Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates

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 obtain...

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Bibliographic Details
Published inarXiv.org
Main Authors Glötzl, Erhard, Richters, Oliver
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 16.07.2021
Subjects
Calculus
Cartesian coordinates
Decay rate
Decomposition
Differential calculus
Differential geometry
Divergence
Fields (mathematics)
Laplace equation
Rotation
Tensors
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Summary: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.
ISSN:2331-8422