn></msub></math> and <math class="ltx_Math" altimg="m72.png" altimg-height="21px" altimg-valign="-5px" altimg-width="22px" alttext="I_{2}" display="inline"><msub><mi href="./1.5#SS1.p2">I</mi><mn>2</mn></msub></math> are intervals, if it is piecewise
continuous in <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math> for each <math class="ltx_Math" altimg="m143.png" altimg-height="21px" altimg-valign="-6px" altimg-width="57px" alttext="y\in I_{2}" display="inline"><mrow><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><msub><mi href="./1.5#SS1.p2">I</mi><mn>2</mn></msub></mrow></math> and piecewise continuous in <math class="ltx_Math" altimg="m142.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi>y</mi></math> for each
<math class="ltx_Math" altimg="m139.png" altimg-height="21px" altimg-valign="-5px" altimg-width="58px" alttext="x\in I_{1}" display="inline"><mrow><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><msub><mi href="./1.5#SS1.p2">I</mi><mn>1</mn></msub></mrow></math>.
</p>
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E3">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="47px" altimg-valign="-16px" altimg-width="34px" alttext="\displaystyle\frac{\partial f}{\partial x}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="49px" altimg-valign="-17px" altimg-width="362px" alttext="\displaystyle=D_{x}f=f_{x}=\lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mo href="./1.5#E3" mathvariant="italic">D</mo><mi>x</mi></msub><mi>f</mi></mrow><mo>=</mo><msub><mi>f</mi><mi>x</mi></msub><mo>=</mo><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>+</mo><mi>h</mi></mrow><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mi>h</mi></mfrac></mstyle></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></span>,
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m65.png" altimg-height="21px" altimg-valign="-5px" altimg-width="31px" alttext="D_{x}" display="inline"><msub><mo href="./1.5#E3" mathvariant="italic">D</mo><mi>x</mi></msub></math>: differential operator (locally)</span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E4">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="50px" altimg-valign="-20px" altimg-width="34px" alttext="\displaystyle\frac{\partial f}{\partial y}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="49px" altimg-valign="-17px" altimg-width="361px" alttext="\displaystyle=D_{y}f=f_{y}=\lim_{h\to 0}\frac{f(x,y+h)-f(x,y)}{h}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mo href="./1.5#E3" mathvariant="italic">D</mo><mi>y</mi></msub><mi>f</mi></mrow><mo>=</mo><msub><mi>f</mi><mi>y</mi></msub><mo>=</mo><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mrow><mi>y</mi><mo>+</mo><mi>h</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mi>h</mi></mfrac></mstyle></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m65.png" altimg-height="21px" altimg-valign="-5px" altimg-width="31px" alttext="D_{x}" display="inline"><msub><mo href="./1.5#E3" mathvariant="italic">D</mo><mi>x</mi></msub></math>: differential operator</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<table id="E5" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="53px" altimg-valign="-20px" altimg-width="56px" alttext="\displaystyle\frac{{\partial}^{2}f}{\partial x\partial y}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="53px" altimg-valign="-21px" altimg-width="125px" alttext="\displaystyle=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}%
\right)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="53px" altimg-valign="-20px" altimg-width="56px" alttext="\displaystyle\frac{{\partial}^{2}f}{\partial y\partial x}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="53px" altimg-valign="-21px" altimg-width="124px" alttext="\displaystyle=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}%
\right)." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle><mo></mo><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle><mo>)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The function <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">continuously differentiable</em> if <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math>,
<math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\ifrac{\partial f}{\partial x}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></math>, and <math class="ltx_Math" altimg="m94.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\ifrac{\partial f}{\partial y}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></math> are continuous, <em class="ltx_emph ltx_font_italic">and
twice-continuously differentiable</em> if also <math class="ltx_Math" altimg="m95.png" altimg-height="28px" altimg-valign="-9px" altimg-width="85px" alttext="\ifrac{{\partial}^{2}f}{{\partial x}^{2}}" display="inline"><mrow><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mrow></math>,
<math class="ltx_Math" altimg="m96.png" altimg-height="28px" altimg-valign="-9px" altimg-width="84px" alttext="\ifrac{{\partial}^{2}f}{{\partial y}^{2}}" display="inline"><mrow><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mrow></math>, <math class="ltx_Math" altimg="m145.png" altimg-height="25px" altimg-valign="-7px" altimg-width="92px" alttext="{\partial}^{2}f/\partial x\partial y" display="inline"><mrow><mrow><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo>/</mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></math>, and
<math class="ltx_Math" altimg="m146.png" altimg-height="25px" altimg-valign="-7px" altimg-width="92px" alttext="{\partial}^{2}f/\partial y\partial x" display="inline"><mrow><mrow><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo>/</mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></math> are continuous. In the latter event</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="53px" altimg-valign="-20px" altimg-width="138px" alttext="\frac{{\partial}^{2}f}{\partial x\partial y}=\frac{{\partial}^{2}f}{\partial y%
\partial x}." display="block"><mrow><mrow><mfrac><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Chain Rule</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="EGx2" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E7">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="47px" altimg-valign="-16px" altimg-width="133px" alttext="\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}f(x(t),y(t))" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="50px" altimg-valign="-20px" altimg-width="167px" alttext="\displaystyle=\frac{\partial f}{\partial x}\frac{\mathrm{d}x}{\mathrm{d}t}+%
\frac{\partial f}{\partial y}\frac{\mathrm{d}y}{\mathrm{d}t}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>x</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>y</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>t</mi></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E8">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="47px" altimg-valign="-16px" altimg-width="184px" alttext="\displaystyle\frac{\partial}{\partial u}f(x(u,v),y(u,v))" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>u</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="50px" altimg-valign="-20px" altimg-width="169px" alttext="\displaystyle=\frac{\partial f}{\partial x}\frac{\partial x}{\partial u}+\frac%
{\partial f}{\partial y}\frac{\partial y}{\partial u}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>u</mi></mrow></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>u</mi></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E9">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="47px" altimg-valign="-16px" altimg-width="248px" alttext="\displaystyle\frac{\partial}{\partial v}f(x(u,v),y(u,v),z(u,v))" display="inline"><mrow><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac></mstyle><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="50px" altimg-valign="-20px" altimg-width="248px" alttext="\displaystyle=\frac{\partial f}{\partial x}\frac{\partial x}{\partial v}+\frac%
{\partial f}{\partial y}\frac{\partial y}{\partial v}+\frac{\partial f}{%
\partial z}\frac{\partial z}{\partial v}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>v</mi></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Implicit Function Theorem</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m69.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="F(x,y)" display="inline"><mrow><mi>F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuously differentiable, <math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="100px" alttext="F(a,b)=0" display="inline"><mrow><mrow><mi>F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>, and
<math class="ltx_Math" altimg="m92.png" altimg-height="23px" altimg-valign="-7px" altimg-width="105px" alttext="\ifrac{\partial F}{\partial y}\not=0" display="inline"><mrow><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>F</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow><mo>≠</mo><mn>0</mn></mrow></math> at
<math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, then in a <em class="ltx_emph ltx_font_italic">neighborhood</em> of <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, that is, an open disk
centered at <math class="ltx_Math" altimg="m117.png" altimg-height="21px" altimg-valign="-6px" altimg-width="32px" alttext="a,b" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></math>, the equation <math class="ltx_Math" altimg="m68.png" altimg-height="23px" altimg-valign="-7px" altimg-width="103px" alttext="F(x,y)=0" display="inline"><mrow><mrow><mi>F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math> defines a continuously
differentiable function <math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-7px" altimg-width="78px" alttext="y=g(x)" display="inline"><mrow><mi>y</mi><mo>=</mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> such that <math class="ltx_Math" altimg="m67.png" altimg-height="23px" altimg-valign="-7px" altimg-width="129px" alttext="F(x,g(x))=0" display="inline"><mrow><mrow><mi>F</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m120.png" altimg-height="23px" altimg-valign="-7px" altimg-width="76px" alttext="b=g(a)" display="inline"><mrow><mi>b</mi><mo>=</mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, and
<math class="ltx_Math" altimg="m127.png" altimg-height="24px" altimg-valign="-8px" altimg-width="144px" alttext="g^{\prime}(x)=-F_{x}/F_{y}" display="inline"><mrow><mrow><msup><mi>g</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo>-</mo><mrow><msub><mi>F</mi><mi>x</mi></msub><mo>/</mo><msub><mi>F</mi><mi>y</mi></msub></mrow></mrow></mrow></math>.</p>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.5(ii) </span>Coordinate Systems</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Davis and Snider (, Chapter 5)</cite>.</dd>
<dt>Referenced by:</dt>
<dd>
,
,
,
,
,
,
,
,
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Clarification (effective with 1.0.5):</dt>
<dd>
The paragraph about notations was added at the beginning of this subsection.
</dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Notations</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<p class="ltx_p">The notations given in this subsection, and also in other coordinate
systems in the DLMF, are those generally used by physicists. For
mathematicians the symbols <math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.5#Px6.p1">θ</mi></math> and <math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math> now are usually interchanged.</p>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Polar Coordinates</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m58.png" altimg-height="19px" altimg-valign="-5px" altimg-width="97px" alttext="0\leq r<\infty" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px5.p1">r</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>, <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="101px" alttext="0\leq\phi\leq 2\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px6.p1">ϕ</mi><mo>≤</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>,
</p>
<table id="E10" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi>x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-6px" altimg-width="87px" alttext="\displaystyle=r\cos\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px5.p1">r</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi>y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-6px" altimg-width="85px" alttext="\displaystyle=r\sin\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px5.p1">r</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a> and
<a href="./1.5#Px5.p1" title="Cylindrical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="EGx3" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="47px" altimg-valign="-16px" altimg-width="34px" alttext="\displaystyle\frac{\partial}{\partial x}" display="inline"><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="50px" altimg-valign="-20px" altimg-width="198px" alttext="\displaystyle=\cos\phi\frac{\partial}{\partial r}-\frac{\sin\phi}{r}\frac{%
\partial}{\partial\phi}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mo></mo><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px5.p1">r</mi></mrow></mfrac></mstyle></mrow><mo>-</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mi href="./1.5#Px5.p1">r</mi></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a> and
<a href="./1.5#Px5.p1" title="Cylindrical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="50px" altimg-valign="-20px" altimg-width="33px" alttext="\displaystyle\frac{\partial}{\partial y}" display="inline"><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="50px" altimg-valign="-20px" altimg-width="198px" alttext="\displaystyle=\sin\phi\frac{\partial}{\partial r}+\frac{\cos\phi}{r}\frac{%
\partial}{\partial\phi}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mo></mo><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px5.p1">r</mi></mrow></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mi href="./1.5#Px5.p1">r</mi></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mfrac></mstyle></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a> and
<a href="./1.5#Px5.p1" title="Cylindrical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">Laplacian</em> is given by</p>
<table id="E13" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E13.png" altimg-height="56px" altimg-valign="-22px" altimg-width="391px" alttext="\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{%
\partial y}^{2}}=\frac{{\partial}^{2}f}{{\partial r}^{2}}+\frac{1}{r}\frac{%
\partial f}{\partial r}+\frac{1}{r^{2}}\frac{{\partial}^{2}f}{{\partial\phi}^{%
2}}." display="block"><mrow><mrow><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px5.p1">r</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mi href="./1.5#Px5.p1">r</mi></mfrac><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px5.p1">r</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><msup><mi href="./1.5#Px5.p1">r</mi><mn>2</mn></msup></mfrac><mo></mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mn>2</mn></msup></mfrac></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a> and
<a href="./1.5#Px5.p1" title="Cylindrical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cylindrical Coordinates</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m58.png" altimg-height="19px" altimg-valign="-5px" altimg-width="97px" alttext="0\leq r<\infty" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px5.p1">r</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>, <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="101px" alttext="0\leq\phi\leq 2\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px6.p1">ϕ</mi><mo>≤</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m57.png" altimg-height="17px" altimg-valign="-4px" altimg-width="123px" alttext="-\infty<z<\infty" display="inline"><mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo><</mo><mi href="./1.1#p2.t1.r2">z</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>,
</p>
<table id="E14" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi>x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-6px" altimg-width="87px" alttext="\displaystyle=r\cos\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px5.p1">r</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi>y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-6px" altimg-width="85px" alttext="\displaystyle=r\sin\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px5.p1">r</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m47.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="14px" altimg-valign="-2px" altimg-width="43px" alttext="\displaystyle=z." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a> and
<a href="./1.5#Px5.p1" title="Cylindrical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Equations () still apply, but</p>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="56px" altimg-valign="-22px" altimg-width="514px" alttext="\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{%
\partial y}^{2}}+\frac{{\partial}^{2}f}{{\partial z}^{2}}=\frac{{\partial}^{2}%
f}{{\partial r}^{2}}+\frac{1}{r}\frac{\partial f}{\partial r}+\frac{1}{r^{2}}%
\frac{{\partial}^{2}f}{{\partial\phi}^{2}}+\frac{{\partial}^{2}f}{{\partial z}%
^{2}}." display="block"><mrow><mrow><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px5.p1">r</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mi href="./1.5#Px5.p1">r</mi></mfrac><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px5.p1">r</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><msup><mi href="./1.5#Px5.p1">r</mi><mn>2</mn></msup></mfrac><mo></mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a> and
<a href="./1.5#Px5.p1" title="Cylindrical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Spherical Coordinates</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m60.png" altimg-height="20px" altimg-valign="-6px" altimg-width="98px" alttext="0\leq\rho<\infty" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px6.p1">ρ</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>, <math class="ltx_Math" altimg="m59.png" altimg-height="21px" altimg-valign="-6px" altimg-width="101px" alttext="0\leq\phi\leq 2\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px6.p1">ϕ</mi><mo>≤</mo><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m61.png" altimg-height="20px" altimg-valign="-5px" altimg-width="89px" alttext="0\leq\theta\leq\pi" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.5#Px6.p1">θ</mi><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>,
</p>
<table id="E16" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi>x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="23px" altimg-valign="-6px" altimg-width="129px" alttext="\displaystyle=\rho\sin\theta\cos\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px6.p1">ρ</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi>y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-6px" altimg-width="127px" alttext="\displaystyle=\rho\sin\theta\sin\phi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px6.p1">ρ</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m47.png" altimg-height="14px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="23px" altimg-valign="-6px" altimg-width="86px" alttext="\displaystyle=\rho\cos\theta." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px6.p1">ρ</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./1.5#Px6.p1">ρ</mi></math>: radius</a> and
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.5#Px6.p1">θ</mi></math>: azimuth</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The Laplacian is given by
</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{%
\partial y}^{2}}+\frac{{\partial}^{2}f}{{\partial z}^{2}}={\frac{1}{\rho^{2}}%
\frac{\partial}{\partial\rho}\left(\rho^{2}\frac{\partial f}{\partial\rho}%
\right)+\frac{1}{\rho^{2}{\sin^{2}}\theta}\frac{{\partial}^{2}f}{{\partial\phi%
}^{2}}}+\frac{1}{\rho^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin%
\theta\frac{\partial f}{\partial\theta}\right)." display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><msup><mo>∇</mo><mn>2</mn></msup><mo></mo><mi>f</mi></mrow></mtd><mtd><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mn>2</mn></msup></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><msup><mi href="./1.5#Px6.p1">ρ</mi><mn>2</mn></msup></mfrac><mo></mo><mrow><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ρ</mi></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><msup><mi href="./1.5#Px6.p1">ρ</mi><mn>2</mn></msup><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ρ</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mrow><msup><mi href="./1.5#Px6.p1">ρ</mi><mn>2</mn></msup><mo></mo><mrow><msup><mi href="./4.14#E1">sin</mi><mn>2</mn></msup><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow></mrow></mfrac><mo></mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">ϕ</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mrow><msup><mi href="./1.5#Px6.p1">ρ</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow></mrow></mfrac><mo></mo><mrow><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">θ</mi></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.5#Px6.p1">θ</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./1.5#Px6.p1">ρ</mi></math>: radius</a> and
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.5#Px6.p1">θ</mi></math>: azimuth</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px6.p2" class="ltx_para">
<p class="ltx_p">For applications and other coordinate systems see §§,
,
. See also
<cite class="ltx_cite ltx_citemacro_citet">Morse and Feshbach (, pp. 655-666)</cite>.</p>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.5(iii) </span>Taylor’s Theorem; Maxima and Minima</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Marsden and Tromba (, Chapter 3)</cite>.</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> is <math class="ltx_Math" altimg="m134.png" altimg-height="18px" altimg-valign="-4px" altimg-width="51px" alttext="n+1" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></math> times continuously differentiable, then</p>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="54px" altimg-valign="-21px" altimg-width="649px" alttext="f(a+\lambda,b+\mu)=f+\left(\lambda\frac{\partial}{\partial x}+\mu\frac{%
\partial}{\partial y}\right)f+\dots+\frac{1}{n!}\left(\lambda\frac{\partial}{%
\partial x}+\mu\frac{\partial}{\partial y}\right)^{n}f+R_{n}," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>a</mi><mo>+</mo><mi>λ</mi></mrow><mo>,</mo><mrow><mi>b</mi><mo>+</mo><mi>μ</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi>f</mi><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mrow><mi>λ</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mi>μ</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mrow></mrow><mo>)</mo></mrow><mo></mo><mi>f</mi></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><mrow><mfrac><mn>1</mn><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mrow><mrow><mi>λ</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mi>μ</mi><mo></mo><mfrac><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mrow></mrow><mo>)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup><mo></mo><mi>f</mi></mrow><mo>+</mo><msub><mi href="./1.4#SS6.p1">R</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m133.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.4#SS6.p1" title="§1.4(vi) Taylor’s Theorem for Real Variables ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="21px" altimg-valign="-5px" altimg-width="30px" alttext="R_{n}" display="inline"><msub><mi href="./1.4#SS6.p1">R</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math>: remainder</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> and its partial derivatives on the right-hand side are evaluated at
<math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, and <math class="ltx_Math" altimg="m76.png" altimg-height="26px" altimg-valign="-7px" altimg-width="189px" alttext="R_{n}/(\lambda^{2}+\mu^{2})^{n/2}\to 0" display="inline"><mrow><mrow><msub><mi href="./1.4#SS6.p1">R</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>/</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>μ</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>→</mo><mn>0</mn></mrow></math> as <math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="128px" alttext="(\lambda,\mu)\to(0,0)" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>λ</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>μ</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo>→</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>.</p>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p"><math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> has a <em class="ltx_emph ltx_font_italic">local minimum</em> (<em class="ltx_emph ltx_font_italic">maximum</em>)
at <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> if
</p>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="50px" altimg-valign="-20px" altimg-width="226px" alttext="\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0\quad\mbox{ at $(%
a,b)$,}" display="block"><mrow><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><mo mathvariant="italic" separator="true"> </mo><mrow><mtext> at </mtext><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mtext>,</mtext></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and the second-order term in () is <em class="ltx_emph ltx_font_italic">positive definite
(negative definite)</em>, that is,
</p>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="51px" altimg-valign="-18px" altimg-width="243px" alttext="\frac{{\partial}^{2}f}{{\partial x}^{2}}>0\;\;\;\mbox{$(<0)$}\quad\mbox{ at $(%
a,b)$,}" display="block"><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>></mo><mrow><mrow><mpadded width="+8.3pt"><mn>0</mn></mpadded><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi></mi><mo><</mo><mn>0</mn></mrow><mo stretchy="false">)</mo></mrow></mrow><mo mathvariant="italic" separator="true"> </mo><mrow><mtext> at </mtext><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mtext>,</mtext></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="60px" altimg-valign="-22px" altimg-width="323px" alttext="\frac{{\partial}^{2}f}{{\partial x}^{2}}\frac{{\partial}^{2}f}{{\partial y}^{2%
}}-\left(\frac{{\partial}^{2}f}{\partial x\partial y}\right)^{2}>0\quad\mbox{ %
at $(a,b)$}." display="block"><mrow><mrow><mrow><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo></mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>-</mo><msup><mrow><mo>(</mo><mfrac><mrow><mpadded width="-1.1pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>></mo><mrow><mn>0</mn><mo mathvariant="italic" separator="true"> </mo><mrow><mtext> at </mtext><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.5(iv) </span>Leibniz’s Theorem for Differentiation of Integrals</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Protter and Morrey (, pp. 288, 298)</cite>.</dd>
<dt>Referenced by:</dt>
<dd>
,
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Finite Integrals</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="61px" altimg-valign="-24px" altimg-width="622px" alttext="\frac{\mathrm{d}}{\mathrm{d}x}\int^{\beta(x)}_{\alpha(x)}f(x,y)\mathrm{d}y={f(%
x,\beta(x))\beta^{\prime}(x)-f(x,\alpha(x))\alpha^{\prime}(x)}+\int^{\beta(x)}%
_{\alpha(x)}\frac{\partial f}{\partial x}\mathrm{d}y." display="block"><mrow><mrow><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>x</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi>α</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>β</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mrow><mi>β</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msup><mi>β</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>-</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mrow><mi>α</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><msup><mi>α</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>+</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><mi>α</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>β</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.3.7</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Sufficient conditions for validity are: (a) <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> and <math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\ifrac{\partial f}{\partial x}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></math> are
continuous on a rectangle <math class="ltx_Math" altimg="m119.png" altimg-height="20px" altimg-valign="-5px" altimg-width="88px" alttext="a\leq x\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mrow></math>, <math class="ltx_Math" altimg="m121.png" altimg-height="21px" altimg-valign="-6px" altimg-width="87px" alttext="c\leq y\leq d" display="inline"><mrow><mi>c</mi><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>d</mi></mrow></math>; (b) when
<math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="x\in[a,b]" display="inline"><mrow><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math> both <math class="ltx_Math" altimg="m81.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\alpha(x)" display="inline"><mrow><mi>α</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m83.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\beta(x)" display="inline"><mrow><mi>β</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> are continuously differentiable
