Differential form valued forms and distributional electromagnetic sources

Abstract

Properties of a fundamental double-form of bi-degree $(p,p)$ for $p\ge 0$ are reviewed in order to establish a distributional framework for analysing equations of the form $$\Delta \Phi + \lambda^2 \Phi = {\cal S} $$ where $\Delta$ is the Hodge-de Rham operator on $p-$forms $ \Phi $ on ${\bf R}^3$. Particular attention is devoted to singular distributional solutions that arise when the source $ {\cal S}$ is a singular $p-$form distribution. A constructive approach to Dirac distributions on (moving) submanifolds embedded in ${\bf R}^3$ is developed in terms of (Leray) forms generated by the geometry of the embedding. This framework offers a useful tool in electromagnetic modeling where the possibly time dependent sources of certain physical attributes, such as electric charge, electric current and polarization or magnetization, are concentrated on localized regions in space.