I'm concerned with the well-known divergence theorem and I'm struggling with the types of feasible domains the theorem is defined, because there are different formulations. More precisely some formulations require a global smooth boundary (e.g. $\partial\Omega\in C^1$). So others require only piecewise smoothness that permits sharp edges.
So I'm wondering, why the global smoothness is justifiable? Might the invariance of the Lebesgue measure in the integral formulation concerning Null sets the reason? It would make sense, but I'm quite uncertain.
I would be grateful for some hints.
The version with smooth boundary is far easier to remember (regarding the prerequisites) and easier to prove. Since it is sufficient in many cases many authors of textbooks are content with this. The proof of the more general versions is often mostly by applying technical reasonings to the smooth version -- there is often no real insight gained by formulating or proving them, unless you are in need for a specific generalization.