How Much Smoke Leaves a Region

A chimney pours out smoke while the wind blows across the town. Draw any loop around the chimney: the net smoke crossing it per second equals the chimney's output, however the wind bends the plume. A loop that misses the chimney has zero net flux.

Click or drag anywhere to start a new trajectory there.

How it works

Flux and the divergence theorem

In one sentence

For a vector field with continuous partials on a region $D$ bounded by a closed curve $C$ with outward normal $\mathbf n$ , $\oint_C \mathbf F\cdot\mathbf n\,ds = \iint_D \nabla\cdot\mathbf F\,dA$ ; in 3D, $\oiint_S \mathbf F\cdot d\mathbf S = \iiint_E \nabla\cdot\mathbf F\,dV$ .

General formula

\oint_C {\htmlData{sym=0}{\mathbf F}}\cdot{\htmlData{sym=1}{\mathbf n}}\,{\htmlData{sym=2}{ds}} = \iint_D {\htmlData{sym=3}{\nabla\cdot\mathbf F}}\,dA, \qquad {\htmlData{sym=0}{\mathbf F}} = ({\htmlData{sym=4}{U}}, 0) + \frac{\htmlData{sym=5}{{Q}}}{r^2 + \varepsilon}\big(x - 1,\; y\big)

Whatever is produced inside must leave through the boundary. Divergence counts production point by point; flux counts what crosses the edge.

What the symbols mean

$\mathbf F$
Smoke flow
How much smoke moves past each point per second, and which way.
$\mathbf n$
Outward normal
Points straight out of the loop at each point.
$ds$
Arc length
A tiny piece of the loop.
$\nabla\cdot\mathbf F$
Divergence
How much smoke is produced at each point: only at the chimney.
$U$
Wind
A steady breeze carrying the smoke east.
$Q$
Output
How strongly the chimney emits.

Flux $\oint_C \mathbf F\cdot\mathbf n\,ds$ adds up how much flow crosses the loop outward per unit length, all the way around.
The divergence $\nabla\cdot\mathbf F$ measures how much flow is created at each point: here only at the chimney, near $(1, 0)$ .
The divergence theorem (in the plane): total outward flux through the loop = total divergence inside it.
The wind $(U, 0)$ has zero divergence: whatever it carries in on one side it carries out on the other, so it doesn't change the net flux.

Learn the math

Curl and DivergenceCalc III · Lesson 45, Surface Integrals Flux IntegralsCalc III · Lesson 48, Surface Integrals The Divergence TheoremCalc III · Lesson 50, Surface Integrals