The integral form of Gauss's Law relates the integral of the electric field over some surface to the charge enclosed by that surface.
where 
is the charge density and the volume integral extends over the
volume
enclosed by the surface in the first integral.
Gauss's Law can also be written in its differential form:
The electric potential 
is related to the electric field:
We can then substitute Eqn. 3 into Eqn. 2
Equation 5 is known as Poisson's Equation. If there is no free
charge,
then 
and Eqn. 5 reduces to Laplace's Eqn.:
In Cartesian coordinates, Laplace's Eqn. reads:
