On Equivalence of Maxwell's Equations in Differential and Integral Forms

Abstract

The analytical tools required to legitimize derivations of electromagnetic boundary conditions on an interface of singularity by using Divergence and Stokes's Theorems are presented in the Space of Schwartz-Sobolev Distributions in two steps. First, the differential and integral forms of Maxwell's Equations are demonstrated to be equally informative. By equal information it is implied that the integral/differential form of field equations can be derived when they are given in differential/integral form. Next, Divergence and Stokes's Theorems of Vector Calculus are shown to be valid in the sense of Schwartz-Sobolev distributions. This reveals that the distributional investigations of differential and integral forms of Maxwell's Equations reveal the same set of boundary conditions.