On Tue, Mar 26, 2013 at 7:09 PM, Dan Asimov <dasimov@earthlink.net> wrote:
No. Surfaces (in 3-space) can have boundaries, and that's what they're talking about.
OK. I thought you were comparing an integral on the interior of the sphere to an integral on its boundary (the sphere), using Stokes' theorem on a 3-manifold. You're comparing an integral on the sphere to an integral on its (null) boundary, using Stokes' theorem on a 2-manifold. In that case, I think the problem is that
(*) div(V) == 0 implies that there exists a vector field W such that V == curl(W).
Is true locally, but not globally. Locally, there are of course many vector fields W such that V == curl(W); you can add any curl-free field to W. But that doesn't mean there is a choice of W that works globally, on the entire 2-manifold. We can look at V as a 2-form, and not every closed 2-form is exact; quotient of the closed 2-forms by the exact 2-forms is the second cohomology group of the surface, which is non-trivial for an orientable compact manifold such as the sphere. Andy