Differential forms in electromagnetic field theory

Abstract

The calculus of differential forms can be presented as concretely as vector analysis; and such a treatment is included. We give Maxwell's laws and their geometrical interpretation and show how Ampere's law and the curl operator become as intuitive as Gauss's law and the divergence. The boundary conditions are quite simple when written using vectors, but there is no obvious graphical interpretation. We show that differential forms make the boundary conditions geometrically obvious. This kind of insight, applied to more complex problems, can guide one to a solution and help interpret the solution when it is discovered. We redevelop the dyadic Green function as a double differential form. The usual results are amplified by the absence of surface normal vectors. The Green forms are extended to anisotropic media. Differential forms are natural for these applications. Compared to the dyadic formulation, the final results are simpler and the derivations more algebraic. The differential forms do not replace vectors; ideally the forms and vectors are used interchangeably as appropriate to a particular problem.