Graphics Reference
In-Depth Information
Consider Figure 4.28 again. The standard orientation of I
2
induces a counter-
clockwise orientation on the edges. Formula (4.36) implies that
2
2
2
2
2
∂I
=-
I
+
I
+
I
-
I
,
(
)
(
)
(
)
(
)
10
,
11
,
20
,
21
,
which specifies that same orientation.
4.9.1.1. Lemma.
∂∂=0.
o
Proof.
This is a straightforward computation.
Lemma 4.9.1.1 is only an aside remark for us here, but it actually is the founda-
tion of an important theory. To do this theory justice would take us far afield and take
too much time. Our objectives in this section are much more limited. Nevertheless,
the reader should return here after reading Chapter 7. We are basically describing a
homology theory for open sets
A
similar to the homology theory for simplicial com-
plexes that will be developed in Chapter 7. The main difference is that we are using
singular k-cubes rather than k-simplices.
With these preliminaries out of the way, we return to the question of integration.
Suppose that w is a k-form on [0,1]
k
. In this case we know that
w=
fdx
Ÿ
dx
Ÿ
...
Ÿ
dx
k
,
1
2
for some unique function f : [0,1]
k
Æ
R
.
Define the
integral of
w
over
[
0
,
1
]
k
by
Definition.
Ú
Ú
w
=
k
f
.
(4.38)
k
[]
[]
01
,
01
,
Of course, the integral on the right-hand side of equation (4.38) is the standard
advanced calculus integral that was defined in Section 4.8. Note how similar all of the
notation is, that is, we are saying that
Ú
Ú
f dx
ŸŸŸ =
dx
...
dx
f dx dx
...
dx
.
1
2
k
12
k
k
k
[]
[]
01
,
01
,
Finally,
Let k > 0. If w is a k-form on
A
and if c : [0,1]
k
Definition.
Æ
A
is a singular k-cube,
then define the
integral of
w
over c
by
Ú
Ú
w
=
[]
*.
c
w
(4.39a)
k
c
01
If w is a 0-form on
A
, then w is just a function f :
A
Æ
R
. Therefore, if c : 0 Æ
A
is a
singular 0-cube, then define
