(Solved): A system of electric charges has a charge density \( \rho(x, y, z) \) and produces an electrostatic ...

A system of electric charges has a charge density \( \rho(x, y, z) \) and produces an electrostatic field \( \mathbf{E}(x, y, z) \) at points \( (x, y, z) \) in space. Gauss' Law states that \[ \iint_{\Sigma} \mathbf{E} \cdot d \sigma=4 \pi \iiint_{S} \rho d V \] for any closed surface \( \Sigma \) which encloses the charges, with \( S \) being the solid region enclosed by \( \Sigma \). Show that \( \nabla \cdot \mathbf{E}=4 \pi \rho \). This is one of Maxwell's Equations.