Since the electric field is (minus) the gradient of the scalar potential, it is conservative. But since \begin{eqnarray*} \grad\times\grad V = \zero \end{eqnarray*} for any function $V$, we can rewrite this as \begin{eqnarray*} \grad\times\EE = \zero \end{eqnarray*} which is the differential form of (the electrostatic version of) Faraday's Law, and another of Maxwell's Equations.

Similarly, we can apply the identity \begin{eqnarray*} \grad\cdot(\grad\times\FF) = 0 \end{eqnarray*} for any vector field $\FF$, to the magnetic vector potential, which yields the fourth and final of Maxwell's Equations, namely \begin{eqnarray*} \grad\cdot\BB = \grad\cdot(\grad\times\AA) = 0 \end{eqnarray*}

In summary, Maxwell's equations for electro- and magnetostatics are: \begin{eqnarray*} \grad\cdot\EE &=& \frac{\rho}{\epsilon_0} \\ \grad\times\EE &=& \zero \\ \noalign{\smallskip} \grad\cdot\BB &=& 0 \\ \noalign{\smallskip} \grad\times\BB &=& \mu_0 \,\JJ \end{eqnarray*}