Divergence theorem

The divergence theorem (or Gauss’s theorem) is the 3D analog of Green’s flux form. It converts a closed-surface flux integral into a triple integral of divergence over the enclosed volume.

Statement

Let $V$ be a solid region in ${\mathbb{R}}^3$ with piecewise-smooth boundary surface $S=\partial V$, oriented outward. Let $\mathbf{F}$ be a $C^1$ vector field on a region containing $V$. Then

$\iint \mathbf{F}\cdot d\mathbf{S}={\iiint }_V\nabla \cdot \mathbf{F}dV.$

Total outward flux across the boundary surface = total divergence over the enclosed volume. The 2D boundary controls a 3D interior; one dimension is consumed by the divergence on the right.

Physical content

If $\mathbf{F}$ is a fluid velocity field, the left side is the net volume of fluid leaving $V$ per unit time, and the right side is the total source strength inside. The theorem is conservation: what flows out equals what is being produced inside.

Why divergence theorem matters

Source-free closed-surface flux vanishes. If $\nabla \cdot \mathbf{F}=0$ throughout $V$, then $\iint \mathbf{F}\cdot d\mathbf{S}=0$. The 3D analog of “conservative ⇒ zero closed-loop circulation.”

Divergence as flux density. Applied to a tiny ball $B$ of volume $\Delta V$ around a point:

$\iint \mathbf{F}\cdot d\mathbf{S}\approx (\nabla \cdot \mathbf{F})\Delta V.$

So $\nabla \cdot \mathbf{F}$ is the flux per unit volume out of an infinitesimal ball — the precise meaning of “source density” claimed when introducing divergence.

Surface deformation principle. If $\nabla \cdot \mathbf{F}=0$ in a region between two closed surfaces $S_1$ and $S_2$ (with $S_2$ inside $S_1$, both oriented outward), then applying the divergence theorem to the region between gives

$\iint \mathbf{F}\cdot d\mathbf{S}=\iint \mathbf{F}\cdot d\mathbf{S}.$

The flux through a closed surface is invariant under deforming the surface through source-free regions. This is the 3D version of the deformation principle for contour integrals — and it sets up the inverse-square result.

Worked examples

Easy. $\mathbf{F}=⟨x,y,z⟩$ outward across the unit sphere.

$\nabla \cdot \mathbf{F}=3$. Volume of unit ball $=4\pi /3$. Flux $=3\cdot 4\pi /3=4\pi$.

Direct verification (hard) vs. div theorem (easy). $\mathbf{F}=⟨x,y,z⟩$ across the closed cylinder $x^2+y^2\le 1$, $0\le z\le 2$.

Direct: three integrals (top disk, bottom disk, lateral cylinder).

By div theorem: $\nabla \cdot \mathbf{F}=3$. Volume of cylinder: $\pi \cdot (1{)}^2\cdot 2=2\pi$. Flux: $3\cdot 2\pi =6\pi$.

In context

The divergence theorem is one of the two grand 3D theorems of vector calculus (the other being Stokes’ theorem), and the foundation of Gauss’s law in electrostatics:

$\iint \mathbf{E}\cdot d\mathbf{S}=\frac{Q_{\mathrm{enc}}}{{ϵ}_0}.$

The left side is the flux of the electric field through any closed surface; the right side is the total enclosed charge divided by ${ϵ}_0$. Derivation: from Maxwell’s equation $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$ (a local statement) and the divergence theorem, $\iint \mathbf{E}\cdot d\mathbf{S}=\iiint (\rho /{ϵ}_0)dV=Q_{\mathrm{enc}}/{ϵ}_0$.

Gauss’s law is the divergence theorem applied to electrostatics — the local-to-global content is exactly the theorem.

Connection to FTC family

Theorem	Boundary	Interior
FTC	$F(b)-F(a)$	$\int F^{\prime }dx$
FTLI	$\varphi (B)-\varphi (A)$	${\int }_C\nabla \varphi \cdot d\mathbf{r}$
Green’s (both)	$\oint$	$\iint$ over plane
Stokes’	$\oint \mathbf{F}\cdot d\mathbf{r}$	$\iint (\nabla \times \mathbf{F})\cdot d\mathbf{S}$
Divergence	$\iint \mathbf{F}\cdot d\mathbf{S}$	$\iiint \nabla \cdot \mathbf{F}dV$

All “integral of derivative on the inside = boundary integral on the outside.” All unified by the generalized Stokes’ theorem ${\int }_{M}d\omega ={\int }_{\partial M}\omega$ on differential forms.

The two parallel stories

The divergence theorem closes out the flux story (zero divergence ⇔ zero closed-surface flux ⇔ has a vector potential), the dual of the circulation story that runs through conservativeness, curl, FTLI, Green’s circulation, and Stokes’. Both stories are completed in vector calculus before complex analysis picks them up in Chapter 10 of Vector Calculus and Complex Analysis via the Cauchy-Riemann equations.