146
IX – Multivariate Differential and Integral Calculus
existence of Bourbaki but not of “ abstract ” and “ modern ” mathematics
which it is far from having invented, returned or converted to this type of
mathematics where the prospects were enormous; for example in algebra,
van der Waerden’s books and later those of Birkhoff-MacLane have probably
exerted a greater worldwide influence than those of Bourbaki; in fact, it is in
van der Waerden that Koszul and I learnt algebra at the ´
Ecole Normale.
As I have explained in the postscript to vol. II, it is also thanks to the support of the military and of industrialists after 1945 that a “ militant gang ”,
far more influential than us and much more keen to propagate applied mathematics was formed; the latter is now on the point of dominating our science.
They are also starting to dominated the minds of some eminent pure
mathematicians. When in the year 2000, “ year of mathematics ”, during
a religious ceremony, a journalist questioned Alain Connes (Fields medallist,Coll` ege de France) he replied as follows according to Le Monde of
25 May 2000 :
Euclid’s questioning was answered by research on non-Euclidean geometry,
which stimulated Riemannian Geometry, which in turn inspired Albert Einstein
for his work on space-time and general relativity used to refine the global
satellite positioning system (GPS).
It would not have been in vain to add that the GPS, like inertial guidance
earlier, was developed by the American military for their own planes, ships
and ground vehicles and for locating with extreme precision enemy objects;
their intention was not to help walkers in the Fontainebleau forest or in the
Great Erg nor to help taxi drivers navigate in Chicago suburbs. Without
American military funding, no civilian firm would have began such an extravagant project needing hundreds of billions in investments and decades of
technical progress regarding missiles, satellites and telecommunications. The
fact that the GPS is now available in the civilian sector does not change anything to the fact that we ought to be able to find better illustrations of the
usefulness of mathematics from Euclid to Einstein than the guidance system
of the most terrifying armaments ever invented by humanity. Because this is
what the GPS also continues to be 2001 even if, for the sake of the cause, we
prefer, as always, to cover up that bosom, which we can’t endure to look on.
2 – Differential Calculus of n Variables
(i) Differential functions. Let f be a function defined on an open subset U
of a n-dimensional real Cartesian space
10 E with values in a p-dimensional
Cartesian space F , for example C if p = 2. Like in the case n = 2 recalled in
chapter VIII, f is said to be differentiable at x ∈ U if there is a linear map
u : E −→ F such that, for sufficiently small h ∈ E,
f (x + h) = f (x) + u(h) + o(h) ,
(2.1)
10 In fact, almost everything applies to functions defined on and with values in
Banach spaces.
IX – Multivariate Differential and Integral Calculus
existence of Bourbaki but not of “ abstract ” and “ modern ” mathematics
which it is far from having invented, returned or converted to this type of
mathematics where the prospects were enormous; for example in algebra,
van der Waerden’s books and later those of Birkhoff-MacLane have probably
exerted a greater worldwide influence than those of Bourbaki; in fact, it is in
van der Waerden that Koszul and I learnt algebra at the ´
Ecole Normale.
As I have explained in the postscript to vol. II, it is also thanks to the support of the military and of industrialists after 1945 that a “ militant gang ”,
far more influential than us and much more keen to propagate applied mathematics was formed; the latter is now on the point of dominating our science.
They are also starting to dominated the minds of some eminent pure
mathematicians. When in the year 2000, “ year of mathematics ”, during
a religious ceremony, a journalist questioned Alain Connes (Fields medallist,Coll` ege de France) he replied as follows according to Le Monde of
25 May 2000 :
Euclid’s questioning was answered by research on non-Euclidean geometry,
which stimulated Riemannian Geometry, which in turn inspired Albert Einstein
for his work on space-time and general relativity used to refine the global
satellite positioning system (GPS).
It would not have been in vain to add that the GPS, like inertial guidance
earlier, was developed by the American military for their own planes, ships
and ground vehicles and for locating with extreme precision enemy objects;
their intention was not to help walkers in the Fontainebleau forest or in the
Great Erg nor to help taxi drivers navigate in Chicago suburbs. Without
American military funding, no civilian firm would have began such an extravagant project needing hundreds of billions in investments and decades of
technical progress regarding missiles, satellites and telecommunications. The
fact that the GPS is now available in the civilian sector does not change anything to the fact that we ought to be able to find better illustrations of the
usefulness of mathematics from Euclid to Einstein than the guidance system
of the most terrifying armaments ever invented by humanity. Because this is
what the GPS also continues to be 2001 even if, for the sake of the cause, we
prefer, as always, to cover up that bosom, which we can’t endure to look on.
2 – Differential Calculus of n Variables
(i) Differential functions. Let f be a function defined on an open subset U
of a n-dimensional real Cartesian space
10 E with values in a p-dimensional
Cartesian space F , for example C if p = 2. Like in the case n = 2 recalled in
chapter VIII, f is said to be differentiable at x ∈ U if there is a linear map
u : E −→ F such that, for sufficiently small h ∈ E,
f (x + h) = f (x) + u(h) + o(h) ,
(2.1)
10 In fact, almost everything applies to functions defined on and with values in
Banach spaces.