Fields and vector calculus
1813–1826
The divergence theorem
Carl Friedrich Gauss; Mikhail Ostrogradsky
Can you count what is inside a box without opening it?
Gauss proved a special case in 1813 while studying gravity; Ostrogradsky stated it in general in 1826. It says you can count what is created inside a region just by watching what crosses its boundary. It is why the gravity outside a round planet is the same as if all its mass sat at the centre.
First, make a guess
In a flow where nothing is created or destroyed inside a loop, the net flow out of the loop is…
- Flow out through the edge ∮F·n ds
- 2.01
- Sources added up inside ∬div F dA
- 2.01
- Agree?
- ✓ Yes, as the theorem promises
Draw any closed loop. Arrows on the edge show flow leaving (green) or entering (pink). Inside, green dots are sources and pink dots are sinks. What flows out through the edge always equals what is created inside: the divergence theorem in two dimensions.
How they figured it out
Gauss proved a version in 1813 while working out the gravitational pull of slightly squashed spheres, the shape of the Earth. The Ukrainian-born mathematician Mikhail Ostrogradsky gave a general proof in 1826; Green had a version too.
Physicists often call it Gauss’s theorem; in Russia it is Ostrogradsky’s. The idea is simple: whatever is created inside a region has to leave through its boundary.
The equation
What flows out through the skin of a region equals what is produced inside it.
See it in real life
Counting a stadium from the gates
To know how many people are inside a stadium, you don’t need to count the seats: count everyone through the gates. The rain falling on a city equals the water leaving through its drains and rivers, plus what soaks in.
Where engineers use it
Gauss’s law
Electric fields of charged spheres, cables and capacitors are found in one line with it.
Fluid flow and CFD
Conservation of mass in pipes and around wings is the divergence theorem.
Worked example: Gravity outside a planet
- Flux through a sphere of radius : .
- Equals inside.
- So , as if all mass sat at the centre.
Where it came from
- Green’s theorem 1828
Same technique
Both trade a total over a boundary for a total over the inside.
Where the comparison stops: Green counts swirl along the edge; the divergence theorem counts flow straight through it. In flat 2D they are the same theorem turned 90°.
- Universal gravitation 1687
Led to
Gauss proved his case while studying the gravity of solid bodies, Newton’s problem.
What it led to
- Light is electromagnetism 1865
Built on
Gauss’s law for electric charge, one of Maxwell’s four equations, is the divergence theorem.
- Galaxies spin too fast 1970
Built on
Only the mass inside an orbit sets its speed: Gauss’s theorem, treating the galaxy as roughly round.