The General Stokes’s Theorem

Abstract

Let ω be a differential form of degree k - 1 and class C ¹ in a neighborhood of a compact regular k-surface with boundary M of class C ². The general Stokes’s theorem gives a relationship between the integral of ω over the boundary of M and the integral of the exterior differential dω over M. It can be viewed as a generalization of Green’s theorem to higher dimensions, and it plays a role not unlike that of the fundamental theorem of calculus in an elementary course of analysis. Particular cases of the general Stokes’s theorem that are of great importance are the divergence theorem, which relates a triple integral with a surface integral and what we know as the classical Stokes’s theorem, which relates a surface integral with a line integral.