Section 14.5 Chain Rule

Assume that all the given functions have continuous second-order partial derivatives. If \(u = f(x, y)\), where \(x = e^s \cos t\) and \(y = e^s \sin t\), show that \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = e^{-2s}\left[\frac{\partial^2 u}{\partial s^2} + \frac{\partial^2 u}{\partial t^2}\right] \]