Stokes' Theorem and de Rham Cohomology through the Language of Forms

Abstract

The Fundamental Theorem of Calculus, Green's theorem, the classical Stokes theorem, and the divergence theorem each relate differentiation on a region to integration on its boundary. This paper develops the language of smooth manifolds and differential forms in which these results become instances of a single statement. After introducing tangent and cotangent spaces, alternating forms, the wedge product, and the exterior derivative, we define integration of top-degree forms on oriented manifolds. We then state the generalized Stokes theorem and give a proof sketch assembled globally by a partition of unity. Finally, we introduce closed and exact forms and de Rham cohomology. The standard angular form in the punctured plane gives a concrete example of a closed form that is not exact, showing how cohomology detects a topological holes.

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