Introduction

What is usually called Helmholtz theorem[1] is the fact that a vector field $\vec{V}$ which vanishes at infinity is completely determined by giving, everywhere, $div\vec{V}$ and $curl\;\vec{V}$. Helmholtz arrived at it in his study of vortices in fluids[4]. Nowadays it is mostly used in Electrodynamics, where the very Maxwell equations express its importance by being nothing else than the specification of what the $div$ and $curl$ of the electric and magnetic fields are.