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the original function. The formalism of differential forms and how they map is set up
the way it is, precisely so as to eliminate these reparameterization factors. Although
manifolds look like Euclidean space locally, the exact identification depends on the
coordinate neighborhood that we choose and so intrinsic properties of manifolds must
be defined so as to be independent of the choice of such neighborhoods.
8.12.7. Theorem.
(Stokes' Theorem) If w is a (k - 1)-form on a manifold
M
and if
c is a k-chain on
M
, then
Ú
Ú
d
ww
=
.
c
∂
Proof.
The proof of this theorem is similar to the one for Theorem 4.9.1.4. See
[Spiv65] or [Spiv70a].
Next, we want to describe how one integrates over a whole manifold. Basically,
since we know how to integrate over
R
n
, we know how to integrate over a coordinate
neighborhood. The problem is that there are many ways to cover a manifold
with coordinate neighborhoods and we must make sure that we make a consistent
definition. First of all, we need to revisit the definition of the orientation of a
manifold
M
n
. The definition that we already have says that it is a continuously vary-
ing choice of orientations of its tangent spaces. We want to relate this to differential
forms.
8.12.8. Lemma.
Let
v
1
,
v
2
,...,
v
n
be a basis of an n-dimensional vector space
V
.
Let wŒL
n
(
V
). If
n
Â
1
w
=
a
v
Œ
V
,
(8.37)
i
ij
j
j
=
then
(
)
=
()(
)
w
ww
,
,...,
w
det
a
w
vv
,
,...,
v
.
(8.38)
12
n
ij
12
n
Define an n-form hŒL
n
(
R
n
) by
Proof.
(
(
) (
)
(
)
)
h
aa
,
,...,
a
,
aa
,
,...,
a
,...,
aa
,
,...,
a
11
12
1
n
21
22
2
n
n
1
n
2
nn
n
n
n
Ê
Á
ˆ
˜
ÂÂ
Â
=
w
av
,
a v
,...,
a v
.
1
jj
2
jj
nj
j
j
=
1
j
=
1
j
=
1
By Proposition 4.9.1(3), h=c det, where det is the determinant map of
R
n
and c is
some constant. Since,
(
)
=
(
)
c
=
h
ee
,
,...,
e
w
vv
,
,...,
v
.
12
n
12
n
we are done.
