The elegance of differential forms in vector calculus and electromagnetics

Abstract

In the chapter one of this text we give an introduction to, and discuss the main integral theorems, of vector calculus; Green's theorem, Stokes' theorem and Gauss' Divergence theorem. Note that the main resource used for this chapter is [8]. Chapter two introduces differential forms and exterior calculus; in it we discuss exterior multiplication and exterior differentiation giving proofs for properties of both. We discuss the integration of differential forms in chapter three and provide definitions of the Divergence, Gradient and Curl and main integral theorems of vector calculus including the Generalised Stokes' theorem that encloses them all in terms of such forms. Further we give a proof of the Generalised Stokes', Green's, Stokes' and Gauss' Divergence theorems. Given the elegance of differential forms that enables us to write the integral theorems of vector calculus as one theorem, the Generalised Stokes' theorem, we show a second elegance by deducing and proving Maxwell's equations, whilst reducing them from four equations to just two. Finally we provide some current research involving differential forms.