and lie in <math class="ltx_Math" altimg="m79.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="[c,d]" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>c</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>d</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></math>.</p>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Infinite Integrals</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<p class="ltx_p">Suppose that <math class="ltx_Math" altimg="m116.png" altimg-height="21px" altimg-valign="-6px" altimg-width="50px" alttext="a,b,c" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></math> are finite, <math class="ltx_Math" altimg="m124.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="d" display="inline"><mi>d</mi></math> is finite or <math class="ltx_Math" altimg="m56.png" altimg-height="17px" altimg-valign="-4px" altimg-width="40px" alttext="+\infty" display="inline"><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></math>, and <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>,
<math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="66px" alttext="\ifrac{\partial f}{\partial x}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></math> are continuous on the partly-closed rectangle or infinite
strip <math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="109px" alttext="[a,b]\times[c,d)" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow><mo>×</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mi>c</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi>d</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math>. Suppose also that <math class="ltx_Math" altimg="m100.png" altimg-height="31px" altimg-valign="-9px" altimg-width="110px" alttext="\int^{d}_{c}f(x,y)\mathrm{d}y" display="inline"><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></math>
converges and <math class="ltx_Math" altimg="m99.png" altimg-height="31px" altimg-valign="-9px" altimg-width="125px" alttext="\int^{d}_{c}(\ifrac{\partial f}{\partial x})\mathrm{d}y" display="inline"><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></math> <em class="ltx_emph ltx_font_italic">converges uniformly</em>
on <math class="ltx_Math" altimg="m119.png" altimg-height="20px" altimg-valign="-5px" altimg-width="88px" alttext="a\leq x\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mrow></math>, that is, given any positive number <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./1.5#Px8.p1">ϵ</mi></math>, however
small, we can find a number <math class="ltx_Math" altimg="m122.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="c_{0}\in[c,d)" display="inline"><mrow><msub><mi>c</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><mi>c</mi><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi>d</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math> that is independent of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math> and is
such that
</p>
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="65px" altimg-valign="-27px" altimg-width="187px" alttext="\left|\int_{c_{1}}^{d}(\ifrac{\partial f}{\partial x})\mathrm{d}y\right|<\epsilon," display="block"><mrow><mrow><mrow><mo>|</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><msub><mi>c</mi><mn>1</mn></msub><mi>d</mi></msubsup><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow><mo>|</mo></mrow><mo><</mo><mi href="./1.5#Px8.p1">ϵ</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#Px8.p1" title="Infinite Integrals ‣ §1.5(iv) Leibniz’s Theorem for Differentiation of Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./1.5#Px8.p1">ϵ</mi></math>: positive number</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for all <math class="ltx_Math" altimg="m123.png" altimg-height="23px" altimg-valign="-7px" altimg-width="96px" alttext="c_{1}\in[c_{0},d)" display="inline"><mrow><msub><mi>c</mi><mn>1</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">[</mo><msub><mi>c</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p2.t1.r1">,</mo><mi>d</mi><mo href="./front/introduction#Sx4.p2.t1.r1" stretchy="false">)</mo></mrow></mrow></math> and all <math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="x\in[a,b]" display="inline"><mrow><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>. Then</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="56px" altimg-valign="-20px" altimg-width="263px" alttext="\frac{\mathrm{d}}{\mathrm{d}x}\int^{d}_{c}f(x,y)\mathrm{d}y=\int^{d}_{c}\frac{%
\partial f}{\partial x}\mathrm{d}y," display="block"><mrow><mrow><mrow><mfrac><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>x</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-3px" altimg-width="88px" alttext="a<x<b" display="inline"><mrow><mi>a</mi><mo><</mo><mi>x</mi><mo><</mo><mi>b</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.5(v) </span>Multiple Integrals</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Marsden and Tromba (, Chapters 5, 6)</cite>.
For ()</cite>.</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<section id="Px9" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Double Integrals</h3>
<div id="Px9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px9.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> be defined on a closed rectangle <math class="ltx_Math" altimg="m74.png" altimg-height="23px" altimg-valign="-7px" altimg-width="148px" alttext="R=[a,b]\times[c,d]" display="inline"><mrow><mi href="./1.5#Px9.p1">R</mi><mo>=</mo><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow><mo>×</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi>c</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi>d</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></mrow></math>. For
</p>
<table id="EGx4" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E25">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle a" display="inline"><mi>a</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="23px" altimg-valign="-6px" altimg-width="234px" alttext="\displaystyle=x_{0}<x_{1}<\cdots<x_{n}=b," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mo><</mo><msub><mi>x</mi><mn>1</mn></msub><mo><</mo><mi mathvariant="normal">⋯</mi><mo><</mo><msub><mi>x</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mi>b</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m135.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E26">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="14px" altimg-valign="-2px" altimg-width="15px" alttext="\displaystyle c" display="inline"><mi>c</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="23px" altimg-valign="-6px" altimg-width="235px" alttext="\displaystyle=y_{0}<y_{1}<\cdots<y_{m}=d," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi>y</mi><mn>0</mn></msub><mo><</mo><msub><mi>y</mi><mn>1</mn></msub><mo><</mo><mi mathvariant="normal">⋯</mi><mo><</mo><msub><mi>y</mi><mi href="./1.1#p2.t1.r5">m</mi></msub><mo>=</mo><mi>d</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m132.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
<p class="ltx_p">let <math class="ltx_Math" altimg="m51.png" altimg-height="24px" altimg-valign="-8px" altimg-width="65px" alttext="(\xi_{j},\eta_{k})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi>ξ</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi>η</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> denote any point in the rectangle
<math class="ltx_Math" altimg="m80.png" altimg-height="24px" altimg-valign="-8px" altimg-width="188px" alttext="[x_{j},x_{j+1}]\times[y_{k},y_{k+1}]" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><msub><mi>x</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><msub><mi>x</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow><mo>×</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><msub><mi>y</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><msub><mi>y</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m128.png" altimg-height="20px" altimg-valign="-6px" altimg-width="141px" alttext="j=0,\dots,n-1" display="inline"><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m130.png" altimg-height="21px" altimg-valign="-6px" altimg-width="148px" alttext="k=0,\dots,m-1" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>-</mo><mn>1</mn></mrow></mrow></mrow></math>. Then
the <em class="ltx_emph ltx_font_italic">double integral</em> of <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> over <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math> is defined by
</p>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="61px" altimg-valign="-31px" altimg-width="492px" alttext="\iint_{R}f(x,y)\mathrm{d}A={\lim\sum_{j,k}f(\xi_{j},\eta_{k})(x_{j+1}-x_{j})(y%
_{k+1}-y_{k})}" display="block"><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.5#Px9.p1">R</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>A</mi></mrow></mrow></mrow><mo>=</mo><mrow><mo movablelimits="false">lim</mo><mo></mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>,</mo><mi href="./1.1#p2.t1.r4">k</mi></mrow></munder><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>ξ</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo>,</mo><msub><mi>η</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>x</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>y</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m129.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.5#Px9.p1" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math>: closed rectangle</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">as <math class="ltx_Math" altimg="m102.png" altimg-height="24px" altimg-valign="-8px" altimg-width="321px" alttext="\max((x_{j+1}-x_{j})+(y_{k+1}-y_{k}))\to 0" display="inline"><mrow><mrow><mi>max</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>x</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>y</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>→</mo><mn>0</mn></mrow></math>. Sufficient conditions for the
limit to exist are that <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous, or piecewise continuous, on
<math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math>.</p>
</div>
<div id="Px9.p2" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> defined on a point set <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math> contained in a rectangle <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math>, let
</p>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="65px" altimg-valign="-27px" altimg-width="347px" alttext="f^{*}(x,y)=\begin{cases}f(x,y),&\mbox{if $(x,y)\in D$},\\
0,&\mbox{if $(x,y)\in R\setminus D$.}\end{cases}" display="block"><mrow><mrow><msup><mi>f</mi><mo>*</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mtext>if </mtext><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.5#Px9.p2">D</mi></mrow></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>if </mtext><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mi href="./1.5#Px9.p1">R</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mi href="./1.5#Px9.p2">D</mi></mrow></mrow><mtext>.</mtext></mrow></mtd></mtr></mtable></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m97.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./front/introduction#Sx4.p2.t1.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m112.png" altimg-height="23px" altimg-valign="-7px" altimg-width="14px" alttext="\setminus" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo></math>: set subtraction</a>,
<a href="./1.5#Px9.p1" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math>: closed rectangle</a> and
<a href="./1.5#Px9.p2" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math>: region contained in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then
</p>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="52px" altimg-valign="-20px" altimg-width="298px" alttext="\iint_{D}f(x,y)\mathrm{d}A=\iint_{R}f^{*}(x,y)\mathrm{d}A," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.5#Px9.p2">D</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>A</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.5#Px9.p1">R</mi></msub><mrow><mrow><msup><mi>f</mi><mo>*</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>A</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#Px9.p1" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math>: closed rectangle</a> and
<a href="./1.5#Px9.p2" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math>: region contained in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">provided the latter integral exists.</p>
</div>
<div id="Px9.p3" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous, and <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math> is the set</p>
<table id="E30" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle a" display="inline"><mi>a</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-6px" altimg-width="79px" alttext="\displaystyle\leq x\leq b," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle\phi_{1}(x)" display="inline"><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="25px" altimg-valign="-7px" altimg-width="117px" alttext="\displaystyle\leq y\leq\phi_{2}(x)," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi>y</mi><mo>≤</mo><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Referenced by:</dt>
<dd>
,
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m106.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\phi_{1}(x)" display="inline"><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\phi_{2}(x)" display="inline"><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> continuous, then</p>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="61px" altimg-valign="-24px" altimg-width="361px" alttext="\iint_{D}f(x,y)\mathrm{d}A=\int^{b}_{a}\int^{\phi_{2}(x)}_{\phi_{1}(x)}f(x,y)%
\mathrm{d}y\mathrm{d}x," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.5#Px9.p2">D</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>A</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.5#Px9.p2" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math>: region contained in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where the right-hand side is interpreted as the repeated integral</p>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="65px" altimg-valign="-27px" altimg-width="243px" alttext="\int^{b}_{a}\left(\int^{\phi_{2}(x)}_{\phi_{1}(x)}f(x,y)\mathrm{d}y\right)%
\mathrm{d}x." display="block"><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mo>(</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">In particular, <math class="ltx_Math" altimg="m106.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\phi_{1}(x)" display="inline"><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m107.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\phi_{2}(x)" display="inline"><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> can be constants.</p>
</div>
<div id="Px9.p4" class="ltx_para">
<p class="ltx_p">Similarly, if <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math> is the set</p>
<table id="E33" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="14px" altimg-valign="-2px" altimg-width="15px" alttext="\displaystyle c" display="inline"><mi>c</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="23px" altimg-valign="-6px" altimg-width="80px" alttext="\displaystyle\leq y\leq d," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>d</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle\psi_{1}(y)" display="inline"><mrow><msub><mi>ψ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="25px" altimg-valign="-7px" altimg-width="118px" alttext="\displaystyle\leq x\leq\psi_{2}(y)," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mrow><msub><mi>ψ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Referenced by:</dt>
<dd>
,
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">with <math class="ltx_Math" altimg="m109.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\psi_{1}(y)" display="inline"><mrow><msub><mi>ψ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m110.png" altimg-height="23px" altimg-valign="-7px" altimg-width="52px" alttext="\psi_{2}(y)" display="inline"><mrow><msub><mi>ψ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> continuous, then</p>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="61px" altimg-valign="-24px" altimg-width="363px" alttext="\iint_{D}f(x,y)\mathrm{d}A=\int^{d}_{c}\int^{\psi_{2}(y)}_{\psi_{1}(y)}f(x,y)%
\mathrm{d}x\mathrm{d}y." display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi href="./1.5#Px9.p2">D</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>A</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi>ψ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msub><mi>ψ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.5#Px9.p2" title="Double Integrals ‣ §1.5(v) Multiple Integrals ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math>: region contained in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.5#Px9.p1">R</mi></math></a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px10" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Change of Order of Integration</h3>
<div id="Px10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px10.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math> can be represented in both forms () and
(), and <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous on <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math>, then</p>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="61px" altimg-valign="-24px" altimg-width="433px" alttext="\int^{b}_{a}\int^{\phi_{2}(x)}_{\phi_{1}(x)}f(x,y)\mathrm{d}y\mathrm{d}x=\int^%
{d}_{c}\int^{\psi_{2}(y)}_{\psi_{1}(y)}f(x,y)\mathrm{d}x\mathrm{d}y." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mrow><msub><mi>ψ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><msub><mi>ψ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px11" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Infinite Double Integrals</h3>
<div id="Px11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px11.p1" class="ltx_para">
<p class="ltx_p">Infinite double integrals occur when <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> becomes infinite at points in <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math>
or when <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.5#Px9.p2">D</mi></math> is unbounded. In the cases () and
() they are defined by taking limits in the repeated
integrals () in an analogous
manner to ().</p>
</div>
<div id="Px11.p2" class="ltx_para">
<p class="ltx_p">Moreover, if <math class="ltx_Math" altimg="m115.png" altimg-height="21px" altimg-valign="-6px" altimg-width="69px" alttext="a,b,c,d" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi></mrow></math> are finite or infinite constants and <math class="ltx_Math" altimg="m125.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="f(x,y)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> is
piecewise continuous on the set <math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="115px" alttext="(a,b)\times(c,d)" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo>×</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>c</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>d</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>, then</p>
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="56px" altimg-valign="-20px" altimg-width="370px" alttext="\int^{b}_{a}\int^{d}_{c}f(x,y)\mathrm{d}y\mathrm{d}x=\int^{d}_{c}\int^{b}_{a}f%
(x,y)\mathrm{d}x\mathrm{d}y," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>c</mi><mi>d</mi></msubsup><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m98.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">whenever both repeated integrals exist and at least one is absolutely
convergent.</p>
</div>
</section>
<section id="Px12" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Triple Integrals</h3>
<div id="Px12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px12.p1" class="ltx_para">
<p class="ltx_p">Finite and infinite integrals can be defined in a similar way. Often the
<math class="ltx_Math" altimg="m55.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="(x,y,z)" display="inline"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></math> sets are of the form</p>
<table id="E37" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.5.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle a" display="inline"><mi>a</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="23px" altimg-valign="-6px" altimg-width="79px" alttext="\displaystyle\leq x\leq b," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle\phi_{1}(x)" display="inline"><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="25px" altimg-valign="-7px" altimg-width="117px" alttext="\displaystyle\leq y\leq\phi_{2}(x)," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi>y</mi><mo>≤</mo><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="25px" altimg-valign="-7px" altimg-width="74px" alttext="\displaystyle\psi_{1}(x,y)" display="inline"><mrow><msub><mi>ψ</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="25px" altimg-valign="-7px" altimg-width="137px" alttext="\displaystyle\leq z\leq\psi_{2}(x,y)." display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>≤</mo><mrow><msub><mi>ψ</mi><mn>2</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m144.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="vi" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.5(vi) </span>Jacobians and Change of Variables</h2>
<div id="SS6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Marsden and Tromba (, pp. 358–371)</cite>.</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<section id="Px13" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Jacobian</h3>
<div id="Px13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px13.p1" class="ltx_para">
<table id="EGx5" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E38">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="53px" altimg-valign="-21px" altimg-width="69px" alttext="\displaystyle\frac{\partial(f,g)}{\partial(x,y)}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>f</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>g</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="53px" altimg-valign="-21px" altimg-width="191px" alttext="\displaystyle=\begin{vmatrix}\ifrac{\partial f}{\partial x}&\ifrac{\partial f}%
{\partial y}\\
\ifrac{\partial g}{\partial x}&\ifrac{\partial g}{\partial y}\end{vmatrix}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo href="./1.3#i">|</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mtd></mtr></mtable><mo href="./1.3#i">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.3#i" title="§1.3(i) Definitions and Elementary Properties ‣ §1.3 Determinants ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\det" display="inline"><mo href="./1.3#i">det</mo></math>: determinant</a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E39">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="53px" altimg-valign="-21px" altimg-width="69px" alttext="\displaystyle\frac{\partial(x,y)}{\partial(r,\phi)}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi href="./1.5#Px4.p1">r</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi href="./1.5#Px4.p1">ϕ</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="228px" alttext="\displaystyle=r\quad\text{(polar coordinates)}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.5#Px4.p1">r</mi><mo separator="true"> </mo><mtext>(polar coordinates)</mtext></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#Px4.p1" title="Polar Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m136.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.5#Px5.p1">r</mi></math>: radius</a> and
<a href="./1.5#Px4.p1" title="Polar Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: angle</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E40">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="53px" altimg-valign="-21px" altimg-width="88px" alttext="\displaystyle\frac{\partial(f,g,h)}{\partial(x,y,z)}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="77px" altimg-valign="-33px" altimg-width="271px" alttext="\displaystyle=\begin{vmatrix}\ifrac{\partial f}{\partial x}&\ifrac{\partial f}%
{\partial y}&\ifrac{\partial f}{\partial z}\\
\ifrac{\partial g}{\partial x}&\ifrac{\partial g}{\partial y}&\ifrac{\partial g%
}{\partial z}\\
\ifrac{\partial h}{\partial x}&\ifrac{\partial h}{\partial y}&\ifrac{\partial h%
}{\partial z}\end{vmatrix}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo href="./1.3#i">|</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>f</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>g</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>h</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>h</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></mtd><mtd columnalign="center"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>h</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>z</mi></mrow></mrow></mtd></mtr></mtable><mo href="./1.3#i">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.3#i" title="§1.3(i) Definitions and Elementary Properties ‣ §1.3 Determinants ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\det" display="inline"><mo href="./1.3#i">det</mo></math>: determinant</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E41">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="53px" altimg-valign="-21px" altimg-width="88px" alttext="\displaystyle\frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo stretchy="false">(</mo><mi href="./1.5#Px6.p1">ρ</mi><mo>,</mo><mi href="./1.5#Px6.p1">θ</mi><mo>,</mo><mi href="./1.5#Px6.p1">ϕ</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="28px" altimg-valign="-7px" altimg-width="311px" alttext="\displaystyle=\rho^{2}\sin\theta\quad\text{(spherical coordinates)}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./1.5#Px6.p1">ρ</mi><mn>2</mn></msup><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.5#Px6.p1">θ</mi></mrow></mrow><mo separator="true"> </mo><mtext>(spherical coordinates)</mtext></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m105.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="\phi" display="inline"><mi href="./1.5#Px6.p1">ϕ</mi></math>: longitude</a>,
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m111.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="\rho" display="inline"><mi href="./1.5#Px6.p1">ρ</mi></math>: radius</a> and
<a href="./1.5#Px6.p1" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.5#Px6.p1">θ</mi></math>: azimuth</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="Px14" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Change of Variables</h3>
<div id="Px14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px14.p1" class="ltx_para">
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E42.png" altimg-height="53px" altimg-valign="-21px" altimg-width="510px" alttext="\iint_{D}f(x,y)\mathrm{d}x\mathrm{d}y=\iint_{D^{*}}f(x(u,v),y(u,v))\left|\frac%
{\partial(x,y)}{\partial(u,v)}\right|\mathrm{d}u\mathrm{d}v," display="block"><mrow><mrow><mrow><msub><mo largeop="true" symmetric="true">∬</mo><mi>D</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow></mrow></mrow><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∬</mo><msup><mi>D</mi><mo>*</mo></msup></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo>|</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></mfrac><mo>|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>v</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r28" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>: open interval</a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math> is the image of <math class="ltx_Math" altimg="m64.png" altimg-height="18px" altimg-valign="-2px" altimg-width="31px" alttext="D^{*}" display="inline"><msup><mi>D</mi><mo>*</mo></msup></math> under a mapping <math class="ltx_Math" altimg="m54.png" altimg-height="23px" altimg-valign="-7px" altimg-width="220px" alttext="(u,v)\to(x(u,v),y(u,v))" display="inline"><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>u</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>v</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo>→</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>
which is one-to-one except perhaps for a set of points of area zero.</p>
<table id="E43" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.5.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\iiint_{D}f(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z=\iiint_{D^{*}}f(x(u,v,w),y%
(u,v,w),z(u,v,w))\*\left|\frac{\partial(x,y,z)}{\partial(u,v,w)}\right|\mathrm%
{d}u\mathrm{d}v\mathrm{d}w." display="block"><mrow><mtable align="baseline 1" columnalign="left" displaystyle="true"><mtr><mtd><mrow><msub><mo largeop="true" symmetric="true">∭</mo><mi>D</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>x</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>y</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mo>=</mo><mrow><msub><mo largeop="true" symmetric="true">∭</mo><msup><mi>D</mi><mo>*</mo></msup></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo>|</mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo>|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>u</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>v</mi></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r3">w</mi></mrow></mrow></mrow><mo>.</mo></mtd></mtr></mtable></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E43.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m101.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m91.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m126.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m138.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m137.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Again the mapping is one-to-one except perhaps for a set of points of volume
zero.
