If z=xe^y, x=u^2+v^2, y=u^2-v^2, find dz/du and dz/dv. these variables are restrticted to domains on which the dunctions are defined

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24-07-2017
Mathematics

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If z=xe^y, x=u^2+v^2, y=u^2-v^2, find dz/du and dz/dv. these variables are restrticted to domains on which the dunctions are defined

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LammettHash LammettHash · Answer 1 · 2017-07-24T19:26:41+02:00

Chain rule:

[tex]\dfrac{\partial z}{\partial u}=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial u}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial u}[/tex]
[tex]\dfrac{\partial z}{\partial u}=e^y(2u)+xe^y(2u)[/tex]
[tex]\dfrac{\partial z}{\partial u}=2ue^y(1+x)[/tex]

[tex]\dfrac{\partial z}{\partial v}=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial v}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial v}[/tex]
[tex]\dfrac{\partial z}{\partial v}=e^y(2v)+xe^y(-2v)[/tex]
[tex]\dfrac{\partial z}{\partial v}=2ve^y(1-x)[/tex]