# How to compute if a function in polar coordinates is volume preserving?

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?

-

The transformation is simply a rotation by angle $k$ about the origin. A rotation is an isometry, so preserves area.
More precisely, this is the transformation given by $$\left[\begin{array}{c}x\\y\end{array}\right]\mapsto\left[\begin{array}{cc}\cos k & \sin k\\- \sin k & \cos k\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right].$$