Representation of vector fields

Abstract

A simple proof is given for the explicit formula which allows one to recover a C2 – smooth vector field A=A(x) in R3, decaying at infinity, from the knowledge of its ∇×A and ∇⋅A. The representation of A as a sum of the gradient field and a divergence-free vector fields is derived from this formula. Similar results are obtained for a vector field in a bounded C2 - smooth domain.