Chapter 14.5, Problem 46E

Assume that all the given functions are differentiable.

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

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