The Helmholtz decomposition revisited

Abstract In electrodynamics it is generally accepted that only vector fields decaying asymptotically more strongly than the inverse of the distance can be split into an irrotational and a solenoidal part, although this decomposition can clearly be applied to all vector fields that vanish asymptotically with a small power. Moreover this decomposition is valid even for vector fields that diverge weaker than linearly. A simple proof is presented.