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 $\begin{pmatrix}\partial_x f \\ \partial_y f \end{pmatrix}$.