  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.
