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In-Depth Information
8.12.3. Lemma.
Equation (8.35) defines a well-defined (k + 1)-form dw on
M
that
is independent of the choice of (
U
,j).
Proof.
This is a straightforward computation.
Definition.
The (k + 1)-form dw is called the
differential
of w. The map wÆdw is
called the
differential operator
d for differential forms on
M
.
8.12.4. Theorem.
(1) If w and h are two k-forms, then d(w+h) = dw+dh.
(2) If w is an r-form and h is a t-form, then
rs
(
)
=Ÿ+
()
Ÿ
d
wh
Ÿ
d
wh
1
w h
d
.
(3) d(dw) = 0 for any k-form w, or simply, d
2
= 0.
(4) If f :
M
n
Æ
W
m
is a differentiable map and w is a k-form on
W
, then f*(dw) =
d(f*w).
Proof.
This is proved just like Theorem 4.9.4.
We are now ready to discuss integration on manifolds. Integrals will be defined
by means of differential forms like we did in Section 4.9.1. We shall continue follow-
ing the presentation given in [Spiv65] and [Spiv70a]. Recall that the theory of inte-
gration developed in Section 4.9.1 used cubes [0,1]
k
. Such spaces are of course not
differentiable manifolds since they have “corners.” However, these spaces are nice
enough so that everything that we did above, such as defining differential forms,
induced maps, etc., could have been done for them and the theorems would also
remain true, and so we shall treat them as if that had been done.
To begin with, we need to extend the definitions from Section 4.9.1 to the context
of manifolds.
A
singular k-cube
in
M
is a C
•
function c : [0,1]
k
Definition.
Æ
M
. Define the
(i,j)-
face
of c to be the singular (n - 1)-cube, c
i,j
, by c
i,j
= cI
k
(i,j)
.
o
Definition.
A formal linear combinations of singular k-cubes in
M
is called a
singular k-chain
and the set of these is denoted by G
k
(
M
). The boundary of a singular
k-chain and the boundary operator
()
Æ
()
∂
: G
M
G
1
M
k
k
-
is defined just like in equations (4.36) and (4.37).
If w is a k-form on
M
and if c : [0,1]
k
Definition.
Æ
M
is a singular k-cube, then
define the
integral of
w
over c
by
Ú
Ú
w
=
c
*.
w
(8.36a)
k
[
]
c
01
