Posted by: Louis Y. Liu | October 5, 2008

On Stokes’ Theorem

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 M is a compact 2-dimensional submanifold with smooth boundary in the plane mathbb{R}^{2}. From symplectic geometry, the natural symplectic form on T^{*}mathbb{R}^{2} can be written as omega=dalpha, where alpha=xi dx+eta dy. If one computes the integral,

int_{D^{*}(partial M)}alphawedgeomega=int_{D^{*}(partial M)}xi dxwedge dywedge deta+int_{D^{*}(partial M)}eta dywedge dxwedge dxi(1)

where D^{*}(partial M) denotes the codisk bundle of partial M, since partial M is an one dimensional curve in mathbb{R}^{2}, it makes the integral containing dxwedge dy measuring the perturbations of base points of cotangent vectors in D^{*}(partial M) be zero. Hence each term of the right hand side of (1) is zero. So one has that int_{D^{*}(partial M)}alphawedgeomega=0, which is also true in general when M is a compact hypersurface contained in some affine n-1-plane in mathbb{R}^{n}.

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 2-dimension of the statement f'=0Longrightarrow f is a constant for f:mathbb{Rrightarrow R}, 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.


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