2020/12/24 by Erhard Glötzl, Glötzl, Erhard, Oliver Richters +1 · 1 citation
Engineering · Physics and Astronomy · #31B99 (Secondary) #35J05 (Primary) 35Q99 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics Simulations and Interactions #Fluid Dynamics and Vibration Analysis #G.1.8 #G.1.9 #Magnetic Bearings and Levitation Dynamics #Mathematical Physics (math-ph) #Scientific Research and Discoveries
paper · pdf · doi:10.48550/arxiv.2012.13157
openalex publication_date 2020/12/24 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
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.