Helmholtz theorem states that given a smooth vector field $\mathbf{H}$, there are a scalar field $\phi$ and a vector field $\mathbf{G}$ such that
$\mathbf{H}=\nabla \phi +\nabla \times \mathbf{G}$
and
$\nabla \mathbf{\cdot G}=0$
Is this decomposition unique? That is, given $\mathbf{H}$, are the fields $\phi$, $\mathbf{G}$ satisfying the above equations unique?
Edit: Unique, up to an additive constant.
Thanks