Distributions in spherical coordinates with applications to classical electrodynamics

Abstract

A general and rigorous method to deal with singularities at the origin of a polar coordinate system is presented. Its power derives from a clear distinction between the radial distance and the radial coordinate variable, which makes that all delta-functions and their derivatives are automatically generated, and insures that the Gauss theorem is correct for any distribution with a finite number of isolated point-like singularities. The method is applied to the Coulomb field, and to show the intrinsic differences between the dipole and dimonopole fields in classical electrodynamics. In all cases the method directly leads to the general expressions required by the internal consistency of classical electrodynamics.