</p>
</div>
</section>
</section>
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<span class="ltx_tag ltx_tag_section">§1.9 </span>Calculus of a Complex Variable</h1>
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<li class="ltx_tocentry"></li>
<li class="ltx_tocentry"></li>
<li class="ltx_tocentry"></li>
<li class="ltx_tocentry"></li>
<li class="ltx_tocentry"></li>
<li class="ltx_tocentry"></li>
</ul>
<section id="i" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.9(i) </span>Complex Numbers</h2>
<div id="SS1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Copson (, Chapter 1)</cite>,
<cite class="ltx_cite ltx_citemacro_citet">Levinson and Redheffer (, Chapter 1)</cite>, or
<cite class="ltx_cite ltx_citemacro_citet">Markushevich (, pp. 14–18)</cite>.</dd>
<dt>Referenced by:</dt>
<dd>
,
,
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<div id="SS1.p1" class="ltx_para">
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="22px" altimg-valign="-6px" altimg-width="101px" alttext="z=x+iy," display="block"><mrow><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m204.png" altimg-height="21px" altimg-valign="-6px" altimg-width="74px" alttext="x,y\in\mathbb{R}" display="inline"><mrow><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./front/introduction#Sx4.p2.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m114.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{R}" display="inline"><mi href="./front/introduction#Sx4.p2.t1.r14">ℝ</mi></math>: real line</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.1</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<section id="Px1" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Real and Imaginary Parts</h3>
<div id="Px1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<table id="E2" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="19px" altimg-valign="-2px" altimg-width="31px" alttext="\displaystyle\Re z" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m19.png" altimg-height="18px" altimg-valign="-6px" altimg-width="44px" alttext="\displaystyle=x," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi>x</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="19px" altimg-valign="-2px" altimg-width="31px" alttext="\displaystyle\Im z" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="18px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=y." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi>y</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</span> and
<span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m80.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</span>
</dd>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.5</span></span> <span class="ltx_origref"><span class="ltx_tag">3.7.6</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px2" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Polar Representation</h3>
<div id="Px2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<table id="E3" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle x" display="inline"><mi>x</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-6px" altimg-width="85px" alttext="\displaystyle=r\cos\theta," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.9#Px2.p1">r</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m43.png" altimg-height="18px" altimg-valign="-6px" altimg-width="17px" alttext="\displaystyle y" display="inline"><mi>y</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-6px" altimg-width="83px" alttext="\displaystyle=r\sin\theta," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.9#Px2.p1">r</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a> and
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.2</span></span>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="29px" altimg-valign="-7px" altimg-width="152px" alttext="r=(x^{2}+y^{2})^{1/2}," display="block"><mrow><mrow><mi href="./1.9#Px2.p1">r</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.3</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and when <math class="ltx_Math" altimg="m220.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="z\neq 0" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>≠</mo><mn>0</mn></mrow></math>,
</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="23px" altimg-valign="-6px" altimg-width="293px" alttext="\theta=\omega,\;\;\pi-\omega,\;\;-\pi+\omega,\mbox{ or }-\omega," display="block"><mrow><mrow><mi href="./1.9#Px2.p1">θ</mi><mo>=</mo><mrow><mi>ω</mi><mo rspace="8.1pt">,</mo><mrow><mi href="./3.12#E1">π</mi><mo>-</mo><mi>ω</mi></mrow><mo rspace="8.1pt">,</mo><mrow><mrow><mo>-</mo><mi href="./3.12#E1">π</mi></mrow><mo>+</mo><mi>ω</mi></mrow><mo>,</mo><mrow><mtext> or </mtext><mo>-</mo><mi>ω</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a> and
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">according as <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> lies in the 1st, 2nd, 3rd, or 4th quadrants. Here
</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="29px" altimg-valign="-9px" altimg-width="256px" alttext="\omega=\operatorname{arctan}\left(|y/x|\right)\in\left[0,\tfrac{1}{2}\pi\right]." display="block"><mrow><mrow><mi>ω</mi><mo>=</mo><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mrow><mo>(</mo><mrow><mo stretchy="false">|</mo><mrow><mi>y</mi><mo>/</mo><mi>x</mi></mrow><mo stretchy="false">|</mo></mrow><mo>)</mo></mrow></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r29">[</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mi href="./3.12#E1">π</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r29">]</mo></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r29" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m78.png" altimg-height="23px" altimg-valign="-7px" altimg-width="43px" alttext="[\NVar{a},\NVar{b}]" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">[</mo><mi class="ltx_nvar">a</mi><mo href="./front/introduction#Sx4.p1.t1.r29">,</mo><mi class="ltx_nvar">b</mi><mo href="./front/introduction#Sx4.p1.t1.r29" stretchy="false">]</mo></mrow></math>: closed interval</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a> and
<a href="./4.23#SS2.p1" title="§4.23(ii) Principal Values ‣ §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m122.png" altimg-height="17px" altimg-valign="-2px" altimg-width="73px" alttext="\operatorname{arctan}\NVar{z}" display="inline"><mrow><mi href="./4.23#SS2.p1">arctan</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: arctangent function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.4</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Modulus and Phase</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<table id="E7" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m47.png" altimg-height="25px" altimg-valign="-7px" altimg-width="27px" alttext="\displaystyle|z|" display="inline"><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="18px" altimg-valign="-6px" altimg-width="41px" alttext="\displaystyle=r," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./1.9#Px2.p1">r</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-6px" altimg-width="42px" alttext="\displaystyle\operatorname{ph}z" display="inline"><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="23px" altimg-valign="-6px" altimg-width="101px" alttext="\displaystyle=\theta+2n\pi," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.9#Px2.p1">θ</mi><mo>+</mo><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi href="./3.12#E1">π</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m186.png" altimg-height="18px" altimg-valign="-3px" altimg-width="54px" alttext="n\in\mathbb{Z}" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m123.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./front/introduction#Sx4.p2.t1.r20" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m115.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="\mathbb{Z}" display="inline"><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></math>: set of all integers</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a> and
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">principal value</em>
of <math class="ltx_Math" altimg="m124.png" altimg-height="21px" altimg-valign="-6px" altimg-width="40px" alttext="\operatorname{ph}z" display="inline"><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></math> corresponds to <math class="ltx_Math" altimg="m178.png" altimg-height="17px" altimg-valign="-2px" altimg-width="53px" alttext="n=0" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow></math>, that is, <math class="ltx_Math" altimg="m55.png" altimg-height="21px" altimg-valign="-6px" altimg-width="133px" alttext="-\pi\leq\operatorname{ph}z\leq\pi" display="inline"><mrow><mrow><mo>-</mo><mi href="./3.12#E1">π</mi></mrow><mo>≤</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>. It
is single-valued on <math class="ltx_Math" altimg="m113.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="\mathbb{C}\setminus\{0\}" display="inline"><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p2.t1.r19">∖</mo><mrow><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow></mrow></math>, except on the interval
<math class="ltx_Math" altimg="m49.png" altimg-height="23px" altimg-valign="-7px" altimg-width="74px" alttext="(-\infty,0)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mn>0</mn><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> where it is discontinuous and two-valued. <em class="ltx_emph ltx_font_italic">Unless indicated
otherwise</em>, these principal values are assumed throughout
the DLMF. (However, if we
require a principal value to be single-valued, then we can restrict
<math class="ltx_Math" altimg="m54.png" altimg-height="21px" altimg-valign="-6px" altimg-width="133px" alttext="-\pi<\operatorname{ph}z\leq\pi" display="inline"><mrow><mrow><mo>-</mo><mi href="./3.12#E1">π</mi></mrow><mo><</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>≤</mo><mi href="./3.12#E1">π</mi></mrow></math>.)
</p>
<table id="E8" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m45.png" altimg-height="25px" altimg-valign="-7px" altimg-width="42px" alttext="\displaystyle|\Re z|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle\leq|z|," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m44.png" altimg-height="25px" altimg-valign="-7px" altimg-width="42px" alttext="\displaystyle|\Im z|" display="inline"><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle\leq|z|," display="inline"><mrow><mrow><mi></mi><mo>≤</mo><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m79.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Im" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℑ</mi><mo></mo></mrow></math>: imaginary part</a>,
<a href="./1.9#E2" title="(1.9.2) ‣ Real and Imaginary Parts ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m80.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\Re" display="inline"><mrow><mi href="./1.9#E2" mathvariant="normal">ℜ</mi><mo></mo></mrow></math>: real part</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="27px" altimg-valign="-6px" altimg-width="82px" alttext="z=re^{i\theta}," display="block"><mrow><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi href="./1.9#Px2.p1">r</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></msup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a> and
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where
</p>
<table id="E10" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E10.png" altimg-height="27px" altimg-valign="-6px" altimg-width="174px" alttext="e^{i\theta}=\cos\theta+i\sin\theta;" display="block"><mrow><mrow><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></msup><mo>=</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></mrow></mrow></mrow><mo>;</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a> and
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">see §.</p>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Complex Conjugate</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="EGx1" class="ltx_equationgroup ltx_eqn_table">
<tbody id="E11">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\displaystyle\overline{z}" display="inline"><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="22px" altimg-valign="-6px" altimg-width="86px" alttext="\displaystyle=x-iy," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi>x</mi><mo>-</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text ltx_font_bold"><math class="ltx_Math" altimg="m125.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="\overline{\NVar{z}}" display="inline"><mover accent="true"><mi class="ltx_nvar" href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover></math>: complex conjugate</span></dd>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.7</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E12">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m46.png" altimg-height="25px" altimg-valign="-7px" altimg-width="27px" alttext="\displaystyle|\overline{z}|" display="inline"><mrow><mo stretchy="false">|</mo><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover><mo stretchy="false">|</mo></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\displaystyle=|z|," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E11" title="(1.9.11) ‣ Complex Conjugate ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="\overline{\NVar{z}}" display="inline"><mover accent="true"><mi class="ltx_nvar" href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover></math>: complex conjugate</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.8</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
<tbody id="E13">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="23px" altimg-valign="-6px" altimg-width="42px" alttext="\displaystyle\operatorname{ph}\overline{z}" display="inline"><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="23px" altimg-valign="-6px" altimg-width="87px" alttext="\displaystyle=-\operatorname{ph}z." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E11" title="(1.9.11) ‣ Complex Conjugate ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="\overline{\NVar{z}}" display="inline"><mover accent="true"><mi class="ltx_nvar" href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover></math>: complex conjugate</a>,
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m123.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.9</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</tbody>
</table>
</div>
</section>
<section id="Px5" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Arithmetic Operations</h3>
<div id="Px5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m227.png" altimg-height="20px" altimg-valign="-6px" altimg-width="119px" alttext="z_{1}=x_{1}+iy_{1}" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>=</mo><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><msub><mi>y</mi><mn>1</mn></msub></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m228.png" altimg-height="20px" altimg-valign="-6px" altimg-width="119px" alttext="z_{2}=x_{2}+iy_{2}" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>=</mo><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><msub><mi>y</mi><mn>2</mn></msub></mrow></mrow></mrow></math>, then</p>
<table id="E14" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="271px" alttext="z_{1}\pm z_{2}=x_{1}\pm x_{2}+\mathrm{i}(y_{1}\pm y_{2})," display="block"><mrow><mrow><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>±</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo>=</mo><mrow><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>±</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>±</mo><msub><mi>y</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E15" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.15</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E15.png" altimg-height="25px" altimg-valign="-7px" altimg-width="326px" alttext="z_{1}z_{2}=x_{1}x_{2}-y_{1}y_{2}+i(x_{1}y_{2}+x_{2}y_{1})," display="block"><mrow><mrow><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo>=</mo><mrow><mrow><mrow><msub><mi>x</mi><mn>1</mn></msub><mo></mo><msub><mi>x</mi><mn>2</mn></msub></mrow><mo>-</mo><mrow><msub><mi>y</mi><mn>1</mn></msub><mo></mo><msub><mi>y</mi><mn>2</mn></msub></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>x</mi><mn>1</mn></msub><mo></mo><msub><mi>y</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><msub><mi>x</mi><mn>2</mn></msub><mo></mo><msub><mi>y</mi><mn>1</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.10</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E16" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.16</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E16.png" altimg-height="53px" altimg-valign="-21px" altimg-width="387px" alttext="\frac{z_{1}}{z_{2}}=\frac{z_{1}\overline{z}_{2}}{|z_{2}|^{2}}=\frac{x_{1}x_{2}%
+y_{1}y_{2}+i(x_{2}y_{1}-x_{1}y_{2})}{x_{2}^{2}+y_{2}^{2}}," display="block"><mrow><mrow><mfrac><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo></mo><msub><mover accent="true"><mi href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover><mn>2</mn></msub></mrow><msup><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo stretchy="false">|</mo></mrow><mn>2</mn></msup></mfrac><mo>=</mo><mfrac><mrow><mrow><msub><mi>x</mi><mn>1</mn></msub><mo></mo><msub><mi>x</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><msub><mi>y</mi><mn>1</mn></msub><mo></mo><msub><mi>y</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo></mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mo>-</mo><mrow><msub><mi>x</mi><mn>1</mn></msub><mo></mo><msub><mi>y</mi><mn>2</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mrow><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>y</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E11" title="(1.9.11) ‣ Complex Conjugate ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="\overline{\NVar{z}}" display="inline"><mover accent="true"><mi class="ltx_nvar" href="./1.1#p2.t1.r2">z</mi><mo href="./1.9#E11">¯</mo></mover></math>: complex conjugate</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.13</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">provided that <math class="ltx_Math" altimg="m229.png" altimg-height="21px" altimg-valign="-6px" altimg-width="59px" alttext="z_{2}\neq 0" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>≠</mo><mn>0</mn></mrow></math>. Also,</p>
<table id="E17" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.17</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="150px" alttext="|z_{1}z_{2}|=|z_{1}|\;|z_{2}|," display="block"><mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo>=</mo><mrow><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo rspace="5.3pt" stretchy="false">|</mo></mrow><mo></mo><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo stretchy="false">|</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.11</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E18" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.18</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E18.png" altimg-height="25px" altimg-valign="-7px" altimg-width="227px" alttext="\operatorname{ph}\left(z_{1}z_{2}\right)=\operatorname{ph}z_{1}+\operatorname{%
ph}z_{2}," display="block"><mrow><mrow><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub></mrow><mo>+</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m123.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.12</span></span>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E19" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.19</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E19.png" altimg-height="53px" altimg-valign="-21px" altimg-width="108px" alttext="\left|\frac{z_{1}}{z_{2}}\right|=\frac{|z_{1}|}{|z_{2}|}," display="block"><mrow><mrow><mrow><mo>|</mo><mfrac><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mfrac><mo>|</mo></mrow><mo>=</mo><mfrac><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo stretchy="false">|</mo></mrow><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo stretchy="false">|</mo></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.14</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E20" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.20</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E20.png" altimg-height="44px" altimg-valign="-19px" altimg-width="198px" alttext="\operatorname{ph}\frac{z_{1}}{z_{2}}=\operatorname{ph}z_{1}-\operatorname{ph}z%
_{2}." display="block"><mrow><mrow><mrow><mi href="./1.9#E7">ph</mi><mo></mo><mfrac><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mfrac></mrow><mo>=</mo><mrow><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub></mrow><mo>-</mo><mrow><mi href="./1.9#E7">ph</mi><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#E7" title="(1.9.7) ‣ Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m123.png" altimg-height="21px" altimg-valign="-6px" altimg-width="26px" alttext="\operatorname{ph}" display="inline"><mi href="./1.9#E7">ph</mi></math>: phase</a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.15</span></span>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Equations () hold for general
