Differential Form Valued Forms and Distributional Electromagnetic Sources

TL;DR

This paper develops a distributional framework using differential forms to analyze electromagnetic equations with singular sources, focusing on solutions involving Dirac distributions on submanifolds in three-dimensional space.

Contribution

It introduces a novel approach to handle singular distributional solutions of electromagnetic equations using differential forms and geometric embeddings.

Findings

Framework for analyzing singular distributional solutions

Constructive method for Dirac distributions on submanifolds

Application to electromagnetic modeling with localized 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.