Graphics Reference
In-Depth Information
The
integral of
w
over singular k-chain
=
Â
c
a
ii
i
is defined by
Â
Ú
Ú
w
=
a
w
.
(8.36b)
i
c
c
i
i
Next, we need some lemmas to show that certain quantities are well defined.
8.12.5. Lemma.
Let c : [0,1]
n
Æ
R
n
be a singular n-cube and let w be an n-form on
R
n
. If c is one-to-one and det c¢≥0 everywhere, then
Ú
Ú
w
=
f
,
(
)
n
[
]
c
c
0,
where f :
R
n
Æ
R
is the unique function with the property that
w=
fdx
Ÿ
dx
Ÿ
...
Ÿ
dx
n
.
1
2
Proof.
This follows immediately from the definitions, Theorem 8.12.2(4), and the
change of variable theorem for integrals.
8.12.6. Lemma.
Let h : [0,1]
k
Æ [0,1]
k
be a one-to-one and onto C
•
map and assume
that det h¢≥0 everywhere. If c is any singular k-cube and if w is any k-form on
M
n
,
then
Ú
Ú
ww
=
.
c
c h
o
Proof.
This is another straightforward computation using Lemma 8.12.5.
Definition.
Given a singular k-cube c on
M
n
and a C
•
one-to-one and onto map
h : [0,1]
k
Æ [0,1]
k
with det h¢π0 everywhere, the map c h is called a
reparameteriza-
tion
of c. The reparameterization is said to be
orientation preserving
if det h¢>0 every-
where and
orientation reversing
if det h¢<0 everywhere.
o
Lemma 8.12.6 is very important for defining an integral on an abstract
manifold. It shows that integrals of differential forms do not change under
orientation-preserving reparameterizations. The reason that we have had to develop
differential forms in preparation for defining integration is that they transform cor-
rectly. If we had defined the integral of a function f :
M
n
Æ
R
over a singular k-cube c
on
M
by
Ú
fc
o
k
[
0,
]
then the result in Lemma 8.12.6 would not always hold. By the change of variable
theorem, the integral of a reparameterized function is not the same as the integral of
