Let $\Omega \subset \mathbb{R}^n$ a compact and starshaped set with a smooth boundary. Furthermore we assume that the barycentre of $\Omega$ is $0$ and $|\Omega| = |B_1(0)|$. Then it is possible to represent the boundary $\partial \Omega$ in polar coordinates. For $x\in\mathbb{R}^n$ and $\zeta\in \Sigma$ (the unit sphere in $\mathbb{R}^n$) these coordinates are give via $$ |x| = 1 + u(\zeta). $$
Now let $\Phi$ be a volume preserving diffeomorphism. Consider the set $\Phi(\Omega)$ and lets assume $\Phi(\Omega)$ is still a compact and starshaped set. My Question now is how do the polar coordinates for the boundary change under such a diffeomorphism? For example are the new polar coordinates give by $$ |y|=1 + u(\Phi(\zeta)), $$ with $y\in\mathbb{R}^n$?