most(or all) proofs i have seen of gauss's law is based on coulumb's law. however coulumb's law is based off of electrostatics which certainly does not hold in electrodynamics. however gauss's law is used extensively in electrodynamics. is gauss's law derived otherwise or is it just a law like an axiom? thanks bigerst
Gauss law is, ∇.E=ρ/ε_{0} .now in most general form E=-∇ψ-∂A/∂t, if you put it into above you get, -∇^{2}ψ-∂/∂t(∇.A)=ρ/ε_{0}............(1) now using lorentz gauge(more preferable) we have ∇.A=-1/c^{2} ∂ψ/∂t if you will put it into (1) you will get the the general form of potential equations.this proof starts with gauss law but the converse can also be done in order to arrive at gauss law.here it does not refer to any coulomb law,and so it is more general.
Although EM started with Coulomb historically, the complete classical theory is based on the four Maxwell equations. Applying the divergence theorem to Maxwell's div D equation gives Gauss's law. Coulomb is the electrostatic case.