Vector circulation. Stokes, Gauss and maybe more?

[AlfredB]

Hello. My first time posting here. So... My question is kinda hard to explain but I will try to. So we all know about the Kelvin-Stokes theorem (not talking about manifolds here) :

And we also know about Ostrogradsky/Gauss Theorem :

[PLAIN]http://upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/23px-OiintLaTeX.svg.png[PLAIN]http://upload.wikimedia.org/math/a/b/9/ab9fd5a4aaa36e402c98cbd36af3a70d.png

I don't have a problem with these 2. I perfectly get them. What bothers me is: can there be found a relationship between the circulation of a vector

and the volume integral ? Just like Stokes relates circulation and curl and Gauss relates flux and div . Now how can a contour enclose a volume ? If you can modify is shape you can(see my attachment ) THat's just an idea, I don't know if it's been done or it can be done. And while you can modifiy the shape of the contour to get different surfaces of different areas you can only get one sphere of only one volume, speciffic to the length of the contour. I'm sorry if I was inchorent, just ask me if you want me to try and formulate my question in a different way although I tried to explain it as good as I could.

 

[HallsofIvy]

Circulation about what loop? A three dimensional object may have a well defined two dimensional surface but there are infinitely many one dimensional loops lying in that surface. (I'm not clear what you picture is supposed to show but it is certainly not a curve completely covering the sphere.)