## Problem on chain rule and change of coordinates

Let $f$ be a function of cartesian coordinates $x,y$. It is possible to express $f$ in terms of polar coordinates $r,\theta$ by $f(r,\theta) = f\bigl(x(r,\theta), y(r, \theta)\bigr)$.

(a) What are the expressions for $x(r, \theta)$ and $y(r, \theta)$? That is, write down the $x$ and $y$ coordinates of a point with polar coordinates $r, \theta$.

(b) Use the chain rule to express $\begin{pmatrix}\partial_r f \\ \partial_\theta f \end{pmatrix}$ as a matrix times $\begin{pmatrix}\partial_x f \\ \partial_y f \end{pmatrix}$.
• ## Solution

(a) Recall that in polar coordinates
\begin{align} \partial_r f &= \partial_x f \ \partial_r x + \partial_y f \ \partial_r y \\ \partial_\theta f &= \partial_x f \ \partial_\theta x + \partial_y f \ \partial_\theta y \end{align}
We end up with
$$\begin{pmatrix} \partial_r w \\ \partial_\theta w \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin \theta \\ -r \sin \theta & r \cos \theta \end{pmatrix} \begin{pmatrix} \partial_x w \\ \partial_y w \end{pmatrix}.$$