generalized Stokes' theorem
/ STOHKS /
This is the summit of the whole field, and it fits on one line: the integral of d omega over a region equals the integral of omega over the boundary of that region. In symbols, the integral over M of d omega equals the integral over the boundary of M of omega. It says that to add up how much a quantity changes throughout the inside of a region, you only need to look at its values around the edge — change inside, total on the rim.
Its power is that it contains, as special cases, every integral theorem of vector calculus. Take the region to be an interval and omega a 0-form: you get the fundamental theorem of calculus, integral of f' equals f at the endpoints. Take a planar region and omega a 1-form: you get Green's theorem. Take a surface in space and omega a 1-form: you get the classical Stokes (curl) theorem. Take a solid and omega a 2-form: you get the divergence theorem. They are not four theorems but one theorem at four degrees, and the reason they look so different in vector calculus is only that gradient, curl, and divergence are the single operator d wearing three costumes.
The honest fine print is real and worth stating. The region must be an oriented manifold-with-boundary, and the boundary must carry the induced orientation (the convention that makes the signs come out right). The form must be smooth, or at least continuously differentiable, on the region including its boundary. There must be no missing points or singularities inside — the angle form shows what goes wrong when the region has a hole the form blows up at. Within these hypotheses the theorem is exact and absolute, and it is the structural reason conservation laws in physics can be stated either as a local differential law or as a global flux balance.
Take the unit disk D and the 1-form omega = x dy. Then d omega = dx^dy, so the integral of d omega over D is the area, pi. The theorem says this equals the integral of omega = x dy around the boundary circle, which we computed as pi in the integration example. The two pi's match — that is Green's theorem and the area formula in one.
The same number, pi, computed once as an area inside and once as a circulation on the rim — Stokes in action.
The theorem needs an orientation and smoothness; it is not a free pass. With a hole or singularity inside the region — like the origin for the angle form — the naive equality fails, and you must either exclude the bad point (adding its own little boundary) or accept a nonzero defect, which is precisely the cohomology class the form represents.