Q.State Gauss's theorem mathematically, explaining the meaning of every symbol in it, and explain why the theorem is true for ANY closed surface even though, in practice, it is applied only to surfaces with a high degree of symmetry.

Gauss's theorem: $\oint\vec{E}\cdot d\vec{A}=q_{\text{enc}}/\epsilon_0$, where $\oint\vec{E}\cdot d\vec{A}$ is the total electric flux through a closed surface (found by summing $\vec{E}\cdot d\vec{A}$ over every infinitesimal area element $d\vec{A}$, taken with outward normal), $q_{\text{enc}}$ is the algebraic sum of all charge strictly inside that closed surface, and $\epsilon_0$ is the permittivity of free space.

The theorem holds for ANY closed surface, of any shape whatsoever, because it is a direct mathematical consequence of Coulomb's law being an exact inverse-square law: every field line leaving an enclosed point charge crosses any surrounding closed surface exactly once, regardless of that surface's shape, so the total flux depends only on the enclosed charge, never on the surface's geometry (a solid-angle argument shows the $1/r^2$ fall-off in $E$ is exactly compensated by the $r^2$ growth of any patch of surface subtending the same solid angle). …

State the theorem and define every symbol, then separate the two distinct ideas: universal TRUTH (any surface) vers …

• Implying the theorem only HOLDS for symmetric surfaces (it holds for all surfaces; symmetry is only needed to actually solve for $E$). …