The Many Faces of Nicolas Bourbaki
Stokes' theorem is a superb generalization of the
fundamental theorem of calculus.
Some Incarnations of Stokes' Theorem
Gradient Conservativity 
ò 
^{ b} _{a} 
grad f . dM
=
f(b)  f(a)


KelvinStokes' Formula 
òò_{S} rot U . dS
=
ò_{C} U . dM

Ostrogradsky's Theorem 
òòò_{V}
div U dV
=
òò_{S} U . dS

Nicolas Bourbaki and this general result are partly due to each other...
Nicolas Bourbaki was
born
on January 14^{th} 1935,
as the collective identity of a group of several highly talented young French mathematicians,
in part from the urge to elucidate the general validity of the above formula
(as reported by the key founder, André Weil, then 28).
The Bourbaki collaboration has been extremely
influential in France and elsewhere.
Bourbaki brought about new riguor based on the
logical foundations of mathematics
(along the way, it also became instrumental in some controversial reforms of mathematical education,
dubiously known as new math in the US,
which originated in a
meeting
at Melun in 1952 under the leadership of
Jean Dieudonné,
Gustave Choquet and
André Lichnerowicz).
The active founding members of the Nicolas Bourbaki group
were:_{ }
The official list of founders includes four members who were less active,
namely:
Szolem Mandelbrojt was the uncle and early mentor of the mathematician
Benoît Mandelbrot
(19242010;
X1944) of fractal fame.
Two other mathematicians had been present at preliminary meetings,
before the actual foundation of the Bourbaki group:
Other noted bourbakists, who joined the group later, include:
The rule was that all members would have to retire from the group at the age
of 50 (Grothendieck and Lang left early, in anger).
All of the above are thus retired.
The ambition of the founders was to put on a fresh solid foundation the
entire mathematical knowledge of their time.
This has taken the form of a collection of books entitled
Elements de Mathématique (note the militant use of the grammatically
incorrect singular form of Mathématiques).
The Association seems alive and well,
although it's not nearly as active and/or influential as it once was
(the latest volumes in the collection were published in 1983 and 1998).
Henri Cartier said that, during his own tenure from 1955 to 1983,
the Bourbaki group was holding three yearly meetings (for a total of about one month per year).
One of the first items on the original agenda was the
aforementioned general Stokes Theorem, which
unifies great results of vector calculus.
However, it sparked a search for rigorous settings which would delay by many uears the publication of the final
presentation by Bourbaki of that particular topic...
