Gauss's Law can be written in two different forms: integral and differential. To relate the two, we use the Divergence theorem.
Gauss's law for the electric field says that the electric flux through any closed surface is proportional to the amount of electric charge contained within that surface. Again let be some volume of space, and let be its surface. Then
where is the total charge contained in . We can write this in terms of the charge density as
so that Gauss's law becomes
where is a fundamental constant. You can look at a not-entirely rigorous proof.