I have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given in polar coordinates by $f(r,\theta) = (r,\theta + k)$ and i want to compute if it is volume-preserving (i.e the determinant of the Jacobian is $+1$ or $-1$).
Can I just compute the partial derivatives of this function as if it were given in euclidean coordinates? Or do I have to compose it with coordinate transformations and then compute the determinant of the Jacobian of the composed function?