Chapter 7 - The Generalized Stokes’s Theorem

Abstract

In this chapter we arrive at our main goal, the Generalized Stokes’s Theorem (GST). We begin by stating this theorem and deriving some of its consequences. We first see how, in the 1-dimensional case, it generalizes the Fundamental Theorem of Calculus (FTC), and we apply the GST to line integrals. We next develop the notion of cap independence, a higher-dimensional analog to path-independence. We then see how in the 2-dimensional case the GST generalizes the classical Green’s and Stokes’s theorems, and how in the 3-dimensional case it generalizes the classical Gauss’s theorem. The GST can be regarded as a vast, higher dimensional generalization of the FTC. We provide a complete and careful proof of the GST, which clearly shows how it is based on the FTC. The GST has a converse, de Rham’s theorem, which we state in general and prove in the important special case of top-dimensional forms.