Electromagnetic Green functions using differential forms

Abstract

In this paper we redevelop the scalar and dyadic Green functions of electromagnetic theory using differential forms. The Green dyadic becomes a double form, which is a differential form in one space with coefficients that are forms in another space, or a differential form-valued form. The results presented here correspond closely with the usual dyadic treatment, but are clearer and more intuitive. Many of the usual expressions using green functions in vector notation require a surface normal; with the Green forms the surface normal is unnecessary. We illustrate the formalism by computing scattering from a randomly rough conducting surface and deriving the Green form for a dielectric half-space. We also define the interior derivative, which is equivalent to the coderivative but for a constant metric has a computational rule dual to that of the exterior derivative and simplifies calculation in coordinates. This work makes available some of the tools that have not yet been presented in the language of differential forms but are essential in applied electromagnetics.