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. We generalize the vector potential from R3 to a rotation potential, an n-dimensional, antisymmetric matrix-valued map describing n(n − 1)/2 rotations within the coordinate planes. We introduce methods using one-dimensional line integrals to calculate gradient and rotation potentials and corresponding fields for n-dimensional analytic vector fields on unbounded domains without restrictions on their behaviour at infinity. Closed-form solutions are obtained for periodic and exponential functions, multivariate polynomials and their linear combinations.

This Mathematica worksheet allows to calculate the Helmholtz decomposition for n-dimensional vector fields. Examples include the Lorenz and Rössler attractor and the competitive Lotka–Volterra equations with n species.