Differential form valued forms and distributional electromagnetic sources

Properties of a fundamental double-form of bidegree $(p,p)$ for $p≥0$ are reviewed in order to establish a distributional framework for analyzing equations of the form $ΔΦ(p)+λ2Φ(p)=S(p)$⁠, where $Δ$ is the Hodge–de Rham operator on $p$-forms $Φ(p)$ on $R3$⁠. Particular attention is devoted to singular distributional solutions that arise when the source $S(p)$ is a singular $p$-form distribution. A constructive approach to Dirac distributions on (moving) submanifolds embedded in $R3$ 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.