226
IX – Multivariate Differential and Integral Calculus
To arrive at a definition of X
(a), what is more generally called a tensor
of type (p, q) at the point a can first be defined by drawing on the Italians.
A priori, we do not know what the concrete nature of such an object is,
but we suspect that it must have “ components ” in each local chart (U, ϕ)
at a ; for example, if (p, q) = (2, 1), these components should be numbers
T
k
ij (ϕ) depending on three indices and on the chart considered. Having admitted this, we stipulate that these components should transform according
to formula (3.9) of § 1 when (U, ϕ) is replaced by another chart (V, ψ) at a :
if coordinates ξ = ϕ(x) and η = ψ(x) of a variable point x ∈ U ∩ V are
connected by
η = θ(ξ) , ξ = ρ(η) ,
where θ maps ϕ(U ∩ V ) diffeomorphically onto ψ(V ∩ U ) and conversely, then
this gives formulas
dη
α = θ
α
i (ξ)dξ
i , dξ
i = ρ
i
α (η)dη
α
with partial derivatives
θ
α
i (ξ) = dη
α /dξ
i , ρ
i
α (η) = dξ
i /dη
α .
(12.1)
Having said this, the numbers T
γ
αβ (ψ) corresponding to the chart (V, ψ) must
satisfy the relation
T
γ
αβ (ψ) = ρ
i
α (η)ρ
j
β (η)θ
γ
k (ξ)T
k
ij (ϕ) ,
(12.2)
where the coefficients are calculated at the points ξ = ϕ(a) and η = ψ(a). At
this stage of the definition, we are reduced to “ absolute differential calculus ” :
we do not know over what we are calculating, but we calculate. We continue
doing so every day in our times, and not only in physics. . .
Then the tangent vectors to X at a are, by definition, the tensors of type
(0, 1) at α. We thus get some h ∈ X
(a) by taking, in each local chart (U, ϕ)
at a, numbers h
i (ϕ) that are subject to the equivalent relations
h
α (ψ) = θ
α
i (ξ)h
i (ϕ) , h
i (ϕ) = ρ
i
α (η)h
α (ψ)
(12.3)
for any local chart ϕ and ψ at α. However, the θ
α
i (ξ) are the entries of the
Jacobian matrix with respect to the canonical basis for R
d at the point ϕ(a),
of the chart change
53
θ : ϕ(x) −→ ψ(x) .
Hence, setting
h(ϕ) = h
i (ϕ)e i ,
(12.4)
53 The notation below replaces the formula θ[ϕ(x)] = ψ(x).
IX – Multivariate Differential and Integral Calculus
To arrive at a definition of X
(a), what is more generally called a tensor
of type (p, q) at the point a can first be defined by drawing on the Italians.
A priori, we do not know what the concrete nature of such an object is,
but we suspect that it must have “ components ” in each local chart (U, ϕ)
at a ; for example, if (p, q) = (2, 1), these components should be numbers
T
k
ij (ϕ) depending on three indices and on the chart considered. Having admitted this, we stipulate that these components should transform according
to formula (3.9) of § 1 when (U, ϕ) is replaced by another chart (V, ψ) at a :
if coordinates ξ = ϕ(x) and η = ψ(x) of a variable point x ∈ U ∩ V are
connected by
η = θ(ξ) , ξ = ρ(η) ,
where θ maps ϕ(U ∩ V ) diffeomorphically onto ψ(V ∩ U ) and conversely, then
this gives formulas
dη
α = θ
α
i (ξ)dξ
i , dξ
i = ρ
i
α (η)dη
α
with partial derivatives
θ
α
i (ξ) = dη
α /dξ
i , ρ
i
α (η) = dξ
i /dη
α .
(12.1)
Having said this, the numbers T
γ
αβ (ψ) corresponding to the chart (V, ψ) must
satisfy the relation
T
γ
αβ (ψ) = ρ
i
α (η)ρ
j
β (η)θ
γ
k (ξ)T
k
ij (ϕ) ,
(12.2)
where the coefficients are calculated at the points ξ = ϕ(a) and η = ψ(a). At
this stage of the definition, we are reduced to “ absolute differential calculus ” :
we do not know over what we are calculating, but we calculate. We continue
doing so every day in our times, and not only in physics. . .
Then the tangent vectors to X at a are, by definition, the tensors of type
(0, 1) at α. We thus get some h ∈ X
(a) by taking, in each local chart (U, ϕ)
at a, numbers h
i (ϕ) that are subject to the equivalent relations
h
α (ψ) = θ
α
i (ξ)h
i (ϕ) , h
i (ϕ) = ρ
i
α (η)h
α (ψ)
(12.3)
for any local chart ϕ and ψ at α. However, the θ
α
i (ξ) are the entries of the
Jacobian matrix with respect to the canonical basis for R
d at the point ϕ(a),
of the chart change
53
θ : ϕ(x) −→ ψ(x) .
Hence, setting
h(ϕ) = h
i (ϕ)e i ,
(12.4)
53 The notation below replaces the formula θ[ϕ(x)] = ψ(x).