Graphics Reference
In-Depth Information
Ú
(
()
0.
ww
=
c
(4.39b)
c
The
integral of
w
over singular k-chain
=
Â
c
a
ii
i
is defined by
Â
Ú
Ú
w
=
a
w
.
(4.39c)
i
c
c
i
i
Again, if one writes out the expressions in detail they will not seem much dis-
similar from those of advanced calculus.
4.9.1.2. Example.
Integrals of singular 1-cubes are nothing but what are called line
integrals. To see this, assume that
A
Õ
R
2
and consider a 1-form w on
A
. We can write
w in the form
w=
a dx
+
b dy,
for some functions a, b :
A
Æ
R
. In classical language, the integral of w along (over)
a singular 1-cube (curve) c : [0,1] Æ
A
is called a
line integral
along the curve c.
Furthermore, if c(t) = (c
1
(t),c
2
(t)), then
1
Ú
Ú
(
¢
()
)
Ú
(
¢
()
)
w
=
w
c t dt
=
w
c t dt
[]
c
01
,
0
1
Ú
[
(
()
)
¢
()
+
(
()
)
¢
()
]
=
act c t
bct c t dt
.
1
2
0
Line integrals are often used in physics. For example, one might want to integrate a
force field along curve.
4.9.1.3. Example.
Consider integrals of 2-cubes, which correspond to classical
surface integrals. Assume that
A
Õ
R
3
and consider a 2-form w on
A
. Let c : [0,1]
2
Æ
A
be a singular 2-cube in
A
.
1
1
ÚÚ
ÚÚ
Ú
Ú
(
)
w
=
c
*
w
=
w
c
,
c
dxdy
,
xy
2
c
I
0
0
where
∂
∂
c
x
∂
∂
c
y
c
=
and
c
=
.
x
y
We can write w in the form
w=
a dx
Ÿ+
dy
b dx
Ÿ+
dz
c dy
Ÿ
dz,
