What Does Stokes' Theorem Reveal About Circulation and Curl?

[Mindscrape]

Just a couple quick conceptual questions about Stokes' Theorem (maybe this belongs in the non-homework math forum?). Does Stokes' theorem say anything about circulation in a field for which the curl is zero? I would think that all it says is that there is no net circulation. Also, if F is a differentiable vector field defined in a region containing a smooth closed surface S, where S is the union of two surfaces S1 and S2, what can be said about
$$\iint_S \nabla \times \mathbf{F} \cdot \mathbf{n} d\sigma$$
?

 

[HallsofIvy]

Stoke's theorem (also called the "curl theorem- in mathematics "Stoke's theorem" is more general), says that
$$\int_S\int \nabla x \vec{f}\cdot d\vec{S}= \int_{\partial S} f\cdot d\vec{\sigma}$$
where S is a surface and $\partial S$ is the boundary curve of that surface.

If $\nabla x \vec{f}$ is 0 then obviously the left side of that is 0 for any S so the right side is 0 for any closed curve. The circulation is 0.

On the other hand, a smooth closed surface has no boundary so the right side of that equation must be 0 and therefore the integral you show is 0.

 

[Mindscrape]

Hah, I know what Stokes' theorem is, but I was looking at some vector calculus books and they had some of these concept questions in them. The first one I thought was obvious, but thought that maybe I had missed something. It was asking if there was anything special about circulation in a field whose curl is zero. I don't see anything special, other than that there could be circulation in areas, but that the net flow on any boundary would have to cancel to be zero.

The second one was meant to be a separate case. I am much more familiar with the divergence theorem, and I know that the divergence theorem would say that the flux depends on the unit normal. For Stokes' theorem it should be the similar case - that the joining of the two surfaces, and the circulation through them will depend on the unit normal vector.
$$\iint_{S1}\nabla \times \mathbf{F} \cdot \mathbf{n_1} d\sigma_1$$ + $$\iint_{S2} \nabla \times \mathbf{F} \cdot \mathbf{n_2} d\sigma_2$$

 

[Mindscrape]

Now that I think about it, the curl of the vector function being zero will mean that the circulation is zero if the curve is simply connected, but if it isn't simply connected then you can't know until you do the flux calculation.