§ 1. Classical Differential Calculus
165
immediately follows. This is a well-known formula allowing the Γ ijk to be
calculated in terms of the derivatives of the g ij ; by (21), we go from this to
the Γ
k
ij by inverting the matrix with entries g ij .
Ricci and his assistant Levi-Civita (who was later to be one of the big
specialists of fluid mechanics) had explained this and many other things in
the more subtle case of arbitrary ds
2 , in particular in a long memoir in French
published in the famous German journal (M´ ethodes du calcul diff´ erentiel absolu, Math. Annalen, 1901) – an excellent example of scientific internationalism at a time when saber rattlers, as they were called, held sway everywhere.
Albert Einstein had to assimilate everything to build his general relativity
about a decade later; not being as great a virtuoso of tensor calculus as the
Italians, he encountered some problems with them.
22 The subsequent development of Riemannian geometry has made it possible to get rid of a large
part of these coordinate calculations from the theory – though this has not
made it easier to understand. . . –, but when he wanted to deduce that light
rays were deviated in the neighbourhood of an intense gravitational field,
Einstein was obliged to make everything explicit to obtain numerical results,
verifiable by experiments, a task that mathematicians are generally exempted
from. Finally, as in many other fields, the modern theory arrived after the
“ profusion of indices ” which, for want of better, can still serve as exercises
for the reader and from which Dieudonn´ e himself did not fully escape at the
end of vol. 3 of his El´ ements d’analyse.
22 For a history of the subject, see Karin Reich, Die Entwicklung des Tensorkalk¨ uls.
Vom absoluten Differentialkalk¨ ul zur Relativit¨ atstheorie (Birkhauser, 1989) ; a
chapter on the subject can also be found in T. Hawkins, Emergence of the Theory
of Lie Groups. An Essay in the History of Mathematics 1869–1926 (SpringerVerlag, 2000), which covers a far wider field. I forego citing modern presentations
of Riemmanian geometry; there are too many and they do not throw any light
on the history of the subject.
165
immediately follows. This is a well-known formula allowing the Γ ijk to be
calculated in terms of the derivatives of the g ij ; by (21), we go from this to
the Γ
k
ij by inverting the matrix with entries g ij .
Ricci and his assistant Levi-Civita (who was later to be one of the big
specialists of fluid mechanics) had explained this and many other things in
the more subtle case of arbitrary ds
2 , in particular in a long memoir in French
published in the famous German journal (M´ ethodes du calcul diff´ erentiel absolu, Math. Annalen, 1901) – an excellent example of scientific internationalism at a time when saber rattlers, as they were called, held sway everywhere.
Albert Einstein had to assimilate everything to build his general relativity
about a decade later; not being as great a virtuoso of tensor calculus as the
Italians, he encountered some problems with them.
22 The subsequent development of Riemannian geometry has made it possible to get rid of a large
part of these coordinate calculations from the theory – though this has not
made it easier to understand. . . –, but when he wanted to deduce that light
rays were deviated in the neighbourhood of an intense gravitational field,
Einstein was obliged to make everything explicit to obtain numerical results,
verifiable by experiments, a task that mathematicians are generally exempted
from. Finally, as in many other fields, the modern theory arrived after the
“ profusion of indices ” which, for want of better, can still serve as exercises
for the reader and from which Dieudonn´ e himself did not fully escape at the
end of vol. 3 of his El´ ements d’analyse.
22 For a history of the subject, see Karin Reich, Die Entwicklung des Tensorkalk¨ uls.
Vom absoluten Differentialkalk¨ ul zur Relativit¨ atstheorie (Birkhauser, 1989) ; a
chapter on the subject can also be found in T. Hawkins, Emergence of the Theory
of Lie Groups. An Essay in the History of Mathematics 1869–1926 (SpringerVerlag, 2000), which covers a far wider field. I forego citing modern presentations
of Riemmanian geometry; there are too many and they do not throw any light
on the history of the subject.