§ 4. Differential Manifolds
273
which h 2 is tangent to the curve limiting Ω . If the vector h 1 “ comes out of ”
Ω, ∂Ω must be oriented like h 2 . But if (h 1 , h 2 ) defines the orientation of X,
this means that the orientation of its normal is that of the vector h 1 ∧ h 2 . In
other words, the vectors h 1 , h 2 and the unit vector of the oriented normal at
x, written in this order, must form a “ direct ” trihedron. We thus recover the
classical rule described in n
◦ 9, (iv) : a passerby following the boundary of
Ω in the direction prescribed by Stokes’ formula and remaining oriented like
the normal to the surface, when looking in front of him or her, must leave Ω
on his or her left, which until further notice is the same for both genders.
In particular, this rule applies to an open subset Ω of C limited by one or
several closed simple curves γ 0 , . . . , γ p , as can be encountered in the theory of holomorphic functions; if Ω is assumed to be interior to γ 0 exterior
71 to γ 1 , . . . , γ p and if Ω is given the usual orientation, then γ 0 must
be “ positively ” oriented (counterclockwise) and the other γ k negatively.
Suppose now that X = R
3 , which is another classical case, so that Ω is an
open bounded set whose border is a 2-dimensional compact submanifold of X,
for example a sphere, a torus, etc. It is natural to orient Ω like the canonical
basis for X. It then remains to orient ∂Ω in such a way that Stokes’ (for that
matter, Ostrogradsky’s) formula holds. The general result shows that if we
choose a basis (h 1 , h 2 , h 3 ) at x ∈ ∂Ω, oriented like the canonical basis and
such that (i) h 1 is the velocity vector at x of a moving object departing from
x and come out of Ω, (ii) (h 2 , h 3 ) form a basis for the tangent plane to ∂Ω at
x, then it needs to be oriented like the basis (h 2 , h 3 ). This amounts to saying
that the normal vector h 2 ∧ h 3 to ∂Ω at x must come out of Ω.
71 In the sense given to these terms in Chapter VIII : Indγ(z) = +1 or −1 according
to whether z is interior or exterior to γ.
273
which h 2 is tangent to the curve limiting Ω . If the vector h 1 “ comes out of ”
Ω, ∂Ω must be oriented like h 2 . But if (h 1 , h 2 ) defines the orientation of X,
this means that the orientation of its normal is that of the vector h 1 ∧ h 2 . In
other words, the vectors h 1 , h 2 and the unit vector of the oriented normal at
x, written in this order, must form a “ direct ” trihedron. We thus recover the
classical rule described in n
◦ 9, (iv) : a passerby following the boundary of
Ω in the direction prescribed by Stokes’ formula and remaining oriented like
the normal to the surface, when looking in front of him or her, must leave Ω
on his or her left, which until further notice is the same for both genders.
In particular, this rule applies to an open subset Ω of C limited by one or
several closed simple curves γ 0 , . . . , γ p , as can be encountered in the theory of holomorphic functions; if Ω is assumed to be interior to γ 0 exterior
71 to γ 1 , . . . , γ p and if Ω is given the usual orientation, then γ 0 must
be “ positively ” oriented (counterclockwise) and the other γ k negatively.
Suppose now that X = R
3 , which is another classical case, so that Ω is an
open bounded set whose border is a 2-dimensional compact submanifold of X,
for example a sphere, a torus, etc. It is natural to orient Ω like the canonical
basis for X. It then remains to orient ∂Ω in such a way that Stokes’ (for that
matter, Ostrogradsky’s) formula holds. The general result shows that if we
choose a basis (h 1 , h 2 , h 3 ) at x ∈ ∂Ω, oriented like the canonical basis and
such that (i) h 1 is the velocity vector at x of a moving object departing from
x and come out of Ω, (ii) (h 2 , h 3 ) form a basis for the tangent plane to ∂Ω at
x, then it needs to be oriented like the basis (h 2 , h 3 ). This amounts to saying
that the normal vector h 2 ∧ h 3 to ∂Ω at x must come out of Ω.
71 In the sense given to these terms in Chapter VIII : Indγ(z) = +1 or −1 according
to whether z is interior or exterior to γ.