On an identity for the volume integral of the square of a vector field

TL;DR

This paper proves a vector identity linking the volume integral of a vector field's square to non-local integrals of its curl and divergence, with applications to magnetic fields of rotating charged shells.

Contribution

It provides a rigorous proof of a proposed vector identity and demonstrates its application to magnetic fields of rotating charged shells.

Findings

The identity relates volume integrals to non-local curl and divergence integrals.

Application to magnetic fields of a rotating shell illustrates the identity's utility.

The proof involves spherical harmonics and non-local integral techniques.

Abstract

A proof is given of the vector identity proposed by Gubarev, Stodolsky and Zakarov that relates the volume integral of the square of a 3-vector field to non-local integrals of the curl and divergence of the field. The identity is applied to the case of the magnetic vector potential and magnetic field of a rotating charged shell. The latter provides a straightforward exercise in the use of the addition theorem of spherical harmonics.