Green's Theorem History question

[jisland85]

I find it a bit interesting that there is a separate theorem for Stokes theorem in a 2D situation. Can someone tell me why this is so? What's the history on these theorems. Did this guy Green come along and generalize Stokes theorem and get credit for it because if this is the case then I will just limit some other theorem to a special case and get my name on it. Silly question I suppose but I was wondering. Thanks . . .

[arildno]

From what I know, Green was active around the 1820's, while Gabriel Stokes was about a generation thereafter.

[HallsofIvy]

There is a difference between "generalize" and "specialize"!!

What happened was Green proved the theorem in the 2-d (flat) case. Later Stokes proved the more general 3-d theorem.

[jisland85]

Ahh, ok, so it was the other way around. Thanks fellows!

[quasar987]

For whoever is interested:

We can read in Stewart's Analysis Vol.2 pp.973 (actually we can't cuz I have it in french so it's a rough translation):

"What we call Stokes' theorem was really discovered by Sir William Thompson (Lord Kelvin). Stokes heard of it through a letter from Thompson in 1850 and asked to his students at Cambridge to demonstrate it during an exam (:surprised). We ignore if one of them succeeded."

[Galileo]

I've got that book too (but in English). There's a footnote about Green too:


Green's Theorem is named after the self-taught English scientist George Green (1793-1841). He worked fulltime in his father's bakery from the age of nine and taught himself mathematics from library books. In 1828 he published privately An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, but only 100 copies were printed and most of those went to his friends.
This pamphlet contained a theorem that is equivalent to what we know as Green's Theorem, but it didn't become widely known at that time. Finally, at age 40, Green entered Cambrigde University as an undergraduate but died four years after graduation. In 1846 William Thompson (Lord Kelvin) located a copy of Green's essay, realized its significance, and had it reprinted.
Green was the first person to try to formulate a mathematical theory of electricity and magnetism. His work was the basis for the subsequent electromagnetic theories of Thomson, Stokes, Rayleigh and Maxwell.


Smart guy, that Green.

[rdt2]

‘The fundamental significance of the vector derivative is revealed by Stokes’ theorem. Incidentally, I think the only virtue of attaching Stokes’ name to the theory is brevity and custom. His only role in originating the theorem was setting it as a problem in a Cambridge exam after learning about it in a letter from Kelvin. He may, however, have been the first person to demonstrate that he did not fully understand the theorem in a published article: where he made the blunder of assuming that the double cross product vanishes for any vector-valued function v = v(x)’.

from Hestenes, ‘Differential Forms in Geometric Calculus’, 1993.

[fourier jr]

isn't the fundamental theorem of calculus a 1-dimensional version of the divergence theorem?

[mathwonk]

yes, all the theorems known as grenns, stokes, gauss, divergence, are just higher diemnsional versions of the FTC. The proof reveals this.

just use repeated integration to express the integrand in these theorems as a repeated integral, and the theorem immediately becomes the FTC.

they also appear in the book by maxwell, elctricity and magnetism. to my knowledge most of these accounts of stokes setting the theorem as a prize problem, come from spivak's introduction to his calculus on manifolds.

[tecumseh]

Thanks for all this history on Green, Stokes', Maxwell, etc. electro and mag. related.

Gives me greater insight.

Lover of the history of math.


Don Wire