The Divergence theorem (Gauss's theorem)

First prove it for a rectangular box $B = [a_1,b_1]\times[a_2,b_2]\times[a_3,b_3]$ with $V = B$, where $\mathbf{F} = (F_1, F_2, F_3)$. By Fubini's theorem, $\iiint_B \frac{\partial F_1}{\partial x}\,dV = \iint \left(F_1(b_1,y,z) - F_1(a_1,y,z)\right) dy\,dz$, and analogous identities hold for the $y$- and $z$-derivative terms.

Each right-hand side is exactly the outward flux of the corresponding component of $\mathbf{F}$ through a pair of opposite faces of $B$, with the outward normal pointing along the positive or negative coordinate direction on each face. Summing the three coordinate contributions gives $\iiint_B (\nabla \cdot \mathbf{F})\,dV = \iint_{\partial B} \mathbf{F} \cdot d\mathbf{S}$.

For a general solid region $V$, approximate it by a fine grid of small rectangular boxes filling $V$. Apply the box case to each box and sum over all boxes. Every interior face shared by two adjacent boxes has outward normals pointing in opposite directions on the two sides, so the corresponding flux contributions across that shared face cancel exactly.

The remaining, uncancelled flux is precisely through the outer boundary surface $\partial V$, while the sum of volume integrals over the small boxes converges to $\iiint_V (\nabla \cdot \mathbf{F})\,dV$ as the grid is refined, establishing the identity for $V$.