Graphics Reference
In-Depth Information
Let f :
M
Æ
R
and let (
U
,j), j :
U
Æ
R
n
8.12.1. Lemma.
be a coordinate neighbor-
hood for
M
. If
()
=
(
()
()
()
)
j
p
uu
p
,
p
,...,
n
p
,
1
2
then
∂
∂
f
u
∂
∂
f
u
++
∂
∂
f
u
df
=
du
+
du
...
du
(8.32)
1
2
n
1
2
n
on
U
.
Proof.
This is a straightforward consequence of the definitions and Theorem
4.9.2.
Given a map f :
M
n
Æ
N
m
Definition.
between differentiable manifolds, there is a
well-defined
induced map
(
)
Æ
(
)
km
k n
f
*: W
N
W
M
(8.33)
that is defined just like the map in equations (4.33) and (4.34) in Section 4.9.
Let f :
M
n
Æ
N
m
and g :
N
m
8.12.2. Theorem.
Æ
R
be differentiable functions. The
map f* on differentiable forms satisfies
(1) f*(w
1
+w
2
) = f*(w
1
) + f*(w
2
)
(2) f*(gw) = (g f) f*w
(3) f*(wŸh) = f*wŸf*h
(4) Assume that n = m and let
p
Œ
M
and
q
= f(
p
). If (
U
,j) and (
V
,y) are coordi-
nate neighborhoods for
p
in
M
and
q
in
N
, respectively, with
o
()
=
(
()
()
()
)
( )
=
(
()
()
()
)
j
p
u
p
,
u
p
,...,
u
p
and
y
q
v
q
,
v
q
,...,
v
q
,
1
2
n
1
2
n
then
∂
(
vf
u
o
)
Ê
Ë
ˆ
¯
i
(
)
=
(
)
f
*
g dv
ŸŸŸ
dv
...
dv
g
o
f
det
du
ŸŸŸ
du
...
du
.
(8.34)
1
2
n
1
2
n
∂
j
Proof.
The theorem basically follows from Theorem 4.9.3. More details can be found
in [Spiv70a].
Now let w be a k-form on
M
. Given any coordinate neighborhood (
U
,j) for
M
,
express w as in equation (8.31) with respect to (
U
,j). Define a (k + 1)-form dw over
U
by
Â
d
w
=
d
w
Ÿ
du
Ÿ
...
Ÿ
du
(8.35a)
i
...
i
i
i
1
k
1
k
1
£< < £
i
...
i
n
1
k
n
∂
w
i
...
i
Â
Â
=
1
k
du
ŸŸŸ
du
...
du
.
(8.35b)
j
i
i
1
k
∂
u
j
1
£< < £
i
...
i
n
j
=
1
1
k
