248
IX – Multivariate Differential and Integral Calculus
M. Mandelbrot seems to think, we do not eliminate “ geometry ” from our preoccupations: our definitions and even our notation directly generalize those
of the 17th century.
But as seen in (ii), the curve μ drawn in X makes it also possible to define
some h ∈ X
(x) independently from the embedding of X into a Cartesian
space: under every local charts (U, ϕ) of X at x, μ becomes a curve t → ϕ[μ(t)]
in R
p and, cf. (12.8),
h(ϕ) =
d
dt
ϕ [μ(t)] for t = 0 .
If, as above, the submanifold X is defined in the neighbourhood of x = (a, b)
by an equation y = f (x) and if μ(t) = (μ 1 (t), μ 2 (t)), the map (x, y) → x from
X to R
p can be chosen as (U, ϕ) since it defines the manifold structure on X.
Then ϕ[μ(t)] = μ 1 (t) and so h(ϕ) = μ
1 (0), where this is the usual derivative.
Hence, by (12),
μ
(0) = (h(ϕ), f
(a)h(ϕ)) ,
(13.13)
where f
(a) is the tangent linear map to f at a in the usual sense of n
◦ 2,
(i). As the map h → h(ϕ) from X
(x) to R
p is linear and bijective, and as
formula (12) likewise defines a linear bijection k → h from R
p onto T x (X), we
thereby obtain by composition an isomorphism h → u from the “ abstract ”
vector space X
(x) onto the “ concrete ” vector subspace T x (X) of R
q . For
any curve μ drawn in X and such that μ(0) = x, this isomorphism maps the
“ abstract ” tangent vector μ
(0) ∈ X
(x) onto the vector
lim [μ(t) − μ(0)] /t
also written μ
(0) by everyone. This also proves that the isomorphism of
vector spaces X
(x) −→ T x (X) defined thereby is absolute, i.e. depends only
on the embedding of X into R
q and not on the choice of the chart.
And it is not understandable why this assimilation is a false trail, in the
words of Dieudonn´ e : it in particular suggests that the vector spaces X
(x) and
X
(y) tangent to X at two different points x and y could, like the subspaces
T x (X) and T y (X) of R
q , have common elements; this is not at all the case:
an element h of X
(x) is a couple consisting of a point x ∈ X and of a family
of vectors h(ϕ) of R
p depending on a local chart at x ; two tangent vectors
at x and y cannot be equal if x = y. In fact, the set of tangent vectors to an
n-dimensional manifold is a 2n-dimensional manifold [n
◦ 12(v)].
(v) Riemann spaces. The definition of tangent spaces makes it possible
to define Riemann spaces. For this, in each X
(x) take an Euclidean scalar
product (h|k) compelled to depend on x in a reasonable manner: suppose
that the functions
g ij (ξ) = (a i (ξ)|a j (ξ))
IX – Multivariate Differential and Integral Calculus
M. Mandelbrot seems to think, we do not eliminate “ geometry ” from our preoccupations: our definitions and even our notation directly generalize those
of the 17th century.
But as seen in (ii), the curve μ drawn in X makes it also possible to define
some h ∈ X
(x) independently from the embedding of X into a Cartesian
space: under every local charts (U, ϕ) of X at x, μ becomes a curve t → ϕ[μ(t)]
in R
p and, cf. (12.8),
h(ϕ) =
d
dt
ϕ [μ(t)] for t = 0 .
If, as above, the submanifold X is defined in the neighbourhood of x = (a, b)
by an equation y = f (x) and if μ(t) = (μ 1 (t), μ 2 (t)), the map (x, y) → x from
X to R
p can be chosen as (U, ϕ) since it defines the manifold structure on X.
Then ϕ[μ(t)] = μ 1 (t) and so h(ϕ) = μ
1 (0), where this is the usual derivative.
Hence, by (12),
μ
(0) = (h(ϕ), f
(a)h(ϕ)) ,
(13.13)
where f
(a) is the tangent linear map to f at a in the usual sense of n
◦ 2,
(i). As the map h → h(ϕ) from X
(x) to R
p is linear and bijective, and as
formula (12) likewise defines a linear bijection k → h from R
p onto T x (X), we
thereby obtain by composition an isomorphism h → u from the “ abstract ”
vector space X
(x) onto the “ concrete ” vector subspace T x (X) of R
q . For
any curve μ drawn in X and such that μ(0) = x, this isomorphism maps the
“ abstract ” tangent vector μ
(0) ∈ X
(x) onto the vector
lim [μ(t) − μ(0)] /t
also written μ
(0) by everyone. This also proves that the isomorphism of
vector spaces X
(x) −→ T x (X) defined thereby is absolute, i.e. depends only
on the embedding of X into R
q and not on the choice of the chart.
And it is not understandable why this assimilation is a false trail, in the
words of Dieudonn´ e : it in particular suggests that the vector spaces X
(x) and
X
(y) tangent to X at two different points x and y could, like the subspaces
T x (X) and T y (X) of R
q , have common elements; this is not at all the case:
an element h of X
(x) is a couple consisting of a point x ∈ X and of a family
of vectors h(ϕ) of R
p depending on a local chart at x ; two tangent vectors
at x and y cannot be equal if x = y. In fact, the set of tangent vectors to an
n-dimensional manifold is a 2n-dimensional manifold [n
◦ 12(v)].
(v) Riemann spaces. The definition of tangent spaces makes it possible
to define Riemann spaces. For this, in each X
(x) take an Euclidean scalar
product (h|k) compelled to depend on x in a reasonable manner: suppose
that the functions
g ij (ξ) = (a i (ξ)|a j (ξ))