As you probably know, there is a formal proof of Gauss's Law for electric fields based on Coulomb's Law and the concept of solid angles. How can one prove Gauss's Law for magnetic fields? Is there a similar proof based on solid angles?
OK, sort of a semantics issue.There are no proofs in physics. The "proof" you cite is a proof that Coloumb's law and Gauss's Law are equivalent.
I know that. I would like to know, as you put it, if there is a way of rigorously showing whether Ampere's Law and Gauss's Law are equivalent.Gauss's Law from magnetism is as it is because a magnetic monopol has never been observed. Is one is observed, the equations will have to change.
I see, but that's not a formal proof. The proof for Gauss's Law for electric fields is quite rigorous and general.You can take a look at common symmetric charge configurations (point charge, line charge, plane change) and calculate the field of each using Gauss's Law and then integrating using ampere's law. You'll get the same result in every case.