Does Stoke's Theorem Apply to All Closed Surfaces?

[Ed Quanta]

If S is a closed surface, then the integral over S of (curlV) dot dn must equal zero.

How could I show this is true in general?

 

[Theelectricchild]

Yikes the proof---

Can't we use a little basic topology to proove this? We discussed homotopic curves or surfaces and I believe when for a force field F. \del X F = 0 then there exists one. I might be able to look this up, in our class actually we were showed the techniques of evaluating surface integrals, but not the rigors of the proofs.

 

[Tide]

I think you need to show that the integral of the vector field over the surface is equivalent to the divergence of the vector field over the volume bounded by the surface. Then, clearly, if the vector field is the curl of a vector then the volume integral is zero.

 

[Theelectricchild]

Oh right if \int\int\int_V div F dV = 0

 

[rgrig]

As Tide said, use Gauss theorem: the flux thru a closed surface is the integral of elementary fluxes within the enclosed volume (i.e. divergence). Then show (simple algebra) that div curl V = 0.