Chapter 14.5, Problem 45E

Assume that all the given functions are differentiable.

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

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