A Discussion of Helmholtz' Theorem

A vector field can be split into an irrotational part (curl = 0) and a solenoidal one (div = 0). This tutorial paper discusses under which conditions such a splitting is unique, both in infinite space and in a finite volume. In the latter case, multiply-bounded and multiply-connected regions must be considered separately, and electric and magnetic splittings are possible, depending on the boundary conditions imposed on the potentials.