# Helmholtz decomposition theorem and Blumenthal's extension by regularization

## TL;DR

This paper discusses the Helmholtz decomposition theorem, extending Blumenthal's regularization approach to include vector fields that grow sublinearly, with applications to time-dependent fields like dipole radiation.

## Contribution

It generalizes Blumenthal's regularization method to prove Helmholtz decomposition for vector fields with sublinear growth, beyond previous restrictions.

## Key findings

- Helmholtz decomposition applies to asymptotically increasing fields with regularization.
- Extension of Blumenthal's method to time-dependent and artificially increasing fields.
- Application to dipole radiation fields and other sublinearly increasing vector fields.

## Abstract

Helmholtz decomposition theorem for vector fields is usually presented with too strong restrictions on the fields and only for time independent fields. Blumenthal showed in 1905 that decomposition is possible for any asymptotically weakly decreasing vector field. He used a regularization method in his proof which can be extended to prove the theorem even for vector fields asymptotically increasing sublinearly. Blumenthal's result is then applied to the time-dependent fields of the dipole radiation and an artificial sublinearly increasing field.