Helmholtz decomposition and potential functions for n-dimensional analytic vector fields

The Helmholtz decomposition splits a sufficiently smooth vector field into a gradient field and a divergence-free rotation field. Existing decomposition methods impose constraints on the behavior of vector fields at infinity and require solving convolution integrals over the entire coordinate space. To allow a Helmholtz decomposition in Rⁿ, we replace the vector potential in R³ by the rotation potential, an n-dimensional, antisymmetric matrix-valued map describing n(n−1)/2 rotations within the coordinate planes.

This Mathematica worksheet calculates closed-form solutions using line-integrals for several unboundedly growing fields including periodic and exponential functions, multivariate polynomials and their linear combinations. Examples include the Lorenz and Rössler attractor and the competitive Lotka-Volterra equations with n species.

A description of this approach can be found in: “Helmholtz decomposition and potential functions for n-dimensional analytic vector fields“, Journal of Mathematical Analysis and Applications, 2023, doi:10.1016/j.jmaa.2023.127138, arXiv:2102.09556. Version 2 of this software has been published as supplementary material for the article.