Representation of vector fields

A simple proof is given for the explicit formula which allows one to recover a $C^2-$smooth vector field $A=A(x)$ in ${\mathbb{R}}^3$, decaying at infinity, from the knowledge of its $\mathrm{\nabla }\times A$ and $\mathrm{\nabla }\cdot A$. 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 $C^2-$smooth domain.