Section 14.5 Chain Rule

Assume that all the given functions have continuous second-order partial derivatives. If \(z = f(x, y)\), where \(x = r \cos \theta\) and \(y = r \sin \theta\), show that \[ \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = \frac{\partial^2 z}{\partial r^2} + \frac{1}{r^2}\frac{\partial^2 z}{\partial \theta^2} + \frac{1}{r}\frac{\partial z}{\partial r} \]