V - Line and Surface Integrals; Differential Forms and Stokes' Theorem

Abstract

This chapter presents an exposition of the machinery that is necessary for the statement, proof, and application of Stokes' theorem. Stokes' theorem is a multi-dimensional generalization of the fundamental theorem of (single-variable) calculus and can be called the fundamental theorem of multi-variable calculus. Among its numerous applications are the classical theorems of vector analysis. Green's theorem is a two-dimensional generalization of the fundamental theorem of calculus. The chapter discusses how to decompose a circular or triangular disk D into oriented two-cells. Upon adding the equations obtained by the application of Green's theorem to each of the oriented two-cells, the line integrals over the interior segments can cancel because each is traversed twice in opposite directions, leaving as a result Green's theorem for D. The multiplication of differentials extends in a natural way to a multiplication of differential forms.