The divergence theorem (Gauss's theorem) is the last and grandest of the
boundary laws. It says the total
flux
of a field out through a closed surface equals the integral of its
divergence
over the solid it encloses.
\oiint_{S} \mathbf{F} \cdot d\mathbf{S} = \iiint_{E} (\nabla \cdot \mathbf{F})\, dV.
Here E is a solid region, S = \partial E
is its closed boundary surface, and \mathbf{n} is the
outward normal. Read it as: the net flow leaving a region equals the sum
of all the little sources inside it. Whatever the
triple integral
of source density adds up to, exactly that much must escape across the skin.
Verifying both sides: the radial field out of a cube
Take \mathbf{F} = (x,\ y,\ z) — the outward explosion field — and
let E be the unit cube
[0,1] \times [0,1] \times [0,1].
Right side — the volume integral (the easy one)
Step 1 — the divergence. As computed on the divergence-and-curl page,
\nabla \cdot \mathbf{F} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 1 + 1 = 3.
Step 2 — integrate the constant over the cube. A constant times the volume:
\iiint_E 3\, dV = 3 \cdot \operatorname{vol}(E) = 3 \cdot 1 = 3.
Left side — the flux through all six faces (the honest one)
Step 3 — handle one pair of faces. On the face
x = 1 the outward normal is \mathbf{n} = (1,0,0),
so \mathbf{F} \cdot \mathbf{n} = x = 1; integrated over that unit
face gives flux 1. On the opposite face
x = 0 the outward normal is (-1,0,0) and
\mathbf{F} \cdot \mathbf{n} = -x = 0; flux
0. The x-pair contributes
1 + 0 = 1.
Step 4 — by symmetry, the other two pairs match. The
y-faces give 1 and the
z-faces give 1, by the identical
argument with y or z in place of
x.
Step 5 — total the flux.
\oiint_S \mathbf{F} \cdot d\mathbf{S} = \underbrace{1}_{x\text{-pair}} + \underbrace{1}_{y\text{-pair}} + \underbrace{1}_{z\text{-pair}} = 3.
Step 6 — compare. Both sides equal 3. Six separate
face integrals collapse into a single line of volume integration — the theorem turns a
boundary tally into an interior one.
The volume side never cares about the shape: for any solid E,
\mathbf{F} = (x,y,z) gives
\iiint_E 3\, dV = 3\,\operatorname{vol}(E). Take the unit ball,
\operatorname{vol} = \tfrac{4}{3}\pi, so the flux must be
3 \cdot \tfrac{4}{3}\pi = 4\pi. Check it directly: on the unit
sphere the outward normal is \mathbf{n} = (x, y, z) itself (the
radial direction), so
\mathbf{F} \cdot \mathbf{n} = x^2 + y^2 + z^2 = 1 on the surface,
and the flux is \iint_S 1\, dS = \operatorname{area} = 4\pi(1)^2 = 4\pi.
The two routes agree — and the volume route was effortless.
The physical reading: sources inside, flow out
Shrink E to a tiny box around a point. The theorem becomes the
definition of divergence: \nabla \cdot \mathbf{F} is the
flux per unit volume escaping an infinitesimal region. Sum those local outflows over a large
region and the interior contributions cancel across every shared internal wall — flow leaving
one cell enters its neighbour — leaving only the flux across the true outer boundary. That
telescoping cancellation is the divergence theorem.
Let E be a solid region whose boundary
S = \partial E is a piecewise-smooth closed surface oriented by
the outward normal, and let \mathbf{F} have
continuous partial derivatives on an open set containing E. Then:
-
Flux out equals divergence within:
\displaystyle \oiint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E (\nabla \cdot \mathbf{F})\, dV.
-
Source-free ⇒ zero net flux: if
\nabla \cdot \mathbf{F} = 0 throughout
E, the total outward flux is 0.
-
It computes volume: with
\mathbf{F} = (x, y, z),
\nabla \cdot \mathbf{F} = 3, so
\operatorname{vol}(E) = \tfrac{1}{3}\oiint_S \mathbf{F} \cdot d\mathbf{S}.
-
Curls leave nothing: since
\nabla \cdot (\nabla \times \mathbf{G}) = 0, the flux of any
curl through a closed surface is 0 — the fact behind
"the surface doesn't matter" in
Stokes' theorem.
Four theorems, one shape. Each equates something on a boundary to something in the
interior:
-
Fundamental theorem of calculus:
\int_a^b f'(x)\, dx = f(b) - f(a) — interior derivative vs the
two boundary endpoints.
-
Green's theorem: a double integral over a plane region vs a line integral
around its boundary curve.
-
Stokes' theorem: curl-flux through a surface vs circulation around its
boundary loop.
-
Divergence theorem: divergence over a solid vs flux through its boundary
surface.
They are one statement in the language of differential forms — the
generalised Stokes' theorem,
\int_{\partial \Omega} \omega = \int_{\Omega} d\omega,
where d is the exterior derivative and
\partial \Omega the boundary of
\Omega. "The integral of a derivative over a region equals the
integral of the original over the boundary" — the FTC made dimension-blind.
Gauss's law of electrostatics. The electric flux out of a closed surface
equals the enclosed charge over \varepsilon_0. Applying the
divergence theorem to \oiint_S \mathbf{E} \cdot d\mathbf{S} = Q_{\text{enc}}/\varepsilon_0
turns the global statement into Maxwell's local one,
\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}.
The continuity equation. If \rho is a density
flowing with current \mathbf{J}, "matter is neither created nor
destroyed" says the rate a region loses mass equals the flux out of its surface. The
divergence theorem converts that integral balance into the pointwise conservation law
\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0.
In both, the theorem is the bridge from an integral law you can measure to a
differential law you can solve.