The Classical Stokes Theorem
At last we come to the version of Stokes’ theorem that people learn with that name in calculus courses. Ironically, unlike the fundamental theorem and divergence theorem special cases, Stokes’ theorem only works in dimension , where the differential can take us straight from a line integral over a -dimensional region to a surface integral over an -dimensional region.
So, let’s say that is some two-dimensional oriented surface inside a three-dimensional manifold , and let be its boundary. On the other side, let be a -form corresponding to a vector field . We can easily define the line integral
and Stokes’ theorem tells us that this is equal to
which means that the line integral of around the (oriented) boundary of is the same as the surface integral of the curl through itself. And this is exactly the old Stokes theorem from multivariable calculus.