266
IX – Multivariate Differential and Integral Calculus
opposite according to whether the determinant of the change of basis matrix
is positive or negative. From this point of view, two bases can be considered
to be equivalent if the determinant of the matrix taking one to the other is
> 0. The set of all bases for E is thereby divided into two equivalence classes,
one which plays a distinguished role; orienting E then amounts to choosing
one of the these classes. The bases belonging to it are said to be direct, by
analogy with physicists’“ direct trihedrons ”. There is a distinguished class of
bases in the spaces R
n : that of the canonical basis. Hence it is possible to
choose a canonical orientation in R
n – but it is impossible to do so in any
other Cartesian space.
The case of a general manifold X can be presented in the same manner.
Let (U, ϕ) be a chart for X. As seen in (i) of n
◦ 13, at each point x of U , it
gives rise to a basis (a i (ξ)) for X
(x) such that
h = h
i (ξ)a i (ξ) for all h ∈ X
(x) .
If (V, ψ) is another chart, then h = h
α (η)b α (η) also holds in the basis corresponding to this chart. But as the formula h
i (ξ) = ρ
i
α (η)h
α (η) whose coefficients are the dξ
i /dη
α transforms the h
i (ξ) to h
α (η), the determinant of the
matrix taking the first basis to the second one is clearly the Jacobian of the
change of basis at x (or of its inverse, which does not change the sign). Hence,
if in every local chart (U, ϕ), the basis (a i (ξ)) is used to orient the tangent
space X
(x) at each x ∈ U as done above to orient a Cartesian space, two
such charts are seen to define the same orientation in U ∩ V if and only if
they orient X
(x) in the same manner for all x ∈ X.
The conclusion is clear : orienting a manifold X amounts to orienting each
tangent space X
(x) so that X can be covered by charts (U, ϕ) satisfying the
following condition: for all x ∈ U , the orientation of X
(x) is defined by the
basis (a i (ξ)) for X
(x) associated to (U, ϕ).
(ii) Integration of differential forms. These arguments show that in order
to define the integral of a differential form ω of maximum degree on a manifold
X, X needs to be assumed to be oriented and ω to have compact support.
For lack of better, the arguments of (1) then show that the integral of ω can
given an absolute meaning in the particular case when the support of ω is
contained in an open subset U of X; apply formula (1) after having chosen
a chart (U, ϕ) compatible with the orientation of X ; the result is the same
for all open Cartesian subsets U such that Supp(ω) ⊂ U and for all possible
diffeomorphisms ϕ.
However, the method supposes that the compact subset K = Supp(ω)
is contained in an open Cartesian set. In the general case, even that of a
sphere in R
3 , only the existence of a covering of K by a finite number of such
open subsets U i can be guaranteed using BL. The method then consists in
constructing forms ω i with compact support satisfying
Supp(ω i ) ⊂ U i & ω =
ω i
(17.2)
IX – Multivariate Differential and Integral Calculus
opposite according to whether the determinant of the change of basis matrix
is positive or negative. From this point of view, two bases can be considered
to be equivalent if the determinant of the matrix taking one to the other is
> 0. The set of all bases for E is thereby divided into two equivalence classes,
one which plays a distinguished role; orienting E then amounts to choosing
one of the these classes. The bases belonging to it are said to be direct, by
analogy with physicists’“ direct trihedrons ”. There is a distinguished class of
bases in the spaces R
n : that of the canonical basis. Hence it is possible to
choose a canonical orientation in R
n – but it is impossible to do so in any
other Cartesian space.
The case of a general manifold X can be presented in the same manner.
Let (U, ϕ) be a chart for X. As seen in (i) of n
◦ 13, at each point x of U , it
gives rise to a basis (a i (ξ)) for X
(x) such that
h = h
i (ξ)a i (ξ) for all h ∈ X
(x) .
If (V, ψ) is another chart, then h = h
α (η)b α (η) also holds in the basis corresponding to this chart. But as the formula h
i (ξ) = ρ
i
α (η)h
α (η) whose coefficients are the dξ
i /dη
α transforms the h
i (ξ) to h
α (η), the determinant of the
matrix taking the first basis to the second one is clearly the Jacobian of the
change of basis at x (or of its inverse, which does not change the sign). Hence,
if in every local chart (U, ϕ), the basis (a i (ξ)) is used to orient the tangent
space X
(x) at each x ∈ U as done above to orient a Cartesian space, two
such charts are seen to define the same orientation in U ∩ V if and only if
they orient X
(x) in the same manner for all x ∈ X.
The conclusion is clear : orienting a manifold X amounts to orienting each
tangent space X
(x) so that X can be covered by charts (U, ϕ) satisfying the
following condition: for all x ∈ U , the orientation of X
(x) is defined by the
basis (a i (ξ)) for X
(x) associated to (U, ϕ).
(ii) Integration of differential forms. These arguments show that in order
to define the integral of a differential form ω of maximum degree on a manifold
X, X needs to be assumed to be oriented and ω to have compact support.
For lack of better, the arguments of (1) then show that the integral of ω can
given an absolute meaning in the particular case when the support of ω is
contained in an open subset U of X; apply formula (1) after having chosen
a chart (U, ϕ) compatible with the orientation of X ; the result is the same
for all open Cartesian subsets U such that Supp(ω) ⊂ U and for all possible
diffeomorphisms ϕ.
However, the method supposes that the compact subset K = Supp(ω)
is contained in an open Cartesian set. In the general case, even that of a
sphere in R
3 , only the existence of a covering of K by a finite number of such
open subsets U i can be guaranteed using BL. The method then consists in
constructing forms ω i with compact support satisfying
Supp(ω i ) ⊂ U i & ω =
ω i
(17.2)