Given a Diffeomorphisem $\Phi$, consider the boundary integral
$ \int_{\partial\Phi(\Omega)}f\cdot\nu ~dS$,
where $\nu$ is the outer unit normal vector on $\Phi(\Omega)$. I now want to do a change of variables for this integarl to get
$ \int_{\partial\Phi(\Omega)}f\cdot\nu ~dS= \int_{\partial\Omega}[f\circ\Phi]\cdot[\nu\circ\Phi]\left|\operatorname{det}D\Phi\right|dS$.
My Problem now is to understand the right hand site of this equation. Since $\nu\circ\Phi$ is not necessary a outer unit normal vector on $\Omega$ I am not sure if this transformation can be done in this way.