264
IX – Multivariate Differential and Integral Calculus
The same difficulty arises in the case of a general manifold. A simple case
is that of a form ω whose compact support is contained in the domain of a
chart (U, ϕ) ; it would be natural to set
X
ω =
U
ω ,
where the right hand side is defined by formula (1) applied to U . If the chart
(U, ϕ) is replaced by the chart (V, ψ) such that Supp(ω) ⊂ V , then, applying
(1) to U or V , it clearly suffices to integrate over U ∩ V ; the arguments used
above then show that, here too, both definitions of the integral coincide only
if the Jacobian of change of coordinates ϕ(x) → ψ(x) is everywhere > 0 in
U ∩ V .
To overcome this difficulty, we are led to allow only changes of local charts
whose Jacobian is everywhere > 0, more precisely, instead of all possible
charts, to use only an atlas all of whose changes of charts have positive
Jacobian. The existence of such an atlas – recall that the charts in an atlas
must cover the manifold – is not obvious and may be false for some manifolds.
When such an atlas exists, the manifold is said to be orientable.
The relation with the classical notion of orientable surface in R
3 recalled
in n
◦ 9, (iv) is easy to see. Indeed, let X be a 2-dimensional submanifold of
R
3 and (U, ϕ) a local chart for X ; if f is the inverse map of ϕ, in the open
subset U , the surface X is given by the parametric representation
x = f
1 (s, t) , y = f
2 (s, t) , z = f
3 (s, t) ,
where (s, t) vary in the open subset ϕ(U ) of the plane. Then the derivatives
D 1 f and D 2 f of f with respect to s and t at each point x ∈ U are two
non-proportional tangent vectors to X at x (in the classical sense); their
classical vector product is a normal vector to X at x, which is a continuous
function of the point x and thus coherently orients the normals to X at
points of U . Similarly, if (V, ψ) is another chart and if g = ψ
−1 , the product
D 1 g∧D 2 g leads to an orientation of normals in V . In U ∩V , the D j g are linear
combinations of the D i f whose coefficients are the entries of the Jacobian
matrix of change of coordinate θ : ϕ(x) −→ ψ(x). As the vector product of
two vectors is an alternating bilinear form of these,
D 1 g(η) ∧ D 2 g(η) = J θ (ξ)D 1 f (ξ) ∧ D 2 f (ξ) ,
which shows that, in U ∩ V , the orientations of normals to X defined by
the charts (U, ϕ) and (V, ψ) are identical only if J θ > 0 everywhere. So if X
can be covered by charts such that the chart change formulas have positive
Jacobians, then the normals at all points of X can be coherently oriented,
which is the classical definition of an orientable surface.
IX – Multivariate Differential and Integral Calculus
The same difficulty arises in the case of a general manifold. A simple case
is that of a form ω whose compact support is contained in the domain of a
chart (U, ϕ) ; it would be natural to set
X
ω =
U
ω ,
where the right hand side is defined by formula (1) applied to U . If the chart
(U, ϕ) is replaced by the chart (V, ψ) such that Supp(ω) ⊂ V , then, applying
(1) to U or V , it clearly suffices to integrate over U ∩ V ; the arguments used
above then show that, here too, both definitions of the integral coincide only
if the Jacobian of change of coordinates ϕ(x) → ψ(x) is everywhere > 0 in
U ∩ V .
To overcome this difficulty, we are led to allow only changes of local charts
whose Jacobian is everywhere > 0, more precisely, instead of all possible
charts, to use only an atlas all of whose changes of charts have positive
Jacobian. The existence of such an atlas – recall that the charts in an atlas
must cover the manifold – is not obvious and may be false for some manifolds.
When such an atlas exists, the manifold is said to be orientable.
The relation with the classical notion of orientable surface in R
3 recalled
in n
◦ 9, (iv) is easy to see. Indeed, let X be a 2-dimensional submanifold of
R
3 and (U, ϕ) a local chart for X ; if f is the inverse map of ϕ, in the open
subset U , the surface X is given by the parametric representation
x = f
1 (s, t) , y = f
2 (s, t) , z = f
3 (s, t) ,
where (s, t) vary in the open subset ϕ(U ) of the plane. Then the derivatives
D 1 f and D 2 f of f with respect to s and t at each point x ∈ U are two
non-proportional tangent vectors to X at x (in the classical sense); their
classical vector product is a normal vector to X at x, which is a continuous
function of the point x and thus coherently orients the normals to X at
points of U . Similarly, if (V, ψ) is another chart and if g = ψ
−1 , the product
D 1 g∧D 2 g leads to an orientation of normals in V . In U ∩V , the D j g are linear
combinations of the D i f whose coefficients are the entries of the Jacobian
matrix of change of coordinate θ : ϕ(x) −→ ψ(x). As the vector product of
two vectors is an alternating bilinear form of these,
D 1 g(η) ∧ D 2 g(η) = J θ (ξ)D 1 f (ξ) ∧ D 2 f (ξ) ,
which shows that, in U ∩ V , the orientations of normals to X defined by
the charts (U, ϕ) and (V, ψ) are identical only if J θ > 0 everywhere. So if X
can be covered by charts such that the chart change formulas have positive
Jacobians, then the normals at all points of X can be coherently oriented,
which is the classical definition of an orientable surface.