values of the phases, but not necessarily for the principal values.</p>
</div>
</section>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Powers</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<table id="E21" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.21</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E21.png" altimg-height="53px" altimg-valign="-21px" altimg-width="751px" alttext="z^{n}=\left(x^{n}-\genfrac{(}{)}{0.0pt}{}{n}{2}x^{n-2}y^{2}+\genfrac{(}{)}{0.0%
pt}{}{n}{4}x^{n-4}y^{4}-\cdots\right)+i\left(\genfrac{(}{)}{0.0pt}{}{n}{1}x^{n%
-1}y-\genfrac{(}{)}{0.0pt}{}{n}{3}x^{n-3}y^{3}+\cdots\right)," display="block"><mrow><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup><mo>=</mo><mrow><mrow><mo>(</mo><mrow><mrow><mrow><msup><mi>x</mi><mi href="./1.1#p2.t1.r5">n</mi></msup><mo>-</mo><mrow><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./1.1#p2.t1.r5">n</mi><mn>2</mn></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><msup><mi>x</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msup><mo></mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./1.1#p2.t1.r5">n</mi><mn>4</mn></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><msup><mi>x</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>4</mn></mrow></msup><mo></mo><msup><mi>y</mi><mn>4</mn></msup></mrow></mrow><mo>-</mo><mi mathvariant="normal">⋯</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mrow><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./1.1#p2.t1.r5">n</mi><mn>1</mn></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><msup><mi>x</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo></mo><mi>y</mi></mrow><mo>-</mo><mrow><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi href="./1.1#p2.t1.r5">n</mi><mn>3</mn></mfrac><mo href="./1.2#i">)</mo></mrow><mo></mo><msup><mi>x</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>3</mn></mrow></msup><mo></mo><msup><mi>y</mi><mn>3</mn></msup></mrow></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m180.png" altimg-height="20px" altimg-valign="-6px" altimg-width="104px" alttext="n=1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m89.png" altimg-height="27px" altimg-valign="-9px" altimg-width="37px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href="./1.2#i">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar" href="./1.1#p2.t1.r5">m</mi><mi class="ltx_nvar" href="./1.1#p2.t1.r5">n</mi></mfrac><mo href="./1.2#i">)</mo></mrow></math>: binomial coefficient</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.22</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px7" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">DeMoivre’s Theorem</h3>
<div id="Px7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px7.p1" class="ltx_para">
<table id="E22" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.22</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E22.png" altimg-height="25px" altimg-valign="-7px" altimg-width="313px" alttext="\cos n\theta+i\sin n\theta=(\cos\theta+i\sin\theta)^{n}," display="block"><mrow><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></mrow></mrow></mrow><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi href="./1.9#Px2.p1">θ</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m186.png" altimg-height="18px" altimg-valign="-3px" altimg-width="54px" alttext="n\in\mathbb{Z}" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./front/introduction#Sx4.p1.t1.r9" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m94.png" altimg-height="15px" altimg-valign="-3px" altimg-width="18px" alttext="\in" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo></math>: element of</a>,
<a href="./front/introduction#Sx4.p2.t1.r20" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m115.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="\mathbb{Z}" display="inline"><mi href="./front/introduction#Sx4.p2.t1.r20">ℤ</mi></math>: set of all integers</a>,
<a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m128.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px8" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Triangle Inequality</h3>
<div id="Px8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px8.p1" class="ltx_para">
<table id="E23" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.23</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E23.png" altimg-height="25px" altimg-valign="-7px" altimg-width="317px" alttext="\left|\left|z_{1}\right|-\left|z_{2}\right|\right|\leq\left|z_{1}+z_{2}\right|%
\leq\left|z_{1}\right|+\left|z_{2}\right|." display="block"><mrow><mrow><mrow><mo>|</mo><mrow><mrow><mo>|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>|</mo></mrow><mo>-</mo><mrow><mo>|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>|</mo></mrow></mrow><mo>|</mo></mrow><mo>≤</mo><mrow><mo>|</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>+</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo>|</mo></mrow><mo>≤</mo><mrow><mrow><mo>|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>|</mo></mrow><mo>+</mo><mrow><mo>|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>|</mo></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.29</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.9(ii) </span>Continuity, Point Sets, and Differentiation</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Levinson and Redheffer (, Chapters 1,2 and pp. 133–138)</cite> and
<cite class="ltx_cite ltx_citemacro_citet">Copson (, Chapters 2,3)</cite>.</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<section id="Px9" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Continuity</h3>
<div id="Px9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px9.p1" class="ltx_para">
<p class="ltx_p">A function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">continuous</em> at a point <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> if
<math class="ltx_Math" altimg="m106.png" altimg-height="34px" altimg-valign="-18px" altimg-width="157px" alttext="\lim\limits_{z\to z_{0}}f(z)=f(z_{0})" display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>→</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></munder><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. That is, given any positive number
<math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi>ϵ</mi></math>, however small, we can find a positive number <math class="ltx_Math" altimg="m83.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\delta" display="inline"><mi href="./1.9#Px9.p1">δ</mi></math> such that
<math class="ltx_Math" altimg="m234.png" altimg-height="23px" altimg-valign="-7px" altimg-width="158px" alttext="|f(z)-f(z_{0})|<\epsilon" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi>ϵ</mi></mrow></math> for all <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> in the open disk <math class="ltx_Math" altimg="m238.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="|z-z_{0}|<\delta" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.9#Px9.p1">δ</mi></mrow></math>.</p>
</div>
<div id="Px9.p2" class="ltx_para">
<p class="ltx_p">A function of two complex variables <math class="ltx_Math" altimg="m162.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="f(z,w)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">continuous</em> at
<math class="ltx_Math" altimg="m53.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="(z_{0},w_{0})" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> if <math class="ltx_Math" altimg="m101.png" altimg-height="38px" altimg-valign="-21px" altimg-width="278px" alttext="\lim\limits_{(z,w)\to(z_{0},w_{0})}f(z,w)=f(z_{0},w_{0})" display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow><mo>→</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></munder><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>,</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo>,</mo><msub><mi href="./1.1#p2.t1.r3">w</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></math>; compare
().</p>
</div>
</section>
<section id="Px10" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Point Sets in <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>
</h3>
<div id="Px10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px10.p1" class="ltx_para">
<p class="ltx_p">A <em class="ltx_emph ltx_font_italic">neighborhood of a point</em> <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> is a disk <math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-7px" altimg-width="104px" alttext="\left|z-z_{0}\right|<\delta" display="inline"><mrow><mrow><mo>|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo>|</mo></mrow><mo><</mo><mi href="./1.9#Px9.p1">δ</mi></mrow></math>. An
<em class="ltx_emph ltx_font_italic">open set</em> in <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> is one in which each point has a neighborhood
that is contained in the set.</p>
</div>
<div id="Px10.p2" class="ltx_para">
<p class="ltx_p">A point <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> is a <em class="ltx_emph ltx_font_italic">limit point</em> (<em class="ltx_emph ltx_font_italic">limiting point</em> or
<em class="ltx_emph ltx_font_italic">accumulation point</em>) of a set of points <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math> in <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> (or
<math class="ltx_Math" altimg="m111.png" altimg-height="18px" altimg-valign="-2px" altimg-width="61px" alttext="\mathbb{C}\cup\infty" display="inline"><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mi mathvariant="normal">∞</mi></mrow></math>) if every neighborhood of <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> contains a point of <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math>
distinct from <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>. (<math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> may or may not belong to <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math>.) As a consequence,
every neighborhood of a limit point of <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math> contains an infinite number of
points of <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math>.
Also, the union of <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math> and its limit points is the <em class="ltx_emph ltx_font_italic">closure</em> of <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi>S</mi></math>.</p>
</div>
<div id="Px10.p3" class="ltx_para">
<p class="ltx_p">A <em class="ltx_emph ltx_font_italic">domain</em>
<math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>, say, is an open set in <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> that is <em class="ltx_emph ltx_font_italic">connected</em>,
that is, any two points can be joined by a polygonal arc (a finite chain of
straight-line segments) lying in the set. Any point whose neighborhoods always
contain members and nonmembers of <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math> is a <em class="ltx_emph ltx_font_italic">boundary point</em>
of <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>. When its boundary points are added the domain is said to be
<em class="ltx_emph ltx_font_italic">closed</em>,
but unless specified otherwise a domain is assumed to be open.
</p>
</div>
<div id="Px10.p4" class="ltx_para">
<p class="ltx_p">A <em class="ltx_emph ltx_font_italic">region</em> is an open domain together with none, some, or all of its
boundary points. Points of a region that are not boundary points are called
<em class="ltx_emph ltx_font_italic">interior points</em>.
</p>
</div>
<div id="Px10.p5" class="ltx_para">
<p class="ltx_p">A function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">continuous on a region</em> <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi>R</mi></math> if for each point <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>
in <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi>R</mi></math> and any given number <math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi>ϵ</mi></math> (<math class="ltx_Math" altimg="m61.png" altimg-height="17px" altimg-valign="-3px" altimg-width="35px" alttext=">0" display="inline"><mrow><mi></mi><mo>></mo><mn>0</mn></mrow></math>) we can find a neighborhood of
<math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> such that <math class="ltx_Math" altimg="m97.png" altimg-height="23px" altimg-valign="-7px" altimg-width="158px" alttext="\left|f(z)-f(z_{0})\right|<\epsilon" display="inline"><mrow><mrow><mo>|</mo><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>|</mo></mrow><mo><</mo><mi>ϵ</mi></mrow></math> for all points <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> in the
intersection of the neighborhood with <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi>R</mi></math>.
</p>
</div>
</section>
<section id="Px11" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Differentiation</h3>
<div id="Px11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px11.p1" class="ltx_para">
<p class="ltx_p">A function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">differentiable</em> at a point <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> if the following
limit exists:</p>
<table id="E24" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.24</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E24.png" altimg-height="49px" altimg-valign="-17px" altimg-width="312px" alttext="f^{\prime}(z)=\frac{\mathrm{d}f}{\mathrm{d}z}=\lim_{h\to 0}\frac{f(z+h)-f(z)}{%
h}." display="block"><mrow><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi>f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mfrac><mo>=</mo><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mo></mo><mfrac><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>+</mo><mi>h</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mi>h</mi></mfrac></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#E4" title="(1.4.4) ‣ §1.4(iii) Derivatives ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="29px" altimg-valign="-9px" altimg-width="27px" alttext="\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.4#E4" lspace="0.8pt" rspace="0pt">d</mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: derivative of <math class="ltx_Math" altimg="m163.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a> and
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px11.p2" class="ltx_para">
<p class="ltx_p">Differentiability automatically implies continuity.</p>
</div>
</section>
<section id="Px12" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cauchy–Riemann Equations</h3>
<div id="Px12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px12.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m167.png" altimg-height="24px" altimg-valign="-7px" altimg-width="47px" alttext="f^{\prime}(z)" display="inline"><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> exists at <math class="ltx_Math" altimg="m216.png" altimg-height="20px" altimg-valign="-6px" altimg-width="94px" alttext="z=x+iy" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m160.png" altimg-height="23px" altimg-valign="-7px" altimg-width="214px" alttext="f(z)=u(x,y)+iv(x,y)" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./1.9#Px14.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./1.9#Px14.p1">v</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow></math>, then
</p>
<table id="E25" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.25</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="47px" altimg-valign="-16px" altimg-width="34px" alttext="\displaystyle\frac{\partial u}{\partial x}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="50px" altimg-valign="-20px" altimg-width="60px" alttext="\displaystyle=\frac{\partial v}{\partial y}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">v</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="50px" altimg-valign="-20px" altimg-width="34px" alttext="\displaystyle\frac{\partial u}{\partial y}" display="inline"><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mfrac></mstyle></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="47px" altimg-valign="-16px" altimg-width="70px" alttext="\displaystyle=-\frac{\partial v}{\partial x}" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">v</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mfrac></mstyle></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m163.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.9#Px14.p1" title="Harmonic Functions ‣ §1.9(ii) Continuity, Point Sets, and Differentiation ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m196.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="u(x,y)" display="inline"><mrow><mi href="./1.9#Px14.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./1.9#Px14.p1" title="Harmonic Functions ‣ §1.9(ii) Continuity, Point Sets, and Differentiation ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m199.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="v(x,y)" display="inline"><mrow><mi href="./1.9#Px14.p1">v</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.30</span></span>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">at <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(x,y)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>.</p>
</div>
<div id="Px12.p2" class="ltx_para">
<p class="ltx_p">Conversely, if at a given point <math class="ltx_Math" altimg="m52.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="(x,y)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>x</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>y</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> the partial derivatives
<math class="ltx_Math" altimg="m90.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="\ifrac{\partial u}{\partial x}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="\ifrac{\partial u}{\partial y}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m92.png" altimg-height="23px" altimg-valign="-7px" altimg-width="64px" alttext="\ifrac{\partial v}{\partial x}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">v</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow></mrow></math>, and <math class="ltx_Math" altimg="m93.png" altimg-height="23px" altimg-valign="-7px" altimg-width="63px" alttext="\ifrac{\partial v}{\partial y}" display="inline"><mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">v</mi></mrow><mo href="./1.5#E3">/</mo><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow></mrow></math>
exist, are continuous, and satisfy (), then <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is
differentiable at <math class="ltx_Math" altimg="m216.png" altimg-height="20px" altimg-valign="-6px" altimg-width="94px" alttext="z=x+iy" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>y</mi></mrow></mrow></mrow></math>.</p>
</div>
</section>
<section id="Px13" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Analyticity</h3>
<div id="Px13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px13.p1" class="ltx_para">
<p class="ltx_p">A function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is said to be <em class="ltx_emph ltx_font_italic">analytic</em> (<em class="ltx_emph ltx_font_italic">holomorphic</em>) at
<math class="ltx_Math" altimg="m217.png" altimg-height="16px" altimg-valign="-5px" altimg-width="59px" alttext="z=z_{0}" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow></math> if it is differentiable in a neighborhood of <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>.</p>
</div>
<div id="Px13.p2" class="ltx_para">
<p class="ltx_p">A function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">analytic in a domain</em>
<math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math> if it is analytic at each point of <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>. A function analytic at every point
of <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> is said to be <em class="ltx_emph ltx_font_italic">entire</em>.
</p>
</div>
<div id="Px13.p3" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic in an open domain <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>, then each of its derivatives
<math class="ltx_Math" altimg="m167.png" altimg-height="24px" altimg-valign="-7px" altimg-width="47px" alttext="f^{\prime}(z)" display="inline"><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m164.png" altimg-height="24px" altimg-valign="-7px" altimg-width="52px" alttext="f^{\prime\prime}(z)" display="inline"><mrow><msup><mi>f</mi><mo>′′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, <math class="ltx_Math" altimg="m84.png" altimg-height="6px" altimg-valign="-2px" altimg-width="28px" alttext="\dots" display="inline"><mi mathvariant="normal">…</mi></math> exists and is analytic in <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>.
