The Helmholtz decomposition of decreasing and weakly increasing vector fields

TL;DR

This paper extends the Helmholtz decomposition theorem to include vector fields that decrease or increase weakly, using a regularized Green's function, with applications demonstrated through examples like electric dipole radiation.

Contribution

It generalizes the Helmholtz decomposition to broader classes of vector fields, including those that grow sublinearly at infinity, based on Blumenthal's work.

Findings

Decomposition possible for asymptotically increasing vector fields

Regularization of Green's function is effective

Applications include electric dipole radiation

Abstract

Helmholtz decomposition theorem for vector fields is presented usually with too strong restrictions on the fields. Based on the work of Blumenthal of 1905 it is shown that the decomposition of vector fields is not only possible for asymptotically weakly decreasing vector fields, but even for vector fields, which asymptotically increase sublinearly. Use is made of a regularizatin of the Greens function and the mathematics of the proof is formulated as simply as possible. We also show a few examples for the decomposition of vector fields including the electric dipole radiation.