Fields and vector calculus

Green’s theorem

George Green, a self-taught miller from Nottingham

Can you measure the area of a shape just by walking around its edge?

George Green ran his father’s windmill and taught himself mathematics. In 1828 he paid to publish an essay on electricity and magnetism; it had only a few dozen subscribers and was almost forgotten. In 1845 William Thomson (later Lord Kelvin) found a copy and realised its importance. Its central idea links what happens along the edge of a region to what happens inside it.

First, make a guess

In a whirlpool that spins the same everywhere, which circle has more swirl around its edge: one at the centre, or the same size off to the side?

How they figured it out

George Green had about a year of school, as a child. He worked in his father’s bakery and then ran the family windmill in Sneinton, Nottingham, teaching himself mathematics from books in a local subscription library.

In 1828, aged 34, he paid to publish his essay on electricity and magnetism. It had 51 subscribers, mostly friends and patrons. At 40 he went to Cambridge as an undergraduate. He died in 1841, almost unknown, until William Thomson found the essay in 1845 and had it reprinted. His windmill still stands, restored as a science centre.

The equation

\oint_C (L\,dx + M\,dy) = \iint_D \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dA

The total swirl around the edge equals all the tiny swirls inside, added up.

Everyday echoA planimeter measures the area of a shape on a map just by tracing its outline: Green’s theorem in brass.

See it in real life

The planimeter

A planimeter is a hinged brass arm with a little wheel. Trace the outline of a field on a map and the wheel’s reading gives the field’s area, without ever looking inside. Surveyors and doctors used them for a century. It works because of Green’s theorem: $A = \tfrac12\oint (x\,dy - y\,dx)$ .

Where engineers use it

Planimeters and CAD
Area $= \tfrac12\oint(x\,dy - y\,dx)$ : how CAD programs compute the area of any outline from its edge.
Lift on wings
Circulation around a wing sets its lift (Kutta–Joukowski theorem).
Electromagnetism
Green’s functions, also from his essay, are used to solve circuit and field problems.

Worked example: Area of a circle from its edge

Edge: $x = R\cos t$ , $y = R\sin t$ .
$\tfrac12\oint (x\,dy - y\,dx) = \tfrac12\int_0^{2\pi} R^2\,dt = \pi R^2$ .

Where it came from

The invention of calculus 1665–1684
Built on
Green’s theorem is the fundamental theorem of calculus, stretched from a line to a flat region.

What it led to

Stokes’ theorem 1850–1854
Led to
Stokes’ theorem is Green’s theorem on curved surfaces in three dimensions.
The divergence theorem 1813–1826
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°.