HelmholtzDecomposition

A Mathematica package for performing analytical Helmholtz decomposition of vector fields in n dimensions, following the approach in Richters and Glotz, "Analytical Helmholtz Decomposition of n-Dimensional Vector Fields" (10.1016/j.jmaa.2023.127138).

Quick Start

(* Load the package *)
Needs["HelmholtzDecomposition`"];

(* Define your coordinate variables *)
x = Subscript[x, 1]; y = Subscript[x, 2];

(* Your vector field *)
f = {x^2 + y, x*y};
coords = {x, y};

(* Complete decomposition with output *)
PrintHelmholtz[f, coords]

(* Direct access to components *)
{gradient, rotational} = HelmholtzDecomposition[f, coords];

Main Functions

One defines variables:

• f: Vector field as a list (e.g., {x^2, y^2})
• xvec: Coordinate variables as a list (e.g., {x, y})

The package as the following main functions:

• HelmholtzDecomposition[f, xvec]: Returns {gradient, rotational} components as a list.

• HelmholtzGradient[f, xvec]: Returns only the gradient component.

• HelmholtzRotational[f, xvec]: Returns only the rotational component.

• PrintHelmholtz[f, xvec]: Displays the complete Helmholtz decomposition as in the original code of Richters and Glotz.

Citation and License

Any intellectual property rights of the original authors are fully respected. Please cite the original paper when using this code.

• Zenodo repository
• Erhard Glötzl, Oliver Richters. Helmholtz decomposition and potential functions for n-dimensional analytic vector fields. 10.1016/j.jmaa.2023.127138