Revisiting the integral form of Gauss' law for a generic case of electrodynamics with arbitrarily moving Gaussian surface

We have re-examined the integral form of Gauss’ law for arbitrarily moving charges inside and outside an arbitrarily expanding (or contracting) and deforming Gaussian surface. We have explicitly calculated the time-dependent Gauss’ flux integral for such a generic non-static case with the Maxwell equations under consideration. We have obtained an evolution equation $\frac{\text{d}}{\text{dt}}{\oint }_{s(t)}\vec{E}\cdot \text{d}\vec{s}(t)=\frac{I_{\text{in}}^{(s)}(t)}{{\epsilon }_0}$ for the time-dependence of the flux-integral. We have pedagogically demonstrated that while the flux integral is dependent on the expansion/contraction of the surface, it is independent of its deformation.