The Classical Version of Stokes’ Theorem Revisited

Abstract. Using only fairly simple and elementary considerations essentially from first year undergraduate mathematics we show how the classical Stokes’ theorem for any given surface and vector field in R follows from an application of Gauss’ divergence theorem to a suitable modification of the vector field in a tubular shell around the given surface. The two stated classical theorems are (like the fundamental theorem of calculus) nothing but shadows of the general version of Stokes’ theorem for differential forms on manifolds. The main points in the present paper, however, is firstly that this latter fact usually does not get within reach for students in first year calculus courses and secondly that calculus textbooks in general only just hint at the correspondence alluded to above. Our proof that Stokes’ theorem follows from Gauss’ divergence theorem goes via a well known and often used exercise, which simply relates the concepts of divergence and curl on the local differential level. The rest of the paper uses only integration in 1, 2, and 3 variables together with a ’fattening’ technique for surfaces and the inverse function theorem.