# Who discovered the general case of Stokes' Theorem?

1. Mar 30, 2006

I know that th three dimensional case was discovered by William Thompson, but who discovered the general case?

2. Mar 30, 2006

### robphy

According to
http://maxwell.byu.edu/~spencerr/phys442/node4.html
which claim to refer to
E. T. Whittaker's A History of the Theories of Aether and Electricity...
says
"1850 - Stokes law is stated without proof by Lord Kelvin (William Thomson). Later Stokes assigns the proof of this theorem as part of the examination for the Smith's Prize. Presumably, he knows how to do the problem. Maxwell, who was a candidate for this prize, later remembers this problem, traces it back to Stokes and calls it Stokes theorem"

http://www.siam.org/siamnews/09-00/green.htm

How "general" do you want?
According to
"The History of Stokes' Theorem", Victor J. Katz
Mathematics Magazine, Vol. 52, No. 3. (May, 1979), pp. 146-156.
(available on JSTOR),
among those that worked on what would become the higher-dimensional "generalized Stokes theorem" is Ostrogradsky (1820s, 1836 for the generalized divergence theorem) and Volterra (1889 for including Green, Div, Stokes as special cases).

3. Mar 30, 2006

Thank you for the article. Interesting stuff.

Aparently Ostrogadkssy discovered Gauss' theorem, Cauchy discovred Green's theorem, Kelvin discovered Stokes' theorem, and some guy named Cartan discovered the general case (which is also called stokes theorem). So none are named correctly. Hmm.

4. Apr 3, 2006

### dextercioby

If you read page #192 of Arnold's book on "Mathematical Methods of Classical Mechanics" you'll see just how big the confusion might be...:tongue2:

Daniel.

5. Apr 7, 2006

### leach

The general case of Stokes Theorem was the first great publication by Nicolas Bourbaki.

Trying google with bourbaki and "stokes theorem" gives some good references.

6. Apr 7, 2006

Wow. Things just keep getting curiouser and curiouser. Thanks all.

7. Apr 7, 2006

### Perturbation

Some guy called Cartan! Some guy! Blasphemy.

Stoke's theorem and Noether's are probably two of my favourite theorems.

$$\int_{U}d\tilde{\omega}=\int_{\partial U}\tilde{\omega}$$

Last edited: Apr 8, 2006
8. Apr 7, 2006

### masudr

Hehe, that's what I was thinking!

9. Apr 7, 2006

My deepest apologies for my ignorance.

Who is Cartan?

10. Apr 8, 2006

### Cexy

Elie Cartan lived in the first half of the 1900s, and made huge discoveries in the study of Lie groups, representation theory, differential geometry, topology and a whole host of others. He gave a complete classification of the simple Lie algebras, and he worked with Einstein on some problems in General Relativity with a non-zero Torsion field. He was also the first person to describe Newton's theory of gravity using the language of differential geometry, and you'll often see it called 'Newton-Cartan geometry' nowadays.

11. Apr 8, 2006

### leach

I read elsewhere that the discoverer of Stokes theorem was Henri Cartan, Elie's Cartan son. Henri Cartan was member of the Bourbaki group.

I agree. These two theorems enclose the true beauty of calculus. I think that the derivation of Euler's variational equation is other good example of that beauty: to obtain such a profound equation using high school math, and of course Euler's geniality.