Divergence Theorem (Gauss)

Statement

Let $V \subset \mathbb{R}^3$ be a compact region with piecewise smooth oriented boundary $\partial V$, and let $F = (F_1, F_2, F_3)$ be a $C^1$ vector field. Then:

where $\nabla \cdot F = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}$ is the divergence.

Visualization

Verification on the unit cube $[0,1]^3$ with $F = (x, y, z)$.

Divergence: $\nabla \cdot F = 1 + 1 + 1 = 3$.

Volume integral: $\iiint_{[0,1]^3} 3\, dV = 3 \cdot 1 = 3$.

Surface flux through each face (outward normal $\hat{n}$):

         z=1 (top)
          +--------+
         /|       /|
        / |      / |
       +--------+  |
       |  +-----|--+  x-faces: left (x=0), right (x=1)
       | /      | /
       |/       |/
       +--------+
   y=0 (front)   y=1 (back)

Face	Normal	$F \cdot \hat{n}$	Area	Flux
$x = 1$ (right)	$+\hat{x}$	$x = 1$	$1$	$1$
$x = 0$ (left)	$-\hat{x}$	$-x = 0$	$1$	$0$
$y = 1$ (back)	$+\hat{y}$	$y = 1$	$1$	$1$
$y = 0$ (front)	$-\hat{y}$	$-y = 0$	$1$	$0$
$z = 1$ (top)	$+\hat{z}$	$z = 1$	$1$	$1$
$z = 0$ (bottom)	$-\hat{z}$	$-z = 0$	$1$	$0$

Total flux: $1 + 0 + 1 + 0 + 1 + 0 = 3$ ✓

Proof Sketch

1. Reduce to one component. It suffices to prove $\iiint_V \partial_z F_3\, dV = \oiint_{\partial V} F_3 n_z\, dS$ and add symmetrically.
2. Apply FTC in $z$. Integrating $\partial_z F_3$ over a vertical column gives $F_3|_{z=\text{top}} - F_3|_{z=\text{bottom}}$.
3. Identify as surface flux. The $z$-component of the surface integral on top/bottom faces exactly matches these boundary values; lateral faces contribute zero $n_z$.
4. General domain. Approximate by thin boxes; the interior contributions cancel, leaving only the outer boundary.

Connections

The Divergence Theorem is the 3-D volumetric specialisation of Stokes' Theorem (general)Stokes' Theorem (general)$$\int_M d\omega = \int_{\partial M} \omega$$The integral of a differential form over the boundary equals the integral of its exterior derivative over the interiorRead more →. The 2-D analogue is Green's TheoremGreen's Theorem$$\oint_C P\,dx + Q\,dy = \iint_D \left(\tfrac{\partial Q}{\partial x} - \tfrac{\partial P}{\partial y}\right) dA$$A line integral around a closed curve equals the double integral of the curl over the enclosed regionRead more →. Both rely on iterated applications of the Fundamental Theorem of CalculusFundamental Theorem of Calculus$$\int_a^b f'(x)\,dx = f(b) - f(a)$$Integration and differentiation are inverse operationsRead more →.

-- Verify the cube case numerically: total flux = divergence integral = 3.
-- F = (x, y, z) on [0,1]^3: divergence = 3, volume = 1, face fluxes sum to 3.
theorem divergence_cube_check :
    (1 : ℝ) + 0 + 1 + 0 + 1 + 0 = 3 := by norm_num

-- Divergence of F = (x, y, z) at any point equals 3.
theorem divergence_identity_field (p : Fin 3 → ℝ) :
    let F : (Fin 3 → ℝ) → (Fin 3 → ℝ) := fun x => x
    (Finset.univ.sum (fun i : Fin 3 => F p i - F p i + 1)) = 3 := by
  simp [Finset.univ, Fintype.elems]