§ 4. Differential Manifolds
231
with a vector h(ϕ) ∈ R
d . To define k, a vector k(ψ) ∈ R
n needs to be derived.
Hence we only lack a map from R
d to R
n , which must be linear in order for
the map X
(a) −→ Y
(b) sought to be so. But in the charts considered, f
can be expressed by a C
r map
F : ϕ(U ) −→ ψ(V )
such that F (ξ) = η ; it has a tangent linear map F
(ξ) : R
d
−→ R
n at ξ.
Hence
k(ψ) = F
(ξ)h(ϕ) where ξ = ϕ(a)
(12.11)
is the unique conceivable solution of the problem; or in coordinates,
k
p (ψ) = D i F
p (ξ).h
i (ϕ) .
(12.11’)
Nonetheless, as always, some verifications need to be made to show the
“ absolute ” character of this construction using charts. The simplest is to
observe that, in relation to charts, the derivatives D i F
p behave in (11’) like
a covector at a does in relation to the index i and like a vector at b = f (a)
in relation to the index p; as the formula respects Einstein’s conventions, it
has an absolute character. . .
Another way of defining f
(a) is to use the construction of tangent vectors
by curves, presented in (ii). If h ∈ X
(a) is a vector μ
(0) of a curve μ
drawn in X and such that μ(0) = a, then the image of μ under f is a
curve ν(t) = f [μ(t)] drawn in Y and such that ν(0) = b . There is vector
ν
(0) = k ∈ Y
(b) corresponding to the latter ; it is just f
(a)h. Indeed, in the
situation used above to define f
(a)h, by (8), the vector h is represented in
the chart (U, ϕ) by the derivative vector h(ϕ) at t = 0 of the function ϕ ◦ μ ;
the vector k ∈ Y
(b) is likewise represented in the chart (V, ψ) by the vector
k(ψ), the derivative at t = 0 of the function ψ ◦ ν; as
ψ ◦ ν = ψ ◦ (f ◦ μ) = (ψ ◦ f ) ◦ μ = (F ◦ ϕ) ◦ μ = F ◦ (ϕ ◦ μ) ,
the classic multivariate chain rule shows that the derivative at t = 0 of ψ ◦ ν
and of ϕ ◦ μ are connected by the formula k(ψ) = F
(ξ)h(ϕ), which reduces
to definition (11).
Consider, for example, the case of a Cartesian space Y = E. At the end of
(ii) above, we saw that tangent spaces to E can be canonically identified with
E; f
(a) can, therefore, be interpreted as a linear map from X
(a) to E. To
write it explicitly, start with some h ∈ X
(a) defined by a curve μ(t) such that
μ(0) = a ; so the image f
(a)h ∈ E
(b), where b = f (a), is defined by the curve
t → f [μ(t)] = ν(t) in E. However, under the above identification of E
(b) with
E, f
(x)h becomes an ordinary vector ν
(0) = lim[ν(t) − ν(0)]/t ; hence, if,
as appropriate, no difference is made between the “ abstract ” vector f
(a)h
and the “ concrete ” vector corresponding to it in E, it can be computed by
the relation
231
with a vector h(ϕ) ∈ R
d . To define k, a vector k(ψ) ∈ R
n needs to be derived.
Hence we only lack a map from R
d to R
n , which must be linear in order for
the map X
(a) −→ Y
(b) sought to be so. But in the charts considered, f
can be expressed by a C
r map
F : ϕ(U ) −→ ψ(V )
such that F (ξ) = η ; it has a tangent linear map F
(ξ) : R
d
−→ R
n at ξ.
Hence
k(ψ) = F
(ξ)h(ϕ) where ξ = ϕ(a)
(12.11)
is the unique conceivable solution of the problem; or in coordinates,
k
p (ψ) = D i F
p (ξ).h
i (ϕ) .
(12.11’)
Nonetheless, as always, some verifications need to be made to show the
“ absolute ” character of this construction using charts. The simplest is to
observe that, in relation to charts, the derivatives D i F
p behave in (11’) like
a covector at a does in relation to the index i and like a vector at b = f (a)
in relation to the index p; as the formula respects Einstein’s conventions, it
has an absolute character. . .
Another way of defining f
(a) is to use the construction of tangent vectors
by curves, presented in (ii). If h ∈ X
(a) is a vector μ
(0) of a curve μ
drawn in X and such that μ(0) = a, then the image of μ under f is a
curve ν(t) = f [μ(t)] drawn in Y and such that ν(0) = b . There is vector
ν
(0) = k ∈ Y
(b) corresponding to the latter ; it is just f
(a)h. Indeed, in the
situation used above to define f
(a)h, by (8), the vector h is represented in
the chart (U, ϕ) by the derivative vector h(ϕ) at t = 0 of the function ϕ ◦ μ ;
the vector k ∈ Y
(b) is likewise represented in the chart (V, ψ) by the vector
k(ψ), the derivative at t = 0 of the function ψ ◦ ν; as
ψ ◦ ν = ψ ◦ (f ◦ μ) = (ψ ◦ f ) ◦ μ = (F ◦ ϕ) ◦ μ = F ◦ (ϕ ◦ μ) ,
the classic multivariate chain rule shows that the derivative at t = 0 of ψ ◦ ν
and of ϕ ◦ μ are connected by the formula k(ψ) = F
(ξ)h(ϕ), which reduces
to definition (11).
Consider, for example, the case of a Cartesian space Y = E. At the end of
(ii) above, we saw that tangent spaces to E can be canonically identified with
E; f
(a) can, therefore, be interpreted as a linear map from X
(a) to E. To
write it explicitly, start with some h ∈ X
(a) defined by a curve μ(t) such that
μ(0) = a ; so the image f
(a)h ∈ E
(b), where b = f (a), is defined by the curve
t → f [μ(t)] = ν(t) in E. However, under the above identification of E
(b) with
E, f
(x)h becomes an ordinary vector ν
(0) = lim[ν(t) − ν(0)]/t ; hence, if,
as appropriate, no difference is made between the “ abstract ” vector f
(a)h
and the “ concrete ” vector corresponding to it in E, it can be computed by
the relation