Textbook Problem

Assume that all the given functions have continuous second-order partial derivatives.

50. If u = f(x, y), where x = e^s cos t and y = e^s sin t, show that

$\frac{{\partial }^{2}u}{\partial x^2}=\frac{{\partial }^{2}u}{\partial y^2}=e^{-2s}\left[\frac{{\partial }^{2}u}{\partial s^2}=\frac{{\partial }^{2}u}{\partial t^2}\right]$