Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates

TL;DR

This paper extends Helmholtz Decomposition to high-dimensional vector fields using convolution with Laplace solutions, providing a straightforward method without differential geometry.

Contribution

It introduces a novel approach to decompose n-dimensional vector fields into source and rotation potentials using elementary calculus techniques.

Findings

Effective decomposition of high-dimensional vector fields.

Simplifies application of Helmholtz Decomposition in n-dimensions.

Avoids complex differential geometry concepts.

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.