§ 4. Differential Manifolds
269
supports, contained in W , are compact; their sum on A is obviously again
equal to 1, qed .
Thanks to Lemma 4, it is possible to give an unambiguous definition of
the integral over an oriented manifold X of a differential form ω of class
C
0 , maximum degree n = dim(X) and with compact support. For this, in
conformity with lemma 4, choose open Cartesian sets U i covering the support
of ω, as well as functions f i and apply formula (3) choosing ω i = f i ω, a form
whose support is a compact subset of U i . If the U i and the f i are replaced by
the V p and the g p satisfying the same condition, then obviously ω i =
g p ω i .
As the support of ω i is a compact subset of U i , the integral of ω i over U i is the
sum of the integrals of g p ω i since, as seen at the start of (ii), the situation
in an open Cartesian set is similar to that in an open subset of R
n . But
the support of g p ω i being contained in the open Cartesian space U i ∩ V p , its
integral over U i , defined as in (i), is equal to its integral over U i ∩ V p . Hence
finally,
Ui
f i ω =
i,p
Ui∩Vp
g p f i ω .
Now, permuting the roles played by the U i and the V p , similarly
Vp
g p ω =
p,i
Vp∩Ui
f i g p ω
follows. So the two partitions of unity used to compute the integral of ω
indeed lead to the same result.
At the same time, if ω and are forms with compact support, then
ω +
=
ω + ;
(17.6)
the union of the supports of ω and being compact, the same partition of
unity can be used to compute the three integrals in question; this reduces to
the trivial case of an open subset of a Cartesian space.
Finally note that if ω is a form of degree p ≤ n and if Y is a p-dimensional
submanifold of X, by inverse image, the immersion Y −→ X leads to a form
of maximum degree on Y . If the support of ω is compact and if Y is closed,
then
Y
ω can be defined.
18 – Stokes’ Formula
It states that if ω is a differential form of degree n − 1 on an n-dimensional
oriented manifold X and if Ω is an open set with compact closure whose
border ∂Ω is an n − 1-dimensional submanifold of X in the sense defined at
the end of n
◦ 12, then
∂Ω
ω =
Ω
dω ,
(18.1)
269
supports, contained in W , are compact; their sum on A is obviously again
equal to 1, qed .
Thanks to Lemma 4, it is possible to give an unambiguous definition of
the integral over an oriented manifold X of a differential form ω of class
C
0 , maximum degree n = dim(X) and with compact support. For this, in
conformity with lemma 4, choose open Cartesian sets U i covering the support
of ω, as well as functions f i and apply formula (3) choosing ω i = f i ω, a form
whose support is a compact subset of U i . If the U i and the f i are replaced by
the V p and the g p satisfying the same condition, then obviously ω i =
g p ω i .
As the support of ω i is a compact subset of U i , the integral of ω i over U i is the
sum of the integrals of g p ω i since, as seen at the start of (ii), the situation
in an open Cartesian set is similar to that in an open subset of R
n . But
the support of g p ω i being contained in the open Cartesian space U i ∩ V p , its
integral over U i , defined as in (i), is equal to its integral over U i ∩ V p . Hence
finally,
Ui
f i ω =
i,p
Ui∩Vp
g p f i ω .
Now, permuting the roles played by the U i and the V p , similarly
Vp
g p ω =
p,i
Vp∩Ui
f i g p ω
follows. So the two partitions of unity used to compute the integral of ω
indeed lead to the same result.
At the same time, if ω and are forms with compact support, then
ω +
=
ω + ;
(17.6)
the union of the supports of ω and being compact, the same partition of
unity can be used to compute the three integrals in question; this reduces to
the trivial case of an open subset of a Cartesian space.
Finally note that if ω is a form of degree p ≤ n and if Y is a p-dimensional
submanifold of X, by inverse image, the immersion Y −→ X leads to a form
of maximum degree on Y . If the support of ω is compact and if Y is closed,
then
Y
ω can be defined.
18 – Stokes’ Formula
It states that if ω is a differential form of degree n − 1 on an n-dimensional
oriented manifold X and if Ω is an open set with compact closure whose
border ∂Ω is an n − 1-dimensional submanifold of X in the sense defined at
the end of n
◦ 12, then
∂Ω
ω =
Ω
dω ,
(18.1)