140
IX – Multivariate Differential and Integral Calculus
I found amusing when eighteen still does not terrify me sixty years later, I do
not see why I should deprive my readers of the pleasure of mastering them
before casting them out into outer darkness instead of comparing the Γ
i
jkh to
an“ unpleasant insect ” like Serge Lang, apparently impressed by Kafka, does
in the preface of his Fundamentals of Differential Geometry. . .
As mentioned earlier about Baire, before the war, the municipal library
of Le Havre used to propose to its readers all the usual French textbooks and
treatises of the time. It also had the complete collection of the M´ emorial des
sciences math´ ematiques, a short monograph series on the most varied topics,
excepting almost all “ modern ” ones that were then being elaborated outside
France. It was in general too difficult or too little interesting for me and
I was anyhow sufficiently busy learning more directly useful mathematics.
Nonetheless, one day I was stunned by the booklet on Absolute Differential
Calculus by Ren´ e Lagrange, professor in Dijon ; the word “ absolute ” that
had intrigued me had been introduced by the Italians who had perhaps read
Balzac. It was a presentation of tensor analysis in Riemann spaces, a vaguely
defined notion: it was more or less possible to understand that it entailed
n-dimensional curved spaces and used curvilinear coordinate systems that
could be changed at will by formulas involving only functions that were as
differentiable as necessary; the square of the distance ds from a point x with
coordinates (x
i ) to an “ infinitely near ” point (x
i + dx
i ) could be calculated
by a formula of type
ds
2 = g ij (x)dx
i dx
j ;
these spaces contained strange objects having an “ absolute ” meaning – what
meaning? a mystery –, tensors represented in each coordinate system by functions of a variable point x in the space assigned inferior and superior indices;
these were supposed to be transformed, under any change of coordinates, by
formulas provided in advance involving the first derivatives of the coordinates
with respect to the old ones; finally, height of virtuosity, the signs
were always omitted. All this was presented without any allusion as to what a curved
space, a vectorial space, a linear functional or multilinear form, etc. were; inventors and users of tensor calculus were still in the position of a physicist
engaged in vector analysis calculations (gradient, rotational, divergence, etc.)
without knowing what a vector is. As will be explained below (n
◦ 12 and 14),
“ tensors ” of the time are just tensor fields, i.e. functions T that, independently from any coordinate system, associate to each point x of the “ curved
space ” X a tensor T (x) of type (p, q) in the vector space X
(x) depending
on x and having the same dimension as X – the “ tangent ” vector space of
X at x, whose somewhat abstract definition will be given in n
◦ 12 –, in the
purely algebraic sense given to this notion above. It is, therefore, a generalization of vector fields of physicists and mathematicians, that are just vector
valued functions, i.e. tensor fields of type (0, 1). But it was marvelous; only
the machinery of traditional differential calculus needed to be known – essentially, the chain rule –, no other idea than that of constructing valid formulas
IX – Multivariate Differential and Integral Calculus
I found amusing when eighteen still does not terrify me sixty years later, I do
not see why I should deprive my readers of the pleasure of mastering them
before casting them out into outer darkness instead of comparing the Γ
i
jkh to
an“ unpleasant insect ” like Serge Lang, apparently impressed by Kafka, does
in the preface of his Fundamentals of Differential Geometry. . .
As mentioned earlier about Baire, before the war, the municipal library
of Le Havre used to propose to its readers all the usual French textbooks and
treatises of the time. It also had the complete collection of the M´ emorial des
sciences math´ ematiques, a short monograph series on the most varied topics,
excepting almost all “ modern ” ones that were then being elaborated outside
France. It was in general too difficult or too little interesting for me and
I was anyhow sufficiently busy learning more directly useful mathematics.
Nonetheless, one day I was stunned by the booklet on Absolute Differential
Calculus by Ren´ e Lagrange, professor in Dijon ; the word “ absolute ” that
had intrigued me had been introduced by the Italians who had perhaps read
Balzac. It was a presentation of tensor analysis in Riemann spaces, a vaguely
defined notion: it was more or less possible to understand that it entailed
n-dimensional curved spaces and used curvilinear coordinate systems that
could be changed at will by formulas involving only functions that were as
differentiable as necessary; the square of the distance ds from a point x with
coordinates (x
i ) to an “ infinitely near ” point (x
i + dx
i ) could be calculated
by a formula of type
ds
2 = g ij (x)dx
i dx
j ;
these spaces contained strange objects having an “ absolute ” meaning – what
meaning? a mystery –, tensors represented in each coordinate system by functions of a variable point x in the space assigned inferior and superior indices;
these were supposed to be transformed, under any change of coordinates, by
formulas provided in advance involving the first derivatives of the coordinates
with respect to the old ones; finally, height of virtuosity, the signs
were always omitted. All this was presented without any allusion as to what a curved
space, a vectorial space, a linear functional or multilinear form, etc. were; inventors and users of tensor calculus were still in the position of a physicist
engaged in vector analysis calculations (gradient, rotational, divergence, etc.)
without knowing what a vector is. As will be explained below (n
◦ 12 and 14),
“ tensors ” of the time are just tensor fields, i.e. functions T that, independently from any coordinate system, associate to each point x of the “ curved
space ” X a tensor T (x) of type (p, q) in the vector space X
(x) depending
on x and having the same dimension as X – the “ tangent ” vector space of
X at x, whose somewhat abstract definition will be given in n
◦ 12 –, in the
purely algebraic sense given to this notion above. It is, therefore, a generalization of vector fields of physicists and mathematicians, that are just vector
valued functions, i.e. tensor fields of type (0, 1). But it was marvelous; only
the machinery of traditional differential calculus needed to be known – essentially, the chain rule –, no other idea than that of constructing valid formulas