I'm trying to prove a result in appendix C.2 of L. Evans, "Partial Differential Equations". Here it is listed as the Gauss-Green theorem and the statement is as follows:
Assume that is a bounded, open subset of and is . Let denote the unit outward normal vector.
(i) Let such that is . Then, for .
(ii) For a vector field we have, .
The book claims that the second result is known as the divergence theorem* and follows from applying the first to each component of . That is, it should be enough to prove (i).
Anyway, I want to prove this for the case where is a simple "box", i.e. . I really have no idea where to start. Could anyone point me in the right direction, give some reference or a sketch of a proof?
* Question: However, my lecturer called this same statement for Stokes theorem. I thought Stokes theorem had something to do with the integral of the curl in the interior? Is this a special case of Stokes theorem, or did he mix them up?