?Assume that all the given functions are differentiable.If \(z=f(x, y)\), where \(x=r

Chapter 12, Problem 49

(choose chapter or problem)

Assume that all the given functions are differentiable.

If \(z=f(x, y)\), where \(x=r \cos \theta\) and \(y=r \sin \theta\), (a) find \(\partial z / \partial r\) and \(\partial z / \partial \theta\) and  show that

                            \(\left(\frac{\partial z}{\partial x}\right)^{2}+\left(\frac{\partial z}{\partial y}\right)^{2}=\left(\frac{\partial z}{\partial r}\right)^{2}+\frac{1}{r^{2}}\left(\frac{\partial z}{\partial \theta}\right)^{2}\)

Equation Transcription:

Text Transcription:

z=f(x,y)

x=r cos⁡ theta

y=r sin⁡ theta

partial z/partial r

partial z/partial theta

(partial z/partial x)^2+(partial z/partial y)^2=(partial z/partial r)^2+1/r^2(partial z/partial theta)^2