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 Sci Advisor P: 1,548 Right. So, with sources, we have $$d * F = * J$$ So now, merely integrate this over some 3-volume V: \begin{align*} \int_V d * F &= \int_V * J \\ \int_{\partial V} * F &= \int_V * J \end{align*} which is Gauss' Law. The reason I focused on the source-free equations, is because it is only when $J = 0$ that the result of integration doesn't care about the choice of surface. Obviously, if $J \neq 0$, then different surfaces might contain different amounts of charge, hence giving different results. Gauss' Law still holds, though.