Determining the surface charge density of a conductor in an electric field mathematically

How can I calculate the charge density of a conductor in an electric field? I have tried many ways but I always seem to not understand how to do it. How can I do this mathematically?

EDIT: The conductor is isolated from other conductors. The external electric field is produced from insulators.(the insulators charge distribution doesnt vary no matter what.) The electric field that's acting on the conductor is given and the potential and charge distribution both have to be found. The surface is parametric. The conductor is hollow and has charges present in it.( again the charges are stationary and do not move). THE TOTAL CHARGE OF THE CONDUCTOR IS ZERO.

• Constant or variable (wrt position) electric field? Commented Dec 23, 2016 at 14:10
• The electric field varies with position and the conductor is of arbitrary shape. Commented Dec 23, 2016 at 14:16
• You are not giving sufficient information in this question. Is the total charge of the conductor given or its potential? Is it far away from other conductors? Commented Dec 23, 2016 at 16:32
• The conductor is isolated from other conductors. The external electric field is produced from insulators.(the insulators charge distribution doesnt vary no matter what.) The electric field that's acting on the conductor is given and the potential and charge distribution both have to be found. The surface is parametric. The conductor is hollow and has charges present in it.( again the charges are stationary and do not move). THE TOTAL CHARGE OF THE CONDUCTOR IS ZERO. Commented Dec 24, 2016 at 6:04

When solving for the potential the simplest general numerical method is often to use Poisson's equation $\nabla^2 V=-\rho_f/\epsilon$, where $\rho_f$ is the local density of free charge. I do not know if this powerful method can be inverted easily to find the densities given the potential (and hence the field). It is easily applied with known charge densities so this could be an obvious starting point.
If you try to invert Poisson's equation, it will have to be interatively and you will benefit from the condition that, to a good approximation $V$ is constant inside a conductor, or alternatively there is no free charge or $\vec E$ field inside the conductor.