Gauss's Law
Gauss's Law states that the electric flux through a closed surface is proporational to the electrical charge inside. This holds true regardless of the volume or shape of the closed surface. This is one of the most fundamental principles of electrodynamics, and is one of Maxwell's Equations. The law is named after Carl Friedrich Gauss. It is useful for calculating the electric field around distributions of charges with lots of symmetry.
In integral form, Gauss's Law is this:
where 
is the electric flux through the surface S, 
is the electric field, 
is a differential area on the closed surface S with an outward facing surface normal defining its direction, 
is the charge enclosed by the surface, 
is the charge density at a point in 
, 
is a constant for the permittivity of free space and the integral 
is over the surface S enclosing volume V.
It can also be written in differential form as:
where is the charge density.
Proof of equivalency
It is easy to show that the differential and integral forms of Gauss's law are equivalent. This can be done by integrating the differential form over a volume:
As integral of charge density over volume is the charge contained within that volume, the integral can be replaced with QA. Using the Divergence theorem, the left hand side can be changed from a volume integral of a divergence into a surface integral with no divergence across the surface of the volume, S. This produces the integral form:
Example of use
Gauss's law is useful when there is a lot of symmetry in a problem. An example of that is a sphere of uniform charge density, and radius R. We choose a surface over which we can easily evaluate:
This surface is known as a "Gaussian surface". We choose it so that 
term in the dot product evaluates simply to either 1 or 0. In our example of a sphere, we choose a spherical Gaussian surface of radius r. This means that all the small vector elements dA point radially outwards and as the electric field due to the sphere will be radial, and will always be parallel. This means the terms will be:
This is the left handside. The right hand side depends on the charge enclosed by the surface and therefore r. This means the electric field takes a different form inside and outside the sphere. Outside the sphere, the charge enclosed is 
while inside it is 
. This means the electric field inside and around the sphere is:
where 
is a unit vector pointing radially outwards. Note that outside the sphere, the field drops off according to an inverse square law. This means that it is the same as if the sphere were a point particle. This is a much simpler calculation than integrating electric field component of each point particle in the sphere.