</p>
</div>
</section>
<section id="Px14" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Harmonic Functions</h3>
<div id="Px14.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px14.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m160.png" altimg-height="23px" altimg-valign="-7px" altimg-width="214px" alttext="f(z)=u(x,y)+iv(x,y)" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi href="./1.9#Px14.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi href="./1.9#Px14.p1">v</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow></math> is analytic in an open domain <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>, then <math class="ltx_Math" altimg="m198.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="u" display="inline"><mi href="./1.9#Px14.p1">u</mi></math> and
<math class="ltx_Math" altimg="m200.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="v" display="inline"><mi href="./1.9#Px14.p1">v</mi></math> are
<em class="ltx_emph ltx_font_italic">harmonic</em> in <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>, that is,</p>
<table id="E26" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.26</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E26.png" altimg-height="56px" altimg-valign="-22px" altimg-width="270px" alttext="\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{\partial}^{2}u}{{\partial y}^{%
2}}=\frac{{\partial}^{2}v}{{\partial x}^{2}}+\frac{{\partial}^{2}v}{{\partial y%
}^{2}}=0," display="block"><mrow><mrow><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>=</mo><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">v</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>x</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">v</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi>y</mi></mrow><mn>2</mn></msup></mfrac></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m163.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.9#Px14.p1" title="Harmonic Functions ‣ §1.9(ii) Continuity, Point Sets, and Differentiation ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m196.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="u(x,y)" display="inline"><mrow><mi href="./1.9#Px14.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a> and
<a href="./1.9#Px14.p1" title="Harmonic Functions ‣ §1.9(ii) Continuity, Point Sets, and Differentiation ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m199.png" altimg-height="23px" altimg-valign="-7px" altimg-width="61px" alttext="v(x,y)" display="inline"><mrow><mi href="./1.9#Px14.p1">v</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.32</span></span>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">or in polar form (()) <math class="ltx_Math" altimg="m198.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="u" display="inline"><mi href="./1.9#Px14.p1">u</mi></math> and <math class="ltx_Math" altimg="m200.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="v" display="inline"><mi href="./1.9#Px14.p1">v</mi></math> satisfy</p>
<table id="E27" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.27</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E27.png" altimg-height="51px" altimg-valign="-18px" altimg-width="231px" alttext="\frac{{\partial}^{2}u}{{\partial r}^{2}}+\frac{1}{r}\frac{\partial u}{\partial
r%
}+\frac{1}{r^{2}}\frac{{\partial}^{2}u}{{\partial\theta}^{2}}=0" display="block"><mrow><mrow><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px2.p1">r</mi></mrow><mn>2</mn></msup></mfrac><mo>+</mo><mrow><mfrac><mn>1</mn><mi href="./1.9#Px2.p1">r</mi></mfrac><mo></mo><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px2.p1">r</mi></mrow></mfrac></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><msup><mi href="./1.9#Px2.p1">r</mi><mn>2</mn></msup></mfrac><mo></mo><mfrac><mrow><mpadded width="-1.7pt"><msup><mo href="./1.5#E3">∂</mo><mn>2</mn></msup></mpadded><mo href="./1.5#E3"></mo><mi href="./1.9#Px14.p1">u</mi></mrow><msup><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi href="./1.9#Px2.p1">θ</mi></mrow><mn>2</mn></msup></mfrac></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m88.png" altimg-height="29px" altimg-valign="-9px" altimg-width="28px" alttext="\frac{\partial\NVar{f}}{\partial\NVar{x}}" display="inline"><mfrac><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">f</mi></mrow><mrow><mo href="./1.5#E3" rspace="0.8pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></mfrac></math>: partial derivative of <math class="ltx_Math" altimg="m163.png" altimg-height="21px" altimg-valign="-6px" altimg-width="16px" alttext="f" display="inline"><mi>f</mi></math> with respect to <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.5#E3" title="(1.5.3) ‣ §1.5(i) Partial Derivatives ‣ §1.5 Calculus of Two or More Variables ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\partial\NVar{x}" display="inline"><mrow><mo href="./1.5#E3" rspace="1.4pt">∂</mo><mo href="./1.5#E3"></mo><mi class="ltx_nvar">x</mi></mrow></math>: partial differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m139.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.9#Px2.p1">θ</mi></math>: angle</a> and
<a href="./1.9#Px14.p1" title="Harmonic Functions ‣ §1.9(ii) Continuity, Point Sets, and Differentiation ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m196.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="u(x,y)" display="inline"><mrow><mi href="./1.9#Px14.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math>: function</a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.7.33</span></span>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">at all points of <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi>D</mi></math>.</p>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.9(iii) </span>Integration</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Levinson and Redheffer (, Chapter 3 and p. 360)</cite>,
<cite class="ltx_cite ltx_citemacro_citet">Copson (, pp. 56–69)</cite>, and
<cite class="ltx_cite ltx_citemacro_citet">Ahlfors (, pp. 168–169)</cite>.
For a proof of the Jordan Curve Theorem see, for example,
<cite class="ltx_cite ltx_citemacro_citet">Dienes (, pp. 177–197)</cite>. The theorem is valid with less
restrictive conditions than those assumed here.</dd>
<dt>Referenced by:</dt>
<dd>
,
,
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<div id="SS3.p1" class="ltx_para">
<p class="ltx_p">An <em class="ltx_emph ltx_font_italic">arc</em> <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> is given by <math class="ltx_Math" altimg="m211.png" altimg-height="23px" altimg-valign="-7px" altimg-width="162px" alttext="z(t)=x(t)+iy(t)" display="inline"><mrow><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow></math>, <math class="ltx_Math" altimg="m149.png" altimg-height="20px" altimg-valign="-5px" altimg-width="84px" alttext="a\leq t\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>b</mi></mrow></math>, where
<math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math> and <math class="ltx_Math" altimg="m208.png" altimg-height="16px" altimg-valign="-6px" altimg-width="15px" alttext="y" display="inline"><mi>y</mi></math> are continuously differentiable. If <math class="ltx_Math" altimg="m203.png" altimg-height="23px" altimg-valign="-7px" altimg-width="38px" alttext="x(t)" display="inline"><mrow><mi>x</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m207.png" altimg-height="23px" altimg-valign="-7px" altimg-width="37px" alttext="y(t)" display="inline"><mrow><mi>y</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> are
continuous and <math class="ltx_Math" altimg="m206.png" altimg-height="24px" altimg-valign="-7px" altimg-width="44px" alttext="x^{\prime}(t)" display="inline"><mrow><msup><mi>x</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m209.png" altimg-height="24px" altimg-valign="-7px" altimg-width="43px" alttext="y^{\prime}(t)" display="inline"><mrow><msup><mi>y</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math> are piecewise continuous, then <math class="ltx_Math" altimg="m212.png" altimg-height="23px" altimg-valign="-7px" altimg-width="37px" alttext="z(t)" display="inline"><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></math>
defines a <em class="ltx_emph ltx_font_italic">contour</em>.</p>
</div>
<div id="SS3.p2" class="ltx_para">
<p class="ltx_p">A contour is <em class="ltx_emph ltx_font_italic">simple</em> if it contains no multiple points, that is, for
every pair of distinct values <math class="ltx_Math" altimg="m195.png" altimg-height="20px" altimg-valign="-6px" altimg-width="45px" alttext="t_{1},t_{2}" display="inline"><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub></mrow></math> of <math class="ltx_Math" altimg="m193.png" altimg-height="17px" altimg-valign="-2px" altimg-width="11px" alttext="t" display="inline"><mi>t</mi></math>, <math class="ltx_Math" altimg="m213.png" altimg-height="23px" altimg-valign="-7px" altimg-width="114px" alttext="z(t_{1})\neq z(t_{2})" display="inline"><mrow><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>≠</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. A
<em class="ltx_emph ltx_font_italic">simple closed contour</em> is a simple contour, except that <math class="ltx_Math" altimg="m210.png" altimg-height="23px" altimg-valign="-7px" altimg-width="101px" alttext="z(a)=z(b)" display="inline"><mrow><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>.
</p>
</div>
<div id="SS3.p3" class="ltx_para">
<p class="ltx_p">Next,</p>
<table id="E28" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.28</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E28.png" altimg-height="56px" altimg-valign="-20px" altimg-width="360px" alttext="\int_{C}f(z)\mathrm{d}z=\int_{a}^{b}f(z(t))(x^{\prime}(t)+iy^{\prime}(t))%
\mathrm{d}t," display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px19.p1">C</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow><mo>=</mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msup><mi>x</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><mrow><msup><mi>y</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.9#Px19.p1" title="Winding Number ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>: closed contour</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for a contour <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> and <math class="ltx_Math" altimg="m157.png" altimg-height="23px" altimg-valign="-7px" altimg-width="65px" alttext="f(z(t))" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math> continuous, <math class="ltx_Math" altimg="m149.png" altimg-height="20px" altimg-valign="-5px" altimg-width="84px" alttext="a\leq t\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>b</mi></mrow></math>. If <math class="ltx_Math" altimg="m158.png" altimg-height="23px" altimg-valign="-7px" altimg-width="120px" alttext="f(z(t_{0}))=\infty" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>t</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mi mathvariant="normal">∞</mi></mrow></math>, <math class="ltx_Math" altimg="m150.png" altimg-height="21px" altimg-valign="-5px" altimg-width="93px" alttext="a\leq t_{0}\leq b" display="inline"><mrow><mi>a</mi><mo>≤</mo><msub><mi>t</mi><mn>0</mn></msub><mo>≤</mo><mi>b</mi></mrow></math>, then the integral is defined analogously to the
infinite integrals in §. Similarly when <math class="ltx_Math" altimg="m148.png" altimg-height="17px" altimg-valign="-4px" altimg-width="77px" alttext="a=-\infty" display="inline"><mrow><mi>a</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math> or
<math class="ltx_Math" altimg="m155.png" altimg-height="19px" altimg-valign="-4px" altimg-width="75px" alttext="b=+\infty" display="inline"><mrow><mi>b</mi><mo>=</mo><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mrow></math>.</p>
</div>
<section id="Px15" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Jordan Curve Theorem</h3>
<div id="Px15.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px15.p1" class="ltx_para">
<p class="ltx_p">Any simple closed contour <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> divides <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> into two open domains that
have <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> as common boundary. One of these domains is bounded and is called the
<em class="ltx_emph ltx_font_italic">interior domain of</em> <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>; the other is unbounded and is called the
<em class="ltx_emph ltx_font_italic">exterior domain of</em> <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>.</p>
</div>
</section>
<section id="Px16" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cauchy’s Theorem</h3>
<div id="Px16.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px16.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous within and on a simple closed contour <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> and analytic
within <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>, then</p>
<table id="E29" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.29</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E29.png" altimg-height="52px" altimg-valign="-20px" altimg-width="135px" alttext="\int_{C}f(z)\mathrm{d}z=0." display="block"><mrow><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px19.p1">C</mi></msub><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E29.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.9#Px19.p1" title="Winding Number ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>: closed contour</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px17" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cauchy’s Integral Formula</h3>
<div id="Px17.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px17.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous within and on a simple closed contour <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> and analytic
within <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>, and if <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn>0</mn></msub></math> is a point within <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>, then
</p>
<table id="E30" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.30</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E30.png" altimg-height="52px" altimg-valign="-20px" altimg-width="227px" alttext="f(z_{0})=\frac{1}{2\pi i}\int_{C}\frac{f(z)}{z-z_{0}}\mathrm{d}z," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px19.p1">C</mi></msub><mrow><mfrac><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E30.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.9#Px19.p1" title="Winding Number ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>: closed contour</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E31" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.31</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E31.png" altimg-height="53px" altimg-valign="-21px" altimg-width="297px" alttext="f^{(n)}(z_{0})=\frac{n!}{2\pi i}\int_{C}\frac{f(z)}{(z-z_{0})^{n+1}}\mathrm{d}z," display="block"><mrow><mrow><mrow><msup><mi>f</mi><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px19.p1">C</mi></msub><mrow><mfrac><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msup></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m179.png" altimg-height="20px" altimg-valign="-6px" altimg-width="123px" alttext="n=1,2,3,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E31.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m177.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px19.p1" title="Winding Number ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>: closed contour</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">provided that in both cases <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> is described in the positive rotational
(anticlockwise) sense.</p>
</div>
</section>
<section id="Px18" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Liouville’s Theorem</h3>
<div id="Px18.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px18.p1" class="ltx_para">
<p class="ltx_p">Any bounded entire function is a constant.</p>
</div>
</section>
<section id="Px19" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Winding Number</h3>
<div id="Px19.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px19.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> is a closed contour, and <math class="ltx_Math" altimg="m225.png" altimg-height="21px" altimg-valign="-6px" altimg-width="63px" alttext="z_{0}\not\in C" display="inline"><mrow><msub><mi>z</mi><mn>0</mn></msub><mo href="./front/introduction#Sx4.p1.t1.r10">∉</mo><mi href="./1.9#Px19.p1">C</mi></mrow></math>, then
</p>
<table id="E32" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.32</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E32.png" altimg-height="52px" altimg-valign="-20px" altimg-width="258px" alttext="\frac{1}{2\pi i}\int_{C}\frac{1}{z-z_{0}}\mathrm{d}z=\mathcal{N}(C,z_{0})," display="block"><mrow><mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi><mo></mo><mi mathvariant="normal">i</mi></mrow></mfrac><mo></mo><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px19.p1">C</mi></msub><mrow><mfrac><mn>1</mn><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow></mfrac><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>z</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><mi class="ltx_font_mathcaligraphic" href="./1.9#E32">𝒩</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.9#Px19.p1">C</mi><mo>,</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E32.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m116.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\mathcal{N}(C,z_{0})" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./1.9#E32">𝒩</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.9#Px19.p1">C</mi><mo>,</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math>: winding number of <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.9#Px19.p1" title="Winding Number ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>: closed contour</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m116.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\mathcal{N}(C,z_{0})" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./1.9#E32">𝒩</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.9#Px19.p1">C</mi><mo>,</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math> is an integer called the <em class="ltx_emph ltx_font_italic">winding number of <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>
with respect to <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn mathvariant="normal">0</mn></msub></math></em>. If <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math> is simple and oriented in the positive
rotational sense, then <math class="ltx_Math" altimg="m116.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="\mathcal{N}(C,z_{0})" display="inline"><mrow><mi class="ltx_font_mathcaligraphic" href="./1.9#E32">𝒩</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.9#Px19.p1">C</mi><mo>,</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow></math> is <math class="ltx_Math" altimg="m59.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="1" display="inline"><mn>1</mn></math> or <math class="ltx_Math" altimg="m57.png" altimg-height="17px" altimg-valign="-2px" altimg-width="14px" alttext="0" display="inline"><mn>0</mn></math> depending whether
<math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi>z</mi><mn>0</mn></msub></math> is inside or outside <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px19.p1">C</mi></math>.</p>
</div>
</section>
<section id="Px20" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Mean Value Property</h3>
<div id="Px20.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px20.p1" class="ltx_para">
<p class="ltx_p">For <math class="ltx_Math" altimg="m197.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="u(z)" display="inline"><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math> harmonic,
</p>
<table id="E33" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.33</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E33.png" altimg-height="56px" altimg-valign="-20px" altimg-width="267px" alttext="u(z)=\frac{1}{2\pi}\int^{2\pi}_{0}u(z+re^{i\phi})\mathrm{d}\phi." display="block"><mrow><mrow><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mrow><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi>z</mi><mo>+</mo><mrow><mi href="./1.9#Px2.p1">r</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>ϕ</mi></mrow></msup></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>ϕ</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E33.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a> and
<a href="./1.9#Px20.p1" title="Mean Value Property ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m197.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="u(z)" display="inline"><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: harmonic function</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px21" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Poisson Integral</h3>
<div id="Px21.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px21.p1" class="ltx_para">
<p class="ltx_p">If <math class="ltx_Math" altimg="m172.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="h(w)" display="inline"><mrow><mi>h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous on <math class="ltx_Math" altimg="m236.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="|w|=R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><mi href="./1.9#Px21.p1">R</mi></mrow></math>, then with <math class="ltx_Math" altimg="m215.png" altimg-height="20px" altimg-valign="-2px" altimg-width="74px" alttext="z=re^{i\theta}" display="inline"><mrow><mi>z</mi><mo>=</mo><mrow><mi href="./1.9#Px2.p1">r</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>θ</mi></mrow></msup></mrow></mrow></math>
</p>
<table id="E34" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.34</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E34.png" altimg-height="55px" altimg-valign="-21px" altimg-width="391px" alttext="u(re^{i\theta})=\frac{1}{2\pi}\int^{2\pi}_{0}\frac{(R^{2}-r^{2})h(Re^{i\phi})%
\mathrm{d}\phi}{R^{2}-2Rr\cos\left(\phi-\theta\right)+r^{2}}" display="block"><mrow><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.9#Px2.p1">r</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>θ</mi></mrow></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></mfrac><mo></mo><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mn>0</mn><mrow><mn>2</mn><mo></mo><mi href="./3.12#E1">π</mi></mrow></msubsup><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><msup><mi href="./1.9#Px21.p1">R</mi><mn>2</mn></msup><mo>-</mo><msup><mi href="./1.9#Px2.p1">r</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mi>h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.9#Px21.p1">R</mi><mo></mo><msup><mi href="./4.2#E11" mathvariant="normal">e</mi><mrow><mi mathvariant="normal">i</mi><mo></mo><mi>ϕ</mi></mrow></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>ϕ</mi></mrow></mrow><mrow><mrow><msup><mi href="./1.9#Px21.p1">R</mi><mn>2</mn></msup><mo>-</mo><mrow><mn>2</mn><mo></mo><mi href="./1.9#Px21.p1">R</mi><mo></mo><mi href="./1.9#Px2.p1">r</mi><mo></mo><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>ϕ</mi><mo>-</mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>+</mo><msup><mi href="./1.9#Px2.p1">r</mi><mn>2</mn></msup></mrow></mfrac></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E34.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./3.12#E1" title="(3.12.1) ‣ §3.12 Mathematical Constants ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><math class="ltx_Math" altimg="m127.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="\pi" display="inline"><mi href="./3.12#E1">π</mi></math>: the ratio of the circumference of a circle to its diameter</a>,
<a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./4.2#E11" title="(4.2.11) ‣ §4.2(ii) Logarithms to a General Base a ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="13px" altimg-valign="-2px" altimg-width="13px" alttext="\mathrm{e}" display="inline"><mi href="./4.2#E11" mathvariant="normal">e</mi></math>: base of natural logarithm</a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a>,
<a href="./1.9#Px20.p1" title="Mean Value Property ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m197.png" altimg-height="23px" altimg-valign="-7px" altimg-width="41px" alttext="u(z)" display="inline"><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: harmonic function</a> and
<a href="./1.9#Px21.p1" title="Poisson Integral ‣ §1.9(iii) Integration ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.9#Px21.p1">R</mi></math>: radius</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">is harmonic in <math class="ltx_Math" altimg="m242.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="|z|<R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.9#Px21.p1">R</mi></mrow></math>. Also with <math class="ltx_Math" altimg="m98.png" altimg-height="23px" altimg-valign="-7px" altimg-width="72px" alttext="\left|w\right|=R" display="inline"><mrow><mrow><mo>|</mo><mi href="./1.1#p2.t1.r3">w</mi><mo>|</mo></mrow><mo>=</mo><mi href="./1.9#Px21.p1">R</mi></mrow></math>,
<math class="ltx_Math" altimg="m105.png" altimg-height="32px" altimg-valign="-16px" altimg-width="149px" alttext="\lim\limits_{z\to w}u(z)=h(w)" display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>z</mi><mo>→</mo><mi href="./1.1#p2.t1.r3">w</mi></mrow></munder><mo></mo><mrow><mi href="./1.9#Px20.p1">u</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi>h</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> as <math class="ltx_Math" altimg="m223.png" altimg-height="13px" altimg-valign="-2px" altimg-width="60px" alttext="z\to w" display="inline"><mrow><mi>z</mi><mo>→</mo><mi href="./1.1#p2.t1.r3">w</mi></mrow></math> within <math class="ltx_Math" altimg="m242.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="|z|<R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.9#Px21.p1">R</mi></mrow></math>.</p>
</div>
</section>
</section>
<section id="iv" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.9(iv) </span>Conformal Mapping</h2>
<div id="SS4.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Markushevich (, pp. 41–46)</cite>,
<cite class="ltx_cite ltx_citemacro_citet">Markushevich (, vol. 1, §34)</cite>, and
<cite class="ltx_cite ltx_citemacro_citet">Levinson and Redheffer (, pp. 259–277)</cite>.</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<div id="SS4.p1" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">extended complex plane</em>,
<math class="ltx_Math" altimg="m110.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="\mathbb{C}\,\cup\,\{\infty\}" display="inline"><mrow><mpadded width="+1.7pt"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></mpadded><mo href="./front/introduction#Sx4.p1.t1.r27" rspace="4.2pt">∪</mo><mrow><mo stretchy="false">{</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">}</mo></mrow></mrow></math>, consists of the points of the complex plane
<math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> together with an ideal point <math class="ltx_Math" altimg="m95.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\infty" display="inline"><mi mathvariant="normal">∞</mi></math> called the <em class="ltx_emph ltx_font_italic">point at
infinity</em>.
