Solenoidal & Irrotational Vector

Solenoidal

If the $\nabla \cdot \vec{D}$ is zero, $\vec{D}$ is solenoidal, then $\vec{D}$ is the curl of some vector filed $\vec{C}$.

Theorem

If
$\nabla\cdot\vec{D}=0\tag{solenoidal}$
there is a $\vec{C}$ such that
$\vec{D}=\nabla \times \vec{C}\tag{1}$

Irrotational

If the $\nabla \times \vec{A}$ is zero, $\vec{A}$ is irrotational, then $\vec{A}$ is the gradient of some scalar field $\psi$.

Theorem

If
$\nabla\times\vec{A}=0\tag{irrotational}$
there is a $\psi$ such that
$\vec{A}=\nabla \psi\tag{2}$