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We use the partition of unity F in (5) to define the integral of w.
Definition.
The
integral of
w
over
M
is defined by
Â
Ú
Ú
w
=
a
◊
w
.
(8.41)
M
M
a
ŒF
8.12.13. Lemma.
If
M
n
is an oriented manifold, then the integral of an n-form w on
M
with compact support is well defined.
Proof.
The first observation is that each term
Ú
M
aw
◊
is defined because a · w is an n-form to which the definition in equation (8.40)
applies. Next, the sum on the right-hand side of equation (8.41) is only a finite sum
since by condition (4) above all but a finite number of terms are zero. This shows that
the right hand side of equation (8.41) makes sense. We need show that it does not
depend on the choice of the partition of unity F. Assume that Y is another partition
of unity. Then
Ê
Á
ˆ
˜
=
Ê
Á
ˆ
˜
=
Â
Â
Â
Â
Â
Â
Ú
Ú
(
)
Ú
(
)
Ú
aw
◊
=
b aw
◊
◊
a bw
◊
◊
bw
◊
,
M
M
M
M
a
Œ
F
a
Œ
F
b
Œ
Y
b
Œ
Y
a
Œ
F
b
Œ
Y
which proves the independence and finishes the proof of the lemma.
8.12.14. Proposition.
(1) Let
M
n
be an oriented manifold. If w
1
and w
2
are n-forms on
M
with compact
support and if a
1
, a
2
Œ
R
, then
Ú
Ú
Ú
a
ww
+
a
=
a
w
+
a
w
,
11
22
1
1
2
2
M
M
M
that is, the integral is linear in the forms.
(2) Let
M
n
and
N
n
be oriented manifolds and f :
M
Æ
N
an orientation-preserving
diffeomorphism. If w is an n-form on
N
with compact support, then
Ú
Ú
w w
NM
=
f*
.
Proof.
This is an easy consequence of what has been proved so far.
Now, if
M
n
is an oriented n-manifold with boundary, then we know that the
orientation on
M
induces a natural orientation on ∂
M
, so that ∂
M
becomes an
oriented (n -
1)-manifold. We always assume that ∂
M
has been given that
orientation.
