A simple proof of the angular momentum Helmholtz theorem and the relation of the theorem to the decomposition of solenoidal vectors into poloidal and toroidal components

Vector spherical harmonics are used in a simple proof of the angular-momentum Helmholtz theorem. The decomposition of vectors defined on a sphere into two components which this theorem gives is carried out explicitly. Furthermore, the potentials which occur in the theorem are given explicitly in terms of the original vector. The decomposition of solenoidal vectors into poloidal and toroidal components is also carried out explicitly. It is shown how these components are related to the components given by the angular-momentum Helmholtz theorem.