Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates
Abstract
This paper introduces a novel method to extend the Helmholtz Decomposition to n-dimensional sufficiently smooth and fast decaying vector fields. The rotation is described by a superposition of n(n-1)/2 rotations within the coordinate planes. The source potential and the rotation potential are obtained by convolving the source and rotation densities with the fundamental solutions of the Laplace equation. The rotation-free gradient of the source potential and the divergence-free rotation of the rotation potential sum to the original vector field. The approach relies on partial derivatives and Newton integrals and allows for a simple application of this standard method to high-dimensional vector fields, without using concepts from differential geometry and tensor calculus.
Cite
@article{arxiv.2012.13157,
title = {Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates},
author = {Erhard Glötzl and Oliver Richters},
journal= {arXiv preprint arXiv:2012.13157},
year = {2021}
}
Comments
v1: 12 pages, 1 figure, 1 table. -- v2: 13 pages, 1 figure, 1 table. Difference to v1: Sign change in the definitions of all rotation operators, nicer diagram, proof that curl g(x) = 0, wording, typos. -- v3: 13 pages. Difference to v2: mainly typos