Let me add something:
I have been watching an MIT OpenCourseWare lecture on the divergence theorem (MIT OpenCourseWare | Mathematics | 18.02 Multivariable Calculus, Fall 2007 | Video Lectures | Lecture 29: Divergence Theorem (cont.)) where Prof. Denis Auroux sketches a proof for a special case of the divergence theorem. That is, under the assumption that the region U is 'vertically simple' and the vector field has only one non-zero component (which makes alot of ugly terms dissappear during the integration).
Now, assuming that is a box should be even easier than assuming it is 'vertically simple', so I suppose I could apply the same ideas as in his proof. However, everywhere I look, I keep seeing the divergence theorem formulated for vector fields in (which I suppose is reasonable in most applications, e.g. in physics). That is, the region is always assumed to be a subset of , rather than the more general case where is a subset of as Evans states.
How should I proceed to prove the theorem for a general vector field on the domain ?