Section 14.5 Chain Rule

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 (b) show that \[ \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 = \left(\frac{\partial z}{\partial r}\right)^2 + \frac{1}{r^2}\left(\frac{\partial z}{\partial \theta}\right)^2 \]