A system of <em class="ltx_emph ltx_font_italic">open disks around infinity</em> is given by
</p>
<table id="E35" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.35</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E35.png" altimg-height="25px" altimg-valign="-7px" altimg-width="245px" alttext="S_{r}=\{z\mid|z|>1/r\}\cup\{\infty\}," display="block"><mrow><mrow><msub><mi href="./1.9#SS4.p1">S</mi><mi href="./1.9#Px2.p1">r</mi></msub><mo>=</mo><mrow><mrow><mo stretchy="false">{</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>∣</mo><mrow><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow><mo>></mo><mrow><mn>1</mn><mo>/</mo><mi href="./1.9#Px2.p1">r</mi></mrow></mrow><mo stretchy="false">}</mo></mrow><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo stretchy="false">{</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">}</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m56.png" altimg-height="17px" altimg-valign="-3px" altimg-width="97px" alttext="0<r<\infty" display="inline"><mrow><mn>0</mn><mo><</mo><mi href="./1.9#Px2.p1">r</mi><mo><</mo><mi mathvariant="normal">∞</mi></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E35.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r27" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m82.png" altimg-height="15px" altimg-valign="-2px" altimg-width="18px" alttext="\cup" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo></math>: union</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.9#Px2.p1" title="Polar Representation ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m189.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="r" display="inline"><mi href="./1.9#Px2.p1">r</mi></math>: radius</a> and
<a href="./1.9#SS4.p1" title="§1.9(iv) Conformal Mapping ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m77.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="S_{r}" display="inline"><msub><mi href="./1.9#SS4.p1">S</mi><mi href="./1.9#Px2.p1">r</mi></msub></math>: neighborhood</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Each <math class="ltx_Math" altimg="m77.png" altimg-height="21px" altimg-valign="-5px" altimg-width="25px" alttext="S_{r}" display="inline"><msub><mi href="./1.9#SS4.p1">S</mi><mi href="./1.9#Px2.p1">r</mi></msub></math> is a <em class="ltx_emph ltx_font_italic">neighborhood</em>
of <math class="ltx_Math" altimg="m95.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\infty" display="inline"><mi mathvariant="normal">∞</mi></math>. Also,
</p>
<table id="E36" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.36</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E36.png" altimg-height="21px" altimg-valign="-6px" altimg-width="193px" alttext="\infty\pm z=z\pm\infty=\infty," display="block"><mrow><mrow><mrow><mi mathvariant="normal">∞</mi><mo>±</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>=</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>±</mo><mi mathvariant="normal">∞</mi></mrow><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E36.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E37" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.37</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E37.png" altimg-height="18px" altimg-valign="-6px" altimg-width="174px" alttext="\infty\cdot z=z\cdot\infty=\infty," display="block"><mrow><mrow><mrow><mi mathvariant="normal">∞</mi><mo>⋅</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>=</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>⋅</mo><mi mathvariant="normal">∞</mi></mrow><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m222.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="z\not=0" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>≠</mo><mn>0</mn></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E37.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E38" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.38</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E38.png" altimg-height="25px" altimg-valign="-7px" altimg-width="88px" alttext="z/\infty=0," display="block"><mrow><mrow><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>/</mo><mi mathvariant="normal">∞</mi></mrow><mo>=</mo><mn>0</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E38.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E39" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.39</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E39.png" altimg-height="25px" altimg-valign="-7px" altimg-width="88px" alttext="z/0=\infty," display="block"><mrow><mrow><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>/</mo><mn>0</mn></mrow><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m220.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="z\neq 0" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>≠</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E39.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">A function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">analytic at</em> <math class="ltx_Math" altimg="m95.png" altimg-height="13px" altimg-valign="-2px" altimg-width="24px" alttext="\infty" display="inline"><mi mathvariant="normal">∞</mi></math> if <math class="ltx_Math" altimg="m171.png" altimg-height="23px" altimg-valign="-7px" altimg-width="124px" alttext="g(z)=f(1/z)" display="inline"><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></math> is analytic
at <math class="ltx_Math" altimg="m214.png" altimg-height="17px" altimg-valign="-2px" altimg-width="51px" alttext="z=0" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mn>0</mn></mrow></math>, and we set <math class="ltx_Math" altimg="m165.png" altimg-height="24px" altimg-valign="-7px" altimg-width="125px" alttext="f^{\prime}(\infty)=g^{\prime}(0)" display="inline"><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi>g</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></mrow></math>.
</p>
</div>
<section id="Px22" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Conformal Transformation</h3>
<div id="Px22.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px22.p1" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is analytic in a domain <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.9#Px22.p1">D</mi></math> and <math class="ltx_Math" altimg="m67.png" altimg-height="21px" altimg-valign="-6px" altimg-width="60px" alttext="C_{1},C_{2}" display="inline"><mrow><msub><mi href="./1.9#Px22.p1">C</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./1.9#Px22.p1">C</mi><mn>2</mn></msub></mrow></math> are two arcs in <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.9#Px22.p1">D</mi></math>
passing through <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>. Let <math class="ltx_Math" altimg="m64.png" altimg-height="23px" altimg-valign="-7px" altimg-width="60px" alttext="C^{\prime}_{1},C^{\prime}_{2}" display="inline"><mrow><msubsup><mi href="./1.9#Px22.p1">C</mi><mn>1</mn><mo>′</mo></msubsup><mo>,</mo><msubsup><mi href="./1.9#Px22.p1">C</mi><mn>2</mn><mo>′</mo></msubsup></mrow></math> be the images of <math class="ltx_Math" altimg="m68.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{1}" display="inline"><msub><mi href="./1.9#Px22.p1">C</mi><mn>1</mn></msub></math> and <math class="ltx_Math" altimg="m69.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{2}" display="inline"><msub><mi href="./1.9#Px22.p1">C</mi><mn>2</mn></msub></math> under
the mapping <math class="ltx_Math" altimg="m201.png" altimg-height="23px" altimg-valign="-7px" altimg-width="83px" alttext="w=f(z)" display="inline"><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>. The <em class="ltx_emph ltx_font_italic">angle between</em> <math class="ltx_Math" altimg="m68.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{1}" display="inline"><msub><mi href="./1.9#Px22.p1">C</mi><mn>1</mn></msub></math> <em class="ltx_emph ltx_font_italic">and</em> <math class="ltx_Math" altimg="m69.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{2}" display="inline"><msub><mi href="./1.9#Px22.p1">C</mi><mn>2</mn></msub></math> at
<math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math> is the angle between the tangents to the two arcs at <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>, that is, the
difference of the signed angles that the tangents make with the positive
direction of the real axis. If <math class="ltx_Math" altimg="m168.png" altimg-height="24px" altimg-valign="-7px" altimg-width="92px" alttext="f^{\prime}(z_{0})\not=0" display="inline"><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>≠</mo><mn>0</mn></mrow></math>, then the angle between <math class="ltx_Math" altimg="m68.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{1}" display="inline"><msub><mi href="./1.9#Px22.p1">C</mi><mn>1</mn></msub></math>
and <math class="ltx_Math" altimg="m69.png" altimg-height="21px" altimg-valign="-5px" altimg-width="27px" alttext="C_{2}" display="inline"><msub><mi href="./1.9#Px22.p1">C</mi><mn>2</mn></msub></math> equals the angle between <math class="ltx_Math" altimg="m65.png" altimg-height="23px" altimg-valign="-7px" altimg-width="27px" alttext="C^{\prime}_{1}" display="inline"><msubsup><mi href="./1.9#Px22.p1">C</mi><mn>1</mn><mo>′</mo></msubsup></math> and <math class="ltx_Math" altimg="m66.png" altimg-height="23px" altimg-valign="-7px" altimg-width="27px" alttext="C^{\prime}_{2}" display="inline"><msubsup><mi href="./1.9#Px22.p1">C</mi><mn>2</mn><mo>′</mo></msubsup></math> both in magnitude and
sense. We then say that the mapping <math class="ltx_Math" altimg="m201.png" altimg-height="23px" altimg-valign="-7px" altimg-width="83px" alttext="w=f(z)" display="inline"><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> is <em class="ltx_emph ltx_font_italic">conformal</em>
(angle-preserving) at <math class="ltx_Math" altimg="m224.png" altimg-height="16px" altimg-valign="-5px" altimg-width="23px" alttext="z_{0}" display="inline"><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></math>.</p>
</div>
<div id="Px22.p2" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">linear transformation</em> <math class="ltx_Math" altimg="m159.png" altimg-height="23px" altimg-valign="-7px" altimg-width="122px" alttext="f(z)=az+b" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>a</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi>b</mi></mrow></mrow></math>, <math class="ltx_Math" altimg="m151.png" altimg-height="21px" altimg-valign="-6px" altimg-width="51px" alttext="a\not=0" display="inline"><mrow><mi>a</mi><mo>≠</mo><mn>0</mn></mrow></math>, has <math class="ltx_Math" altimg="m166.png" altimg-height="24px" altimg-valign="-7px" altimg-width="85px" alttext="f^{\prime}(z)=a" display="inline"><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mi>a</mi></mrow></math>
and <math class="ltx_Math" altimg="m201.png" altimg-height="23px" altimg-valign="-7px" altimg-width="83px" alttext="w=f(z)" display="inline"><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> maps <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math> conformally onto <math class="ltx_Math" altimg="m112.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></math>.
</p>
</div>
</section>
<section id="Px23" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Bilinear Transformation</h3>
<div id="Px23.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px23.p1" class="ltx_para">
<table id="E40" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.40</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E40.png" altimg-height="48px" altimg-valign="-17px" altimg-width="175px" alttext="w=f(z)=\frac{az+b}{cz+d}," display="block"><mrow><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>=</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi>a</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi>b</mi></mrow><mrow><mrow><mi>c</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi>d</mi></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m154.png" altimg-height="21px" altimg-valign="-6px" altimg-width="103px" alttext="ad-bc\not=0" display="inline"><mrow><mrow><mrow><mi>a</mi><mo></mo><mi>d</mi></mrow><mo>-</mo><mrow><mi>b</mi><mo></mo><mi>c</mi></mrow></mrow><mo>≠</mo><mn>0</mn></mrow></math>, <math class="ltx_Math" altimg="m156.png" altimg-height="21px" altimg-valign="-6px" altimg-width="49px" alttext="c\not=0" display="inline"><mrow><mi>c</mi><mo>≠</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E40.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m202.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E41" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.41</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="25px" altimg-valign="-7px" altimg-width="78px" alttext="\displaystyle f(-d/c)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mrow><mi>d</mi><mo>/</mo><mi>c</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="18px" altimg-valign="-6px" altimg-width="53px" alttext="\displaystyle=\infty," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="25px" altimg-valign="-7px" altimg-width="53px" alttext="\displaystyle f(\infty)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="62px" alttext="\displaystyle=a/c." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi>a</mi><mo>/</mo><mi>c</mi></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E41.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E42" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.42</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E42.png" altimg-height="51px" altimg-valign="-21px" altimg-width="164px" alttext="f^{\prime}(z)=\frac{ad-bc}{(cz+d)^{2}}," display="block"><mrow><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi>a</mi><mo></mo><mi>d</mi></mrow><mo>-</mo><mrow><mi>b</mi><mo></mo><mi>c</mi></mrow></mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>c</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi>d</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m221.png" altimg-height="23px" altimg-valign="-7px" altimg-width="85px" alttext="z\not=-d/c" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>≠</mo><mrow><mo>-</mo><mrow><mi>d</mi><mo>/</mo><mi>c</mi></mrow></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E42.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E43" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.43</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E43.png" altimg-height="47px" altimg-valign="-16px" altimg-width="158px" alttext="f^{\prime}(\infty)=\frac{bc-ad}{c^{2}}." display="block"><mrow><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mrow><mi>b</mi><mo></mo><mi>c</mi></mrow><mo>-</mo><mrow><mi>a</mi><mo></mo><mi>d</mi></mrow></mrow><msup><mi>c</mi><mn>2</mn></msup></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E43.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E44" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.44</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E44.png" altimg-height="48px" altimg-valign="-17px" altimg-width="127px" alttext="z=\frac{dw-b}{-cw+a}." display="block"><mrow><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mfrac><mrow><mrow><mi>d</mi><mo></mo><mi href="./1.1#p2.t1.r3">w</mi></mrow><mo>-</mo><mi>b</mi></mrow><mrow><mrow><mo>-</mo><mrow><mi>c</mi><mo></mo><mi href="./1.1#p2.t1.r3">w</mi></mrow></mrow><mo>+</mo><mi>a</mi></mrow></mfrac></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E44.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r3" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m202.png" altimg-height="13px" altimg-valign="-2px" altimg-width="19px" alttext="w" display="inline"><mi href="./1.1#p2.t1.r3">w</mi></math>: variable</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px23.p2" class="ltx_para">
<p class="ltx_p">The transformation () is a one-to-one conformal mapping
of <math class="ltx_Math" altimg="m110.png" altimg-height="23px" altimg-valign="-7px" altimg-width="87px" alttext="\mathbb{C}\,\cup\,\{\infty\}" display="inline"><mrow><mpadded width="+1.7pt"><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></mpadded><mo href="./front/introduction#Sx4.p1.t1.r27" rspace="4.2pt">∪</mo><mrow><mo stretchy="false">{</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">}</mo></mrow></mrow></math> onto itself.
</p>
</div>
<div id="Px23.p3" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">cross ratio</em>
of <math class="ltx_Math" altimg="m226.png" altimg-height="23px" altimg-valign="-7px" altimg-width="204px" alttext="z_{1},z_{2},z_{3},z_{4}\in\mathbb{C}\cup\{\infty\}" display="inline"><mrow><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>,</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msub><mo>,</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>4</mn></msub></mrow><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi><mo href="./front/introduction#Sx4.p1.t1.r27">∪</mo><mrow><mo stretchy="false">{</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">}</mo></mrow></mrow></mrow></math> is defined by
</p>
<table id="E45" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.45</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E45.png" altimg-height="53px" altimg-valign="-21px" altimg-width="169px" alttext="\frac{(z_{1}-z_{2})(z_{3}-z_{4})}{(z_{1}-z_{4})(z_{3}-z_{2})}," display="block"><mrow><mfrac><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msub><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>4</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>4</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msub><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E45.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">or its limiting form, and is invariant under bilinear transformations.</p>
</div>
<div id="Px23.p4" class="ltx_para">
<p class="ltx_p">Other names for the bilinear transformation are <em class="ltx_emph ltx_font_italic">fractional linear
transformation</em>, <em class="ltx_emph ltx_font_italic">homographic transformation</em>, and <em class="ltx_emph ltx_font_italic">Möbius
transformation</em>.</p>
</div>
</section>
</section>
<section id="v" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.9(v) </span>Infinite Sequences and Series</h2>
<div id="SS5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Copson (, pp. 19–24, 92–98)</cite>.</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<div id="SS5.p1" class="ltx_para">
<p class="ltx_p">A sequence <math class="ltx_Math" altimg="m147.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\{z_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math> <em class="ltx_emph ltx_font_italic">converges</em>
to <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> if <math class="ltx_Math" altimg="m103.png" altimg-height="32px" altimg-valign="-16px" altimg-width="106px" alttext="\lim\limits_{n\to\infty}z_{n}=z" display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>=</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></math>. For <math class="ltx_Math" altimg="m230.png" altimg-height="20px" altimg-valign="-6px" altimg-width="125px" alttext="z_{n}=x_{n}+iy_{n}" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><msub><mi>x</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>+</mo><mrow><mi mathvariant="normal">i</mi><mo></mo><msub><mi>y</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow></mrow></math>, the
sequence <math class="ltx_Math" altimg="m147.png" altimg-height="23px" altimg-valign="-7px" altimg-width="44px" alttext="\{z_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math> converges iff the sequences <math class="ltx_Math" altimg="m144.png" altimg-height="23px" altimg-valign="-7px" altimg-width="46px" alttext="\{x_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi>x</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math> and <math class="ltx_Math" altimg="m145.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="\{y_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi>y</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math>
separately converge. A <em class="ltx_emph ltx_font_italic">series</em> <math class="ltx_Math" altimg="m136.png" altimg-height="26px" altimg-valign="-8px" altimg-width="80px" alttext="\sum^{\infty}_{n=0}z_{n}" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> <em class="ltx_emph ltx_font_italic">converges</em>
if the sequence <math class="ltx_Math" altimg="m191.png" altimg-height="26px" altimg-valign="-8px" altimg-width="124px" alttext="s_{n}=\sum^{n}_{k=0}z_{k}" display="inline"><mrow><msub><mi href="./1.9#SS5.p1">s</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./1.1#p2.t1.r5">n</mi></msubsup><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow></mrow></math> converges. The series is
<em class="ltx_emph ltx_font_italic">divergent</em> if <math class="ltx_Math" altimg="m192.png" altimg-height="16px" altimg-valign="-5px" altimg-width="24px" alttext="s_{n}" display="inline"><msub><mi href="./1.9#SS5.p1">s</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></math> does not converge. The series converges
<em class="ltx_emph ltx_font_italic">absolutely</em> if <math class="ltx_Math" altimg="m137.png" altimg-height="26px" altimg-valign="-8px" altimg-width="91px" alttext="\sum^{\infty}_{n=0}|z_{n}|" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">|</mo></mrow></mrow></math> converges. A series
<math class="ltx_Math" altimg="m136.png" altimg-height="26px" altimg-valign="-8px" altimg-width="80px" alttext="\sum^{\infty}_{n=0}z_{n}" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> converges (diverges) absolutely when
<math class="ltx_Math" altimg="m104.png" altimg-height="35px" altimg-valign="-16px" altimg-width="144px" alttext="\lim\limits_{n\to\infty}|z_{n}|^{1/n}<1" display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><msup><mrow><mo stretchy="false">|</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">|</mo></mrow><mrow><mn>1</mn><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msup></mrow><mo><</mo><mn>1</mn></mrow></math> (<math class="ltx_Math" altimg="m62.png" altimg-height="17px" altimg-valign="-3px" altimg-width="35px" alttext=">1" display="inline"><mrow><mi></mi><mo>></mo><mn>1</mn></mrow></math>), or when
<math class="ltx_Math" altimg="m102.png" altimg-height="32px" altimg-valign="-16px" altimg-width="172px" alttext="\lim\limits_{n\to\infty}\left|\ifrac{z_{n+1}}{z_{n}}\right|<1" display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mrow><mo>|</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>|</mo></mrow></mrow><mo><</mo><mn>1</mn></mrow></math> (<math class="ltx_Math" altimg="m62.png" altimg-height="17px" altimg-valign="-3px" altimg-width="35px" alttext=">1" display="inline"><mrow><mi></mi><mo>></mo><mn>1</mn></mrow></math>).
