Gauss's Law from Coulomb's Law: A Direct Sum over Ray Intersections

Abstract We derive the electrostatic form of Gauss’s law for a bounded region with a smooth closed boundary. Starting with Coulomb’s force and the area-sum definition of flux, we express the surface integral as a sum over the intersections of each ray with the boundary. The inverse-square field factor cancels the radial area factor, leaving the sign of the cosine of the angle between the ray and the outward normal. Successive signs alternate, and the last sign is positive. Pairing adjacent terms reduces the sum to a criterion determined by the first intersection. Summing the charges and integrating over the angular coordinates gives the enclosed charge divided by the vacuum permittivity. The appendices establish the area formula, the geometric pairing rules, and the treatment of tangential directions.