Let $\mathbf F$ be a smooth vector field in $\mathbb R^3$, which is null outside a finite compact domain $V$. By the Helmholtz decomposition theorem, there exist a scalar field $\Phi$ and a vector field $\mathbf A$ such that $${\mathbf F} = \nabla \Phi + \nabla \times {\mathbf A}.$$

Do there always exist smooth fields $\Phi$ and $\mathbf A$, which are null outside $V$ and which satisfy this identity?