Fields and vector calculus
1850–1854
Stokes’ theorem
William Thomson (Lord Kelvin) and George Gabriel Stokes
Why does spinning a magnet near a coil of wire make electricity?
Thomson described the result in a letter to Stokes in 1850. Stokes then set it as a question in Cambridge’s Smith’s Prize exam of 1854, and among the students sitting it was James Clerk Maxwell. It is Green’s theorem lifted from flat paper onto any curved surface.
First, make a guess
Around a plughole vortex, the swirl is zero everywhere except at the centre. A loop that encloses the plughole has circulation…
- Swirl around the edge ∮F·dr
- -1.06e-5
- Tiny swirls added up inside ∬curl F dA
- 0
- Agree?
- ✓ Yes, as the theorem promises
- Area from the edge alone ½∮(x dy − y dx)
- 2.01
Draw any closed loop with your finger or mouse (or move it with the arrow keys). Green arrows on the edge go with the loop, pink ones against. Inside, each paddle wheel spins at the local curl. Both totals always agree: that is Green’s theorem.
How they figured it out
In July 1850 William Thomson (later Lord Kelvin) mentioned the theorem in a postscript to a letter to George Gabriel Stokes, who held Newton’s old professorship at Cambridge.
In February 1854 Stokes set it as a question in the Smith’s Prize exam, Cambridge’s hardest mathematics competition. One of the students was James Clerk Maxwell, who shared the prize that year. The theorem became central to his equations of electricity and magnetism.
The equation
Swirl around the rim of any surface equals the curl poking through the surface.
See it in real life
An induction cooktop
An induction cooktop never gets hot itself. A rapidly changing magnetic field passes through the base of your pan, and by Faraday’s law (written with Stokes’ theorem) that drives swirling currents around the metal, which heat it.
Where engineers use it
Generators
Faraday’s law in integral form, via Stokes, gives the voltage of every generator and transformer.
Ampère’s law
Magnetic field around a current-carrying wire follows in one line.
Where it came from
- Green’s theorem 1828
Led to
Stokes’ theorem is Green’s theorem on curved surfaces in three dimensions.
What it led to
- Light is electromagnetism 1865
Built on
Maxwell sat the exam with Stokes’ question; his equations switch between edge and inside forms using Stokes’ theorem.