230
IX – Multivariate Differential and Integral Calculus
that Leibniz, a philosopher mathematician, believed in the existence of a preestablished harmony
55 governing Creation and hence differential geometry.
(iii) Differential of a map. In the general case, the construction of X
(a)
makes it possible to define the differential df (a) of a numerical function differentiable at a. For this, choose a chart (U, ϕ) at a, write that f (x) = F [ϕ(x)]
where the expression F of f in the chart considered is differentiable at
ξ = ϕ(a), and, for all h ∈ X
(a), set
df (a; h) = dF [ξ; h(ϕ)] = D i F (ξ)h
i (ϕ) ,
(12.9)
where D i = d/dξ
i . The multivariate chain rule immediately shows that, under
chart change, the D i F (ξ) and h
i (ϕ) are inversely transformed, in other words,
that the D i F (ξ) are transformed like the components of a tensor of type
(0, 1) ; so the left hand side does not depend on the chosen chart, which
legitimizes definition (9). In the particular case of the coordinate function
x → ϕ
i (x), the function F is (ξ
1 , . . . , ξ
d ) → ξ
i , and so
dϕ
i (a; h) = h
i (ϕ) .
(12.9’)
Like in a Cartesian space,
df (a) = D i f (a).dξ
i
(12.9”)
is a shorthand version of formula (5), where D i f denotes the derivatives of f
considered as a function of local coordinates ξ
i = ϕ
i (x) and the differential
dϕ
i (a; h) is shortened to dξ
i . This gives a linear functional on X
(a), i.e. a
covector df (a) at a. It is obvious that
p = fg =⇒ dp(a; h) = df (a; h)g(a) + f (a)dg(a; h)
for all h ∈ X
(a).
More generally, if f is a map from a d-dimensional manifold X to a ndimensional manifold Y , for all a ∈ X, a tangent linear map
f
(a) : X
(a) −→ Y
(b) ,
(12.10)
where b = f (a), may be defined, obviously by making a differentiability
assumption about f . As always, the method is imposed by the data of the
situation. Indeed, the aim is to associate to each vector h ∈ X
(a) a vector
k ∈ Y
(b). For this, choose charts (U, ϕ) and (V, ψ) of X and Y , with a ∈ U ,
b ∈ V , f (U ) ⊂ V , ϕ(a) = ξ and ψ(b) = η. The construction of tangent spaces
then furnishes us with bijective maps from X
(a) and Y
(b) to R
d and R
n and
55 Pre-established harmony is a theory of Leibnitz according to which the spiritual
and the physical world are like two perfect, but independent, clocks, always
indicating the same time. Littr´ e.
IX – Multivariate Differential and Integral Calculus
that Leibniz, a philosopher mathematician, believed in the existence of a preestablished harmony
55 governing Creation and hence differential geometry.
(iii) Differential of a map. In the general case, the construction of X
(a)
makes it possible to define the differential df (a) of a numerical function differentiable at a. For this, choose a chart (U, ϕ) at a, write that f (x) = F [ϕ(x)]
where the expression F of f in the chart considered is differentiable at
ξ = ϕ(a), and, for all h ∈ X
(a), set
df (a; h) = dF [ξ; h(ϕ)] = D i F (ξ)h
i (ϕ) ,
(12.9)
where D i = d/dξ
i . The multivariate chain rule immediately shows that, under
chart change, the D i F (ξ) and h
i (ϕ) are inversely transformed, in other words,
that the D i F (ξ) are transformed like the components of a tensor of type
(0, 1) ; so the left hand side does not depend on the chosen chart, which
legitimizes definition (9). In the particular case of the coordinate function
x → ϕ
i (x), the function F is (ξ
1 , . . . , ξ
d ) → ξ
i , and so
dϕ
i (a; h) = h
i (ϕ) .
(12.9’)
Like in a Cartesian space,
df (a) = D i f (a).dξ
i
(12.9”)
is a shorthand version of formula (5), where D i f denotes the derivatives of f
considered as a function of local coordinates ξ
i = ϕ
i (x) and the differential
dϕ
i (a; h) is shortened to dξ
i . This gives a linear functional on X
(a), i.e. a
covector df (a) at a. It is obvious that
p = fg =⇒ dp(a; h) = df (a; h)g(a) + f (a)dg(a; h)
for all h ∈ X
(a).
More generally, if f is a map from a d-dimensional manifold X to a ndimensional manifold Y , for all a ∈ X, a tangent linear map
f
(a) : X
(a) −→ Y
(b) ,
(12.10)
where b = f (a), may be defined, obviously by making a differentiability
assumption about f . As always, the method is imposed by the data of the
situation. Indeed, the aim is to associate to each vector h ∈ X
(a) a vector
k ∈ Y
(b). For this, choose charts (U, ϕ) and (V, ψ) of X and Y , with a ∈ U ,
b ∈ V , f (U ) ⊂ V , ϕ(a) = ξ and ψ(b) = η. The construction of tangent spaces
then furnishes us with bijective maps from X
(a) and Y
(b) to R
d and R
n and
55 Pre-established harmony is a theory of Leibnitz according to which the spiritual
and the physical world are like two perfect, but independent, clocks, always
indicating the same time. Littr´ e.