Hodge Theory and Electromagnetism

Abstract

Let M be a compact domain in R3. The Hodge Decomposition Theorem yields a decomposition of the space of vector elds on M into ve mutually orthogonal subspaces that encode geometric and topological features of M. This decomposition is useful in many branches of mathematics, physics, and engineering. In this paper, we study the general version of this theorem, valid for di erential forms on smooth, compact, oriented manifolds with boundary, in any dimension, and deduce from it the particular ve-term decomposition for compact domains in 3-space. We do this by using basic notions from multivariable calculus, linear algebra, di erential forms, and algebraic topology, following the article [CDTG], by Cantarella, DeTurck and Gluck, and the book of Schwarz [S]. Furthermore, we present some applications of the notions developed in this paper to the formulation of Maxwell's equations and to the graphical representations of di erential forms in Rn.