How to differentiate multivariable calculus with parametric equations?

Question

$u=u(x,y)$ and $x=e^s\cos t, y=e^s\sin t$

How do I work with this type of multivariable calculus?

E.g. how do I find $u_{ss}$, $u_{tt}$, $u_{xx}$ etc?

I don't know where to start exactly, I had a few attempts but not sure.

Answer 1

Simple (in theory…): the chain rule for multivariable functions yields \begin{align} \frac{\partial u}{\partial s}&=\frac{\partial u}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial s} \\ \frac{\partial^2 u}{\partial s^2}&=\frac{\partial}{\partial x}\biggl(\frac{\partial u}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial s}\biggr)\frac{\partial x}{\partial s}+\frac{\partial}{\partial y}\biggl(\frac{\partial u}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial s}\biggr)\frac{\partial y}{\partial s} \end{align} &c. &c.