Stokes’ Theorem

Abstract

Stokes’ theorem equates the integral of one expression over a surface to the integral of a related expression over the curve that bounds the surface. A similar result, called Gauss’s theorem, or the divergence theorem, equates the integral of a function over a 3-dimensional region to the integral of a related expression over the surface that bounds the region. The similarities are not accidental. Using the language of differential forms, we show these two theorems are instances (along with Green’s theorem and the fundamental theorem of calculus) of a single theorem that connects one integral over a domain to a related one over its boundary. To explore the connections, we combine the “modern” approach, using differential forms to clarify statements and proofs, with the “classical” appoach, using vector fields to understand the individual theorems in the physical terms in which they arose.