Gauss's law: The net electric flux through any closed surface S is Phi_E = integral(E . dA) = Q_enclosed/epsilon_0. The integral is over the entire closed surface. Key points: (1) Only the charge enclosed by the Gaussian surface contributes to the flux — external charges contribute zero net flux; (2) E on the Gaussian surface is due to ALL charges (inside and outside), but the NET flux depends only on enclosed charge; (3) Gauss's law is always true but only useful for calculating E when there is sufficient symmetry; (4) Phi_E is positive if net enclosed charge is positive, negative if negative, zero if no net enclosed charge. Proof follows from the fact that flux through a sphere around a point charge is Q/epsilon_0, and any closed surface can be deformed to a sphere without changing the enclosed charge.