TITLE: Interesting Applications of the Classical Stokes Theorem?
QUESTION [23 upvotes]: When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y by evaluating the surface integral of the curl of the vector field over the surface X" or vice versa. The trouble is that the vector fields, curves and surfaces are pretty much arbitrary except for being chosen so that one or both of the integrals are computationally tractable.
One more interesting application of the classical Stokes theorem is that it allows one to interpret the curl of a vector field as a measure of swirling about an axis. But aside from that I'm unable to recall any other applications which are especially surprising, deep or interesting. 
I would like use Stokes theorem show my multivariable calculus students something that they enjoyable. Any suggestions?

REPLY [7 votes]: You can tell your students that a clever use of Stokes theorem can give you a Fields medal. Indeed, the proof that the formality map given by M. Kontsevich is a $L_\infty$-morphism, is nothing else than Stokes theorem. A detailed account of this can be found in  Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66 (2003), no. 3, 157–216 or with more details in Déformation, quantification, théorie de Lie, 123–164, Panor. Synthèses, 20, Soc. Math. France, Paris, 2005  which is in English, contrary to its title.