Generalized second-order partial derivatives of 1/r

TL;DR

This paper derives generalized second-order derivatives of 1/r in 3D, introduces non-spherical regularizations, and demonstrates their application in classical electrodynamics.

Contribution

It presents new non-spherical regularization methods for second-order derivatives of 1/r, expanding beyond standard spherical approaches.

Findings

Derived generalized second-order derivatives of 1/r.

Introduced non-spherical regularization alternatives.

Illustrated utility in classical electrodynamics.

Abstract

The generalized second-order partial derivatives of 1/r, where r is the radial distance in 3D, are obtained using a result of the potential theory of classical analysis. Some non-spherical regularization alternatives to the standard spherical-regularization expression for the derivatives are derived. The utility of a spheroidal-regularization expression is illustrated on an example from classical electrodynamics.