To avoid dealing with electric fields that come from charges which are infinitely far and non-volumetric charge distributions, I stated the problem in the following way: assume a vector field E defined inside an sphere S. If div E = 0 and curl E = 0, can I find a charge arrangement outside the sphere that will have E as the resultant electric field? I will try to state it in a more rigorous way: can I always find a function [imath]\rho(x, y, z)[/imath] that satisfies the identity

[math]\forall{(x_0, y_0,z_0)\in S}\left(\int_{-\infty }^{\infty }\int_{-\infty }^{\infty}\int_{-\infty}^{\infty} \frac{\rho (x,y,z)\cdot (x_0-x, y_0-y, z_0-z)}{|(x_0-x, y_0-y, z_0-z)|^{3}}dxdydz = \vec{\mathbf{E}}(x_0,y_0,z_0)\right)[/math]no matter how E is defined, as long as div(E) = 0 and curl E = 0?

_{Did I do it right?}

I am using the fact that div E = 0 because it seems to simplify the problem; if there is always a solution when div E = 0, I can use superposition to find the solutions when div E !=0 by dividing the field into a "source only" field and a "source free" field.