I'm looking for rigorous hypothesis for the application of Reynolds' transport theorem :
$$ \frac{\mathrm{d}}{\mathrm{d}t}\left[ \int_{\Omega(t)} \phi({\bf x},t) \mathrm{d}{\bf x} \right] = \int_{\Omega(t)} \frac{\partial}{\partial t}\phi({\bf x},t) \mathrm{d}{\bf x} + \int_{\partial\Omega(t)} \phi({\bf x},t)\frac{\mathrm{d} {\bf x}}{\mathrm{d} t}.{\bf n}_b \mathrm{d}{\bf x}, $$ where $\Omega(t)$ is a piecewice smooth manifold with boundaries (a portion of a polyhedron for instance), $\partial\Omega(t)$ is the boundary of $\Omega(t)$ and ${\bf n}_b$ is the normal of $\partial\Omega(t)$ at ${\bf x}$. In particular, provided all the integrals are well defined of course, I'm interested in the case when $\phi$ is only piecewise smooth, and when the parameterizations I can find of the boundary $\partial\Omega(t)$ are only piecewise continuously differentiable.
Thanks for any help.