Absolutely convergent series are also convergent.</p>
</div>
<div id="SS5.p2" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m143.png" altimg-height="23px" altimg-valign="-7px" altimg-width="70px" alttext="\{f_{n}(z)\}" display="inline"><mrow><mo stretchy="false">{</mo><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">}</mo></mrow></math> be a sequence of functions defined on a set <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math>. This
sequence <em class="ltx_emph ltx_font_italic">converges pointwise</em>
to a function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> if</p>
<table id="E46" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.46</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E46.png" altimg-height="34px" altimg-valign="-16px" altimg-width="161px" alttext="f(z)=\lim_{n\to\infty}f_{n}(z)" display="block"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E46.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for each <math class="ltx_Math" altimg="m219.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="z\in S" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.9#SS5.p2">S</mi></mrow></math>. The sequence <em class="ltx_emph ltx_font_italic">converges uniformly</em>
on <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math>, if for every <math class="ltx_Math" altimg="m85.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon>0" display="inline"><mrow><mi href="./1.9#SS5.p2">ϵ</mi><mo>></mo><mn>0</mn></mrow></math> there exists an integer <math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="N" display="inline"><mi>N</mi></math>, independent of
<math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>, such that</p>
<table id="E47" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.47</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E47.png" altimg-height="25px" altimg-valign="-7px" altimg-width="160px" alttext="|f_{n}(z)-f(z)|<\epsilon" display="block"><mrow><mrow><mo stretchy="false">|</mo><mrow><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.9#SS5.p2">ϵ</mi></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E47.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#SS5.p2" title="§1.9(v) Infinite Sequences and Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./1.9#SS5.p2">ϵ</mi></math>: positive number</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for all <math class="ltx_Math" altimg="m219.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="z\in S" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.9#SS5.p2">S</mi></mrow></math> and <math class="ltx_Math" altimg="m185.png" altimg-height="20px" altimg-valign="-5px" altimg-width="61px" alttext="n\geq N" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>≥</mo><mi>N</mi></mrow></math>.</p>
</div>
<div id="SS5.p3" class="ltx_para">
<p class="ltx_p">A series
<math class="ltx_Math" altimg="m135.png" altimg-height="26px" altimg-valign="-8px" altimg-width="106px" alttext="\sum^{\infty}_{n=0}f_{n}(z)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> <em class="ltx_emph ltx_font_italic">converges uniformly</em> on <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math>, if the sequence
<math class="ltx_Math" altimg="m190.png" altimg-height="26px" altimg-valign="-8px" altimg-width="176px" alttext="s_{n}(z)=\sum^{n}_{k=0}f_{k}(z)" display="inline"><mrow><mrow><msub><mi href="./1.9#SS5.p1">s</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./1.1#p2.t1.r5">n</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math> converges uniformly on <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math>.</p>
</div>
<section id="Px24" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Weierstrass <math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="M" display="inline"><mi href="./1.9#Px24">M</mi></math>-test</h3>
<div id="Px24.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
<a class="ltx_keyword" href="./idx/M#Mtestforuniformconvergence"><math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="M" display="inline"><mi href="./1.9#Px24">M</mi></math>-test for uniform convergence</a>, <a class="ltx_keyword" href="./idx/W#WeierstrassMtest">Weierstrass <math class="ltx_Math" altimg="m72.png" altimg-height="18px" altimg-valign="-2px" altimg-width="26px" alttext="M" display="inline"><mi href="./1.9#Px24">M</mi></math>-test</a>, </dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px24.p1" class="ltx_para">
<p class="ltx_p">Suppose <math class="ltx_Math" altimg="m142.png" altimg-height="23px" altimg-valign="-7px" altimg-width="54px" alttext="\{M_{n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.9#Px24">M</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">}</mo></mrow></math> is a sequence of real numbers such that
<math class="ltx_Math" altimg="m132.png" altimg-height="26px" altimg-valign="-8px" altimg-width="90px" alttext="\sum^{\infty}_{n=0}M_{n}" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><msub><mi href="./1.9#Px24">M</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> converges and <math class="ltx_Math" altimg="m235.png" altimg-height="23px" altimg-valign="-7px" altimg-width="118px" alttext="|f_{n}(z)|\leq M_{n}" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><msub><mi href="./1.9#Px24">M</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> for all <math class="ltx_Math" altimg="m219.png" altimg-height="18px" altimg-valign="-3px" altimg-width="52px" alttext="z\in S" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./1.9#SS5.p2">S</mi></mrow></math>
and all <math class="ltx_Math" altimg="m182.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 0" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>≥</mo><mn>0</mn></mrow></math>. Then the series <math class="ltx_Math" altimg="m135.png" altimg-height="26px" altimg-valign="-8px" altimg-width="106px" alttext="\sum^{\infty}_{n=0}f_{n}(z)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> converges
uniformly on <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math>.
</p>
</div>
<div id="Px24.p2" class="ltx_para">
<p class="ltx_p">A doubly-infinite series
<math class="ltx_Math" altimg="m131.png" altimg-height="27px" altimg-valign="-10px" altimg-width="126px" alttext="\sum^{\infty}_{n=-\infty}f_{n}(z)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> converges (uniformly) on <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math> iff each of the
series <math class="ltx_Math" altimg="m135.png" altimg-height="26px" altimg-valign="-8px" altimg-width="106px" alttext="\sum^{\infty}_{n=0}f_{n}(z)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m138.png" altimg-height="26px" altimg-valign="-8px" altimg-width="118px" alttext="\sum^{\infty}_{n=1}f_{-n}(z)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mrow><mo>-</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> converges
(uniformly) on <math class="ltx_Math" altimg="m76.png" altimg-height="18px" altimg-valign="-2px" altimg-width="18px" alttext="S" display="inline"><mi href="./1.9#SS5.p2">S</mi></math>.
</p>
</div>
</section>
</section>
<section id="vi" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.9(vi) </span>Power Series</h2>
<div id="SS6.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Copson (, pp. 37–40)</cite>,
<cite class="ltx_cite ltx_citemacro_citet">Levinson and Redheffer (, pp. 349–351)</cite>, or
<cite class="ltx_cite ltx_citemacro_citet">Markushevich (, pp. 131–135)</cite>. For the operations on
series, see <cite class="ltx_cite ltx_citemacro_citet">Henrici (, Chapter 1)</cite> or
<cite class="ltx_cite ltx_citemacro_citet">Olver (, pp. 19–22)</cite>.</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<div id="SS6.p1" class="ltx_para">
<p class="ltx_p">For a series <math class="ltx_Math" altimg="m133.png" altimg-height="26px" altimg-valign="-8px" altimg-width="160px" alttext="\sum^{\infty}_{n=0}a_{n}(z-z_{0})^{n}" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></math> there is a number <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.9#E49">R</mi></math>, <math class="ltx_Math" altimg="m58.png" altimg-height="20px" altimg-valign="-5px" altimg-width="103px" alttext="0\leq R\leq\infty" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi href="./1.9#E49">R</mi><mo>≤</mo><mi mathvariant="normal">∞</mi></mrow></math>, such that the series converges for all <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> in <math class="ltx_Math" altimg="m237.png" altimg-height="23px" altimg-valign="-7px" altimg-width="110px" alttext="|z-z_{0}|<R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.9#E49">R</mi></mrow></math> and
diverges for <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> in <math class="ltx_Math" altimg="m240.png" altimg-height="23px" altimg-valign="-7px" altimg-width="110px" alttext="|z-z_{0}|>R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo>></mo><mi href="./1.9#E49">R</mi></mrow></math>. The circle <math class="ltx_Math" altimg="m239.png" altimg-height="23px" altimg-valign="-7px" altimg-width="110px" alttext="|z-z_{0}|=R" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo>=</mo><mi href="./1.9#E49">R</mi></mrow></math> is called the
<em class="ltx_emph ltx_font_italic">circle of convergence</em>
of the series, and <math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.9#E49">R</mi></math> is the <em class="ltx_emph ltx_font_italic">radius of convergence</em>. Inside the circle
the sum of the series is an analytic function <math class="ltx_Math" altimg="m161.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>. For <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> in
<math class="ltx_Math" altimg="m241.png" altimg-height="23px" altimg-valign="-7px" altimg-width="105px" alttext="|z-z_{0}|\leq\rho" display="inline"><mrow><mrow><mo stretchy="false">|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>ρ</mi></mrow></math> (<math class="ltx_Math" altimg="m60.png" altimg-height="18px" altimg-valign="-3px" altimg-width="41px" alttext="<R" display="inline"><mrow><mi></mi><mo><</mo><mi href="./1.9#E49">R</mi></mrow></math>), the convergence is absolute and
uniform. Moreover,</p>
<table id="E48" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.48</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E48.png" altimg-height="50px" altimg-valign="-16px" altimg-width="133px" alttext="a_{n}=\frac{f^{(n)}(z_{0})}{n!}," display="block"><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mfrac><mrow><msup><mi>f</mi><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">n</mi><mo stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E48.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p1.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m48.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></math>: factorial (as in <math class="ltx_Math" altimg="m177.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href="./front/introduction#Sx4.p1.t1.r14" lspace="0pt" rspace="3.5pt">!</mo></mrow></math>)</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E49" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.49</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E49.png" altimg-height="38px" altimg-valign="-16px" altimg-width="184px" alttext="R=\liminf_{n\to\infty}|a_{n}|^{-1/n}." display="block"><mrow><mrow><mi href="./1.9#E49">R</mi><mo>=</mo><mrow><munder><mo href="./front/introduction#Sx4.p2.t1.r4" movablelimits="false">lim inf</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><msup><mrow><mo stretchy="false">|</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo stretchy="false">|</mo></mrow><mrow><mo>-</mo><mrow><mn>1</mn><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mrow></msup></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E49.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.9#E49">R</mi></math>: radius of convergence (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./front/introduction#Sx4.p2.t1.r4" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><math class="ltx_Math" altimg="m107.png" altimg-height="18px" altimg-valign="-2px" altimg-width="60px" alttext="\liminf" display="inline"><mo href="./front/introduction#Sx4.p2.t1.r4">lim inf</mo></math>: least limit point</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="SS6.p2" class="ltx_para">
<p class="ltx_p">For the converse of this result see §.</p>
</div>
<section id="Px25" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Operations</h3>
<div id="Px25.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px25.p1" class="ltx_para">
<p class="ltx_p">When <math class="ltx_Math" altimg="m129.png" altimg-height="23px" altimg-valign="-7px" altimg-width="71px" alttext="\sum a_{n}z^{n}" display="inline"><mrow><mo largeop="true" symmetric="true">∑</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></math> and <math class="ltx_Math" altimg="m130.png" altimg-height="23px" altimg-valign="-7px" altimg-width="69px" alttext="\sum b_{n}z^{n}" display="inline"><mrow><mo largeop="true" symmetric="true">∑</mo><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></math> both converge</p>
<table id="E50" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.50</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E50.png" altimg-height="64px" altimg-valign="-27px" altimg-width="348px" alttext="\sum^{\infty}_{n=0}(a_{n}\pm b_{n})z^{n}=\sum^{\infty}_{n=0}a_{n}z^{n}\pm\sum^%
{\infty}_{n=0}b_{n}z^{n}," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>±</mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow><mo>=</mo><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow><mo>±</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E50.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E51" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.51</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E51.png" altimg-height="66px" altimg-valign="-27px" altimg-width="328px" alttext="\left(\sum^{\infty}_{n=0}a_{n}z^{n}\right)\left(\sum^{\infty}_{n=0}b_{n}z^{n}%
\right)=\sum^{\infty}_{n=0}c_{n}z^{n}," display="block"><mrow><mrow><mrow><mrow><mo>(</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mi>c</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E51.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E52" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.52</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E52.png" altimg-height="64px" altimg-valign="-28px" altimg-width="151px" alttext="c_{n}=\sum^{n}_{k=0}a_{k}b_{n-k}." display="block"><mrow><mrow><msub><mi>c</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mn>0</mn></mrow><mi href="./1.1#p2.t1.r5">n</mi></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo></mo><msub><mi>b</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mi href="./1.1#p2.t1.r4">k</mi></mrow></msub></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E52.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m173.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px25.p2" class="ltx_para">
<p class="ltx_p">Next, let</p>
<table id="E53" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.53</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E53.png" altimg-height="28px" altimg-valign="-7px" altimg-width="263px" alttext="f(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m153.png" altimg-height="21px" altimg-valign="-6px" altimg-width="60px" alttext="a_{0}\not=0" display="inline"><mrow><msub><mi>a</mi><mn>0</mn></msub><mo>≠</mo><mn>0</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E53.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then the expansions (), and
() hold for all sufficiently small <math class="ltx_Math" altimg="m243.png" altimg-height="23px" altimg-valign="-7px" altimg-width="26px" alttext="|z|" display="inline"><mrow><mo stretchy="false">|</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">|</mo></mrow></math>.</p>
<table id="E54" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.54</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E54.png" altimg-height="51px" altimg-valign="-21px" altimg-width="262px" alttext="\frac{1}{f(z)}=b_{0}+b_{1}z+b_{2}z^{2}+\cdots," display="block"><mrow><mrow><mfrac><mn>1</mn><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><msub><mi>b</mi><mn>1</mn></msub><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mrow><msub><mi>b</mi><mn>2</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E54.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where</p>
<table id="E55" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.55</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="22px" altimg-valign="-5px" altimg-width="24px" alttext="\displaystyle b_{0}" display="inline"><msub><mi>b</mi><mn>0</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="25px" altimg-valign="-7px" altimg-width="72px" alttext="\displaystyle=1/a_{0}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><msub><mi>a</mi><mn>0</mn></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="22px" altimg-valign="-5px" altimg-width="24px" alttext="\displaystyle b_{1}" display="inline"><msub><mi>b</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="28px" altimg-valign="-7px" altimg-width="97px" alttext="\displaystyle=-a_{1}/a_{0}^{2}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>/</mo><msubsup><mi>a</mi><mn>0</mn><mn>2</mn></msubsup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="22px" altimg-valign="-5px" altimg-width="24px" alttext="\displaystyle b_{2}" display="inline"><msub><mi>b</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="28px" altimg-valign="-7px" altimg-width="160px" alttext="\displaystyle=(a_{1}^{2}-a_{0}a_{2})/a_{0}^{3}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><msubsup><mi>a</mi><mn>1</mn><mn>2</mn></msubsup><mo>-</mo><mrow><msub><mi>a</mi><mn>0</mn></msub><mo></mo><msub><mi>a</mi><mn>2</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><msubsup><mi>a</mi><mn>0</mn><mn>3</mn></msubsup></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E55.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E56" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.56</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E56.png" altimg-height="25px" altimg-valign="-7px" altimg-width="372px" alttext="b_{n}=-(a_{1}b_{n-1}+a_{2}b_{n-2}+\dots+a_{n}b_{0})/a_{0}," display="block"><mrow><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>b</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi>b</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msub><mi>b</mi><mn>0</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><msub><mi>a</mi><mn>0</mn></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m183.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>≥</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E56.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px25.p3" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m152.png" altimg-height="20px" altimg-valign="-5px" altimg-width="60px" alttext="a_{0}=1" display="inline"><mrow><msub><mi>a</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn></mrow></math>,</p>
<table id="E57" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.57</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E57.png" altimg-height="28px" altimg-valign="-7px" altimg-width="297px" alttext="\ln f(z)=q_{1}z+q_{2}z^{2}+q_{3}z^{3}+\cdots," display="block"><mrow><mrow><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi href="./1.9#Px25.p3">q</mi><mn>1</mn></msub><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mrow><msub><mi href="./1.9#Px25.p3">q</mi><mn>2</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><msub><mi href="./1.9#Px25.p3">q</mi><mn>3</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E57.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./4.2#E2" title="(4.2.2) ‣ §4.2(i) The Logarithm ‣ §4.2 Definitions ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m109.png" altimg-height="18px" altimg-valign="-2px" altimg-width="34px" alttext="\ln\NVar{z}" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: principal branch of logarithm function</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m188.png" altimg-height="18px" altimg-valign="-8px" altimg-width="22px" alttext="q_{j}" display="inline"><msub><mi href="./1.9#Px25.p3">q</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: coefficients</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(principal value), where
</p>
<table id="E58" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.58</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m39.png" altimg-height="18px" altimg-valign="-6px" altimg-width="24px" alttext="\displaystyle q_{1}" display="inline"><msub><mi href="./1.9#Px25.p3">q</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="18px" altimg-valign="-6px" altimg-width="52px" alttext="\displaystyle=a_{1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m40.png" altimg-height="18px" altimg-valign="-6px" altimg-width="24px" alttext="\displaystyle q_{2}" display="inline"><msub><mi href="./1.9#Px25.p3">q</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="28px" altimg-valign="-7px" altimg-width="141px" alttext="\displaystyle=(2a_{2}-a_{1}^{2})/2," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>2</mn><mo></mo><msub><mi>a</mi><mn>2</mn></msub></mrow><mo>-</mo><msubsup><mi>a</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>2</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex18" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="18px" altimg-valign="-6px" altimg-width="24px" alttext="\displaystyle q_{3}" display="inline"><msub><mi href="./1.9#Px25.p3">q</mi><mn>3</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="28px" altimg-valign="-7px" altimg-width="215px" alttext="\displaystyle=(3a_{3}-3a_{1}a_{2}+a_{1}^{3})/3," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>3</mn><mo></mo><msub><mi>a</mi><mn>3</mn></msub></mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi>a</mi><mn>2</mn></msub></mrow></mrow><mo>+</mo><msubsup><mi>a</mi><mn>1</mn><mn>3</mn></msubsup></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>3</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E58.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m188.png" altimg-height="18px" altimg-valign="-8px" altimg-width="22px" alttext="q_{j}" display="inline"><msub><mi href="./1.9#Px25.p3">q</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: coefficients</a></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E59" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.59</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E59.png" altimg-height="25px" altimg-valign="-7px" altimg-width="552px" alttext="q_{n}=(na_{n}-(n-1)a_{1}q_{n-1}-(n-2)a_{2}q_{n-2}-\cdots-a_{n-1}q_{1})/n," display="block"><mrow><mrow><msub><mi href="./1.9#Px25.p3">q</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi href="./1.9#Px25.p3">q</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi href="./1.9#Px25.p3">q</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow><mo>-</mo><mi mathvariant="normal">⋯</mi><mo>-</mo><mrow><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi href="./1.9#Px25.p3">q</mi><mn>1</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m184.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 2" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>≥</mo><mn>2</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E59.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m188.png" altimg-height="18px" altimg-valign="-8px" altimg-width="22px" alttext="q_{j}" display="inline"><msub><mi href="./1.9#Px25.p3">q</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: coefficients</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Also,</p>
