While searching for the statements of divergence theorem, I came across two different conditions:
$(1)$ In wikipedia and proofwiki, it is said that the volume should be compact and boundary should be piecewise smooth.
$(2)$ In Stewart's calculus book and this website and a number of websites, it is said that the volume must be simple solid and nothing is said about boundary.
So why are there two statements? Which one is more general?
Also what exactly is "piecewise smooth boundary"?
I guess it means that the surface $S$ can be broken up into finite number of surfaces $S_1,S_2,S_3,...$ or $S_n$ all of which (considered separately) have a unique tangent plane at every point on each of the surfaces $S_n$.