10.2 Stoke's Theorem

What is even more important about Green's Theorem is that it applies just as well for regions R on surfaces that are locally planar: that is, that look like a plane in tiny pieces of them, (like the surface of the earth.) We can break R up into tiny pieces each one looking planar, apply Green's theorem on each and add up the mess to get the same theorem on all of R. The result is called Stokes' Theorem, and it reads,

Theorem: With R an arbitrary reasonably smooth and nice region on a surface that is locally planar, and with R its boundary, we have

This theorem is extremely important in the history of science, as we shall see. It not only supplies the interpretation of the curl of a vector field previously mentioned, but (along with the Divergence Theorem) gives a direct connection between macroscopic integral laws and differential equations that played an important role in the discovery of the laws of electrodynamics and the development of radio and wireless communication.
We can use it as a tool for rewriting integrals of one kind as integrals of another and for proving peculiar facts about irrotational fields (those for which curl = 0).
It can be stated in the following form: the circulation integral of v over a bounding curve is the flux integral of the curl of v over any surface it bounds.
Notice that this theorem immediately implies that if R is simply connected and curl v = 0 throughout it, then the potential f (with v =f) defined as path integral is path independent.