<table id="E60" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.60</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E60.png" altimg-height="28px" altimg-valign="-7px" altimg-width="287px" alttext="(f(z))^{\nu}=p_{0}+p_{1}z+p_{2}z^{2}+\cdots," display="block"><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.9#Px25.p3">ν</mi></msup><mo>=</mo><mrow><msub><mi href="./1.9#Px25.p3">p</mi><mn>0</mn></msub><mo>+</mo><mrow><msub><mi href="./1.9#Px25.p3">p</mi><mn>1</mn></msub><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mrow><msub><mi href="./1.9#Px25.p3">p</mi><mn>2</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E60.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./1.9#Px25.p3">ν</mi></math>: complex</a> and
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m187.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="p_{j}" display="inline"><msub><mi href="./1.9#Px25.p3">p</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: coefficients</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">(principal value), where <math class="ltx_Math" altimg="m121.png" altimg-height="18px" altimg-valign="-3px" altimg-width="54px" alttext="\nu\in\mathbb{C}" display="inline"><mrow><mi href="./1.9#Px25.p3">ν</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mi href="./front/introduction#Sx4.p1.t1.r1">ℂ</mi></mrow></math>,
</p>
<table id="E61" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex19" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.61</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="18px" altimg-valign="-6px" altimg-width="25px" alttext="\displaystyle p_{0}" display="inline"><msub><mi href="./1.9#Px25.p3">p</mi><mn>0</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="22px" altimg-valign="-6px" altimg-width="43px" alttext="\displaystyle=1," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex20" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m37.png" altimg-height="18px" altimg-valign="-6px" altimg-width="25px" alttext="\displaystyle p_{1}" display="inline"><msub><mi href="./1.9#Px25.p3">p</mi><mn>1</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m11.png" altimg-height="18px" altimg-valign="-6px" altimg-width="63px" alttext="\displaystyle=\nu a_{1}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mi href="./1.9#Px25.p3">ν</mi><mo></mo><msub><mi>a</mi><mn>1</mn></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex21" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m38.png" altimg-height="18px" altimg-valign="-6px" altimg-width="25px" alttext="\displaystyle p_{2}" display="inline"><msub><mi href="./1.9#Px25.p3">p</mi><mn>2</mn></msub></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m12.png" altimg-height="28px" altimg-valign="-7px" altimg-width="213px" alttext="\displaystyle=\nu((\nu-1)a_{1}^{2}+2a_{2})/2," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi href="./1.9#Px25.p3">ν</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.9#Px25.p3">ν</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msubsup><mi>a</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><msub><mi>a</mi><mn>2</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E61.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./1.9#Px25.p3">ν</mi></math>: complex</a> and
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m187.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="p_{j}" display="inline"><msub><mi href="./1.9#Px25.p3">p</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: coefficients</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">and</p>
<table id="E62" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.62</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E62.png" altimg-height="25px" altimg-valign="-7px" altimg-width="771px" alttext="p_{n}=((\nu-n+1)a_{1}p_{n-1}+(2\nu-n+2)a_{2}p_{n-2}+\dots+((n-1)\nu-1)a_{n-1}p%
_{1}+n\nu a_{n})/n," display="block"><mrow><mrow><msub><mi href="./1.9#Px25.p3">p</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./1.9#Px25.p3">ν</mi><mo>-</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msub><mi href="./1.9#Px25.p3">p</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><mi href="./1.9#Px25.p3">ν</mi></mrow><mo>-</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>a</mi><mn>2</mn></msub><mo></mo><msub><mi href="./1.9#Px25.p3">p</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mi href="./1.9#Px25.p3">ν</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo></mo><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msub><mi href="./1.9#Px25.p3">p</mi><mn>1</mn></msub></mrow><mo>+</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo></mo><mi href="./1.9#Px25.p3">ν</mi><mo></mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m183.png" altimg-height="19px" altimg-valign="-5px" altimg-width="53px" alttext="n\geq 1" display="inline"><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>≥</mo><mn>1</mn></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E62.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>,
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\nu" display="inline"><mi href="./1.9#Px25.p3">ν</mi></math>: complex</a> and
<a href="./1.9#Px25.p3" title="Operations ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m187.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="p_{j}" display="inline"><msub><mi href="./1.9#Px25.p3">p</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: coefficients</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">For the definitions of the principal values of <math class="ltx_Math" altimg="m108.png" altimg-height="23px" altimg-valign="-7px" altimg-width="62px" alttext="\ln f(z)" display="inline"><mrow><mi href="./4.2#E2">ln</mi><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> and <math class="ltx_Math" altimg="m51.png" altimg-height="23px" altimg-valign="-7px" altimg-width="67px" alttext="(f(z))^{\nu}" display="inline"><msup><mrow><mo stretchy="false">(</mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.9#Px25.p3">ν</mi></msup></math>
see §§.</p>
</div>
<div id="Px25.p4" class="ltx_para">
<p class="ltx_p">Lastly, a power series can be differentiated any number of times within its
circle of convergence:</p>
<table id="E63" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.63</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E63.png" altimg-height="64px" altimg-valign="-27px" altimg-width="340px" alttext="f^{(m)}(z)=\sum_{n=0}^{\infty}{\left(n+1\right)_{m}}a_{n+m}(z-z_{0})^{n}," display="block"><mrow><mrow><mrow><msup><mi>f</mi><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r5">m</mi><mo stretchy="false">)</mo></mrow></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mrow><mo href="./5.2#iii">(</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mn>1</mn></mrow><mo href="./5.2#iii">)</mo></mrow><mi href="./1.1#p2.t1.r5">m</mi></msub><mo></mo><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>+</mo><mi href="./1.1#p2.t1.r5">m</mi></mrow></msub><mo></mo><msup><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m99.png" altimg-height="23px" altimg-valign="-7px" altimg-width="110px" alttext="\left|z-z_{0}\right|<R" display="inline"><mrow><mrow><mo>|</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>0</mn></msub></mrow><mo>|</mo></mrow><mo><</mo><mi href="./1.9#E49">R</mi></mrow></math>, <math class="ltx_Math" altimg="m175.png" altimg-height="20px" altimg-valign="-6px" altimg-width="128px" alttext="m=0,1,2,\dots" display="inline"><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>=</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi mathvariant="normal">…</mi></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E63.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m231.png" altimg-height="24px" altimg-valign="-8px" altimg-width="41px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href="./5.2#iii">(</mo><mi class="ltx_nvar" href="./5.1#p2.t1.r5">a</mi><mo href="./5.2#iii">)</mo></mrow><mi class="ltx_nvar" href="./5.1#p2.t1.r1">n</mi></msub></math>: Pochhammer’s symbol (or shifted factorial)</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#E49" title="(1.9.49) ‣ §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m75.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="R" display="inline"><mi href="./1.9#E49">R</mi></math>: radius of convergence</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="vii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.9(vii) </span>Inversion of Limits</h2>
<div id="SS7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>Notes:</dt>
<dd>See <cite class="ltx_cite ltx_citemacro_citet">Copson (, pp. 27–30, 95–97)</cite>. For
(), see
<cite class="ltx_cite ltx_citemacro_citet">Titchmarsh (, §1.77)</cite>.</dd>
<dt>Referenced by:</dt>
<dd>
,
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>See also:</dt>
<dd>Annotations for and
</dd>
</dl>
</div>
</div>
<section id="Px26" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Double Sequences and Series</h3>
<div id="Px26.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px26.p1" class="ltx_para">
<p class="ltx_p">A set of complex numbers <math class="ltx_Math" altimg="m146.png" altimg-height="24px" altimg-valign="-8px" altimg-width="63px" alttext="\{z_{m,n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub><mo stretchy="false">}</mo></mrow></math> where <math class="ltx_Math" altimg="m176.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math> and <math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math> take all positive
integer values is called a <em class="ltx_emph ltx_font_italic">double sequence</em>. It <em class="ltx_emph ltx_font_italic">converges to</em> <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>
if for every <math class="ltx_Math" altimg="m85.png" altimg-height="17px" altimg-valign="-3px" altimg-width="49px" alttext="\epsilon>0" display="inline"><mrow><mi href="./1.9#Px26.p1">ϵ</mi><mo>></mo><mn>0</mn></mrow></math>, there is an integer <math class="ltx_Math" altimg="m74.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="N" display="inline"><mi>N</mi></math> such that
</p>
<table id="E64" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.64</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E64.png" altimg-height="26px" altimg-valign="-8px" altimg-width="125px" alttext="|z_{m,n}-z|<\epsilon" display="block"><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub><mo>-</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi href="./1.9#Px26.p1">ϵ</mi></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E64.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px26.p1" title="Double Sequences and Series ‣ §1.9(vii) Inversion of Limits ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="13px" altimg-valign="-2px" altimg-width="12px" alttext="\epsilon" display="inline"><mi href="./1.9#Px26.p1">ϵ</mi></math>: positive number</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for all <math class="ltx_Math" altimg="m174.png" altimg-height="21px" altimg-valign="-6px" altimg-width="87px" alttext="m,n\geq N" display="inline"><mrow><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo>≥</mo><mi>N</mi></mrow></math>. Suppose <math class="ltx_Math" altimg="m146.png" altimg-height="24px" altimg-valign="-8px" altimg-width="63px" alttext="\{z_{m,n}\}" display="inline"><mrow><mo stretchy="false">{</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub><mo stretchy="false">}</mo></mrow></math> converges to <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math> and the repeated
limits</p>
<table id="E65" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex22" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.65</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E65a.png" altimg-height="24px" altimg-valign="-8px" altimg-width="219px" alttext="\lim_{m\to\infty}\left(\lim_{n\to\infty}z_{m,n}\right)," display="inline"><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mrow><mo>(</mo><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex23" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E65b.png" altimg-height="24px" altimg-valign="-8px" altimg-width="210px" alttext="\lim_{n\to\infty}\left(\lim_{m\to\infty}z_{m,n}\right)" display="inline"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><mrow><mo>(</mo><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E65.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">exist. Then both repeated limits equal <math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>.</p>
</div>
<div id="Px26.p2" class="ltx_para">
<p class="ltx_p">A <em class="ltx_emph ltx_font_italic">double series</em>
is the limit of the double sequence
</p>
<table id="E66" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.66</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E66.png" altimg-height="65px" altimg-valign="-27px" altimg-width="179px" alttext="z_{p,q}=\sum^{p}_{m=0}\sum^{q}_{n=0}\zeta_{m,n}." display="block"><mrow><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>=</mo><mn>0</mn></mrow><mi>p</mi></munderover><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi>q</mi></munderover><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E66.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px26.p2" title="Double Sequences and Series ‣ §1.9(vii) Inversion of Limits ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-8px" altimg-width="35px" alttext="\zeta_{p,q}" display="inline"><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></math>: sum</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">If the limit exists, then the double series is <em class="ltx_emph ltx_font_italic">convergent</em>; otherwise it
is <em class="ltx_emph ltx_font_italic">divergent</em>. The double series is <em class="ltx_emph ltx_font_italic">absolutely convergent</em> if it is
convergent when <math class="ltx_Math" altimg="m140.png" altimg-height="23px" altimg-valign="-8px" altimg-width="43px" alttext="\zeta_{m,n}" display="inline"><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></math> is replaced by <math class="ltx_Math" altimg="m233.png" altimg-height="24px" altimg-valign="-8px" altimg-width="54px" alttext="|\zeta_{m,n}|" display="inline"><mrow><mo stretchy="false">|</mo><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub><mo stretchy="false">|</mo></mrow></math>.
</p>
</div>
<div id="Px26.p3" class="ltx_para">
<p class="ltx_p">If a double series is absolutely convergent, then it is also convergent and its
sum is given by either of the repeated sums</p>
<table id="E67" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex24" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.9.67</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E67a.png" altimg-height="26px" altimg-valign="-8px" altimg-width="182px" alttext="\sum^{\infty}_{m=0}\left(\sum^{\infty}_{n=0}\zeta_{m,n}\right)," display="inline"><mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex25" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E67b.png" altimg-height="26px" altimg-valign="-8px" altimg-width="182px" alttext="\sum^{\infty}_{n=0}\left(\sum^{\infty}_{m=0}\zeta_{m,n}\right)." display="inline"><mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mrow><mo>(</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi href="./1.1#p2.t1.r5">m</mi><mo>,</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E67.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m176.png" altimg-height="13px" altimg-valign="-2px" altimg-width="22px" alttext="m" display="inline"><mi href="./1.1#p2.t1.r5">m</mi></math>: nonnegative integer</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px26.p2" title="Double Sequences and Series ‣ §1.9(vii) Inversion of Limits ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m141.png" altimg-height="23px" altimg-valign="-8px" altimg-width="35px" alttext="\zeta_{p,q}" display="inline"><msub><mi href="./1.9#Px26.p2">ζ</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></math>: sum</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
<section id="Px27" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Term-by-Term Integration</h3>
<div id="Px27.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px27.p1" class="ltx_para">
<p class="ltx_p">Suppose the series <math class="ltx_Math" altimg="m135.png" altimg-height="26px" altimg-valign="-8px" altimg-width="106px" alttext="\sum^{\infty}_{n=0}f_{n}(z)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math>, where <math class="ltx_Math" altimg="m170.png" altimg-height="23px" altimg-valign="-7px" altimg-width="51px" alttext="f_{n}(z)" display="inline"><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is continuous,
converges uniformly on every <em class="ltx_emph ltx_font_italic">compact set</em>
of a domain <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.9#Px27.p1">D</mi></math>, that is, every closed and bounded set in <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.9#Px27.p1">D</mi></math>. Then</p>
<table id="E68" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.68</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E68.png" altimg-height="64px" altimg-valign="-27px" altimg-width="294px" alttext="\int_{C}\sum^{\infty}_{n=0}f_{n}(z)\mathrm{d}z=\sum^{\infty}_{n=0}\int_{C}f_{n%
}(z)\mathrm{d}z" display="block"><mrow><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px27.p1">C</mi></msub><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msub><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi href="./1.9#Px27.p1">C</mi></msub><mrow><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow></mrow></mrow></mrow></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E68.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a>,
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m218.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a> and
<a href="./1.9#Px27.p1" title="Term-by-Term Integration ‣ §1.9(vii) Inversion of Limits ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px27.p1">C</mi></math>: finite contour in <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.9#Px27.p1">D</mi></math></a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">for any finite contour <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="C" display="inline"><mi href="./1.9#Px27.p1">C</mi></math> in <math class="ltx_Math" altimg="m71.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.9#Px27.p1">D</mi></math>.</p>
</div>
</section>
<section id="Px28" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Dominated Convergence Theorem</h3>
<div id="Px28.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Keywords:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
and
</dd>
</dl>
</div>
</div>
<div id="Px28.p1" class="ltx_para">
<p class="ltx_p">Let <math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math> be a finite or infinite interval, and <math class="ltx_Math" altimg="m169.png" altimg-height="23px" altimg-valign="-7px" altimg-width="128px" alttext="f_{0}(t),f_{1}(t),\dots" display="inline"><mrow><mrow><msub><mi>f</mi><mn>0</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mrow><msub><mi>f</mi><mn>1</mn></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo><mi mathvariant="normal">…</mi></mrow></math>
be real or complex continuous functions, <math class="ltx_Math" altimg="m194.png" altimg-height="23px" altimg-valign="-7px" altimg-width="79px" alttext="t\in(a,b)" display="inline"><mrow><mi>t</mi><mo href="./front/introduction#Sx4.p1.t1.r9">∈</mo><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></mrow></math>. Suppose
<math class="ltx_Math" altimg="m134.png" altimg-height="26px" altimg-valign="-8px" altimg-width="103px" alttext="\sum^{\infty}_{n=0}f_{n}(t)" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> converges uniformly in any compact interval in
<math class="ltx_Math" altimg="m50.png" altimg-height="23px" altimg-valign="-7px" altimg-width="48px" alttext="(a,b)" display="inline"><mrow><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">(</mo><mi>a</mi><mo href="./front/introduction#Sx4.p1.t1.r28">,</mo><mi>b</mi><mo href="./front/introduction#Sx4.p1.t1.r28" stretchy="false">)</mo></mrow></math>, and at least one of the following two conditions is satisfied:</p>
<table id="E69" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.69</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E69.png" altimg-height="64px" altimg-valign="-27px" altimg-width="200px" alttext="\int^{b}_{a}\sum^{\infty}_{n=0}|f_{n}(t)|\mathrm{d}t<\infty," display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo><</mo><mi mathvariant="normal">∞</mi></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E69.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E70" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.70</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E70.png" altimg-height="64px" altimg-valign="-27px" altimg-width="200px" alttext="\sum^{\infty}_{n=0}\int^{b}_{a}|f_{n}(t)|\mathrm{d}t<\infty." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo><</mo><mi mathvariant="normal">∞</mi></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E70.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E71" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.9.71</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E71.png" altimg-height="64px" altimg-valign="-27px" altimg-width="295px" alttext="\int^{b}_{a}\sum^{\infty}_{n=0}f_{n}(t)\mathrm{d}t=\sum^{\infty}_{n=0}\int^{b}%
_{a}f_{n}(t)\mathrm{d}t." display="block"><mrow><mrow><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mrow><msubsup><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mrow><msub><mi>f</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi>t</mi></mrow></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E71.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m118.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="\mathrm{d}\NVar{x}" display="inline"><mrow><mo href="./1.4#iv" rspace="0.8pt">d</mo><mi class="ltx_nvar">x</mi></mrow></math>: differential of <math class="ltx_Math" altimg="m205.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math></a>,
<a href="./1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m96.png" altimg-height="26px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href="./1.4#iv" largeop="true" symmetric="true">∫</mo></math>: integral</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m181.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
<dt>Referenced by:</dt>
<dd></dd>
<dt>Permalink:</dt>
<dd></dd>
<dt>Encodings:</dt>
<dd>
</dd>
<dt>See also:</dt>
<dd>Annotations for ,
,
and
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
</section>
</div>
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