and it is now a matter of (tedious) calculus to arrive at the right result.

This is the task of exercises 3 and 4, which finally compute the Laplacian

in polar coordinates.

!!! warning Inverse function theorem

In this calculation one might be tempted to use the inverse

function theorem to compute derivatives like

$\frac{\partial \varphi}{\partial x}$ from the much simpler

$\frac{\partial x}{\partial \varphi}$. Note though that here we

are dealing with functions depending on several variables, so the

*Jacobian* has to be used (see [Wikipedia](https://en.wikipedia.org/wiki/Inverse_function_theorem). A direct calculation is in this particular case more easy.

Note that this procedure also carries over to other coordinate systems,

although the calculations can become quite tedious. In these cases,