Stokes’ theorem plays an important role in computing integrals of differential forms on manifolds, and its consequence, the Cauchy integral formula, is quite fundamental in complex analysis in one variable as well.

Suppose is a compact -dimensional submanifold with smooth boundary in the plane . From symplectic geometry, the natural symplectic form on can be written as , where . If one computes the integral,

(1)

where denotes the codisk bundle of , since is an one dimensional curve in , it makes the integral containing measuring the perturbations of base points of cotangent vectors in be zero. Hence each term of the right hand side of (1) is zero. So one has that , which is also true in general when is a compact hypersurface contained in some affine -plane in .

By the way, Gowers wrote a post, one way of looking at Cauchy theorem, which is very interesting and tells us that the Cauchy’s theorem is the natural generalization in -dimension of the statement is a constant for , that, we know, is a direct consequence of the second fundamnetal theorem of calculus which is the special case of the Stokes’ theorem in one dimension. So in this sense, Stokes’ theorem is the “generator” of all the theorems and statements of this kind.