data/mathoid/52.mml
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xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="ModifyingBelow u 1 left-parenthesis bold x comma z 1 right-parenthesis equals v 1 plus ModifyingAbove u With dot Subscript x Baseline With bottom-brace Underscript By definition of v 1 Endscripts equals ModifyingAbove minus StartFraction partial-differential upper V Subscript x Baseline Over partial-differential bold x EndFraction g Subscript x Baseline left-parenthesis bold x right-parenthesis minus k 1 left-parenthesis ModifyingBelow z 1 minus u Subscript x Baseline left-parenthesis bold x right-parenthesis With bottom-brace Underscript e 1 Endscripts right-parenthesis With top-brace Overscript v 1 Endscripts plus ModifyingAbove StartFraction partial-differential u Subscript x Baseline Over partial-differential bold x EndFraction left-parenthesis ModifyingBelow f Subscript x Baseline left-parenthesis bold x right-parenthesis plus g Subscript x Baseline left-parenthesis bold x right-parenthesis z 1 With bottom-brace Underscript ModifyingAbove bold x With dot left-parenthesis i period e period comma StartFraction normal d bold x Over normal d t EndFraction right-parenthesis Endscripts right-parenthesis With top-brace Overscript ModifyingAbove u With dot Subscript x Baseline left-parenthesis i period e period comma StartFraction normal d u Subscript x Baseline Over normal d t EndFraction right-parenthesis Endscripts">
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<annotation encoding="application/x-tex">\underbrace {u_{1}(\mathbf {x} ,z_{1})=v_{1}+{\dot {u}}_{x}} _{{\text{By definition of }}v_{1}}=\overbrace {-{\frac {\partial V_{x}}{\partial \mathbf {x} }}g_{x}(\mathbf {x} )-k_{1}(\underbrace {z_{1}-u_{x}(\mathbf {x} )} _{e_{1}})} ^{v_{1}}\,+\,\overbrace {{\frac {\partial u_{x}}{\partial \mathbf {x} }}(\underbrace {f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1}} _{{\dot {\mathbf {x} }}{\text{ (i.e., }}{\frac {\operatorname {d} \mathbf {x} }{\operatorname {d} t}}{\text{)}}})} ^{{\dot {u}}_{x}{\text{ (i.e., }}{\frac {\operatorname {d} u_{x}}{\operatorname {d} t}}{\text{)}}}</annotation>
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</math>