Graphics Reference
In-Depth Information
Definition.
The map
(
)
Æ
(
)
km
kn
f
*: W
R
W
R
defined by
(
)( )
=
(
(
()
)
(4.33)
f
*
w
p
f
*
w
f
p
,
is called the
induced map
on differential forms. (The map f* on the right-hand side of
equation (4.33) is the induced map referred to by expression (4.32).) The form f*w is
often called the
pullback
to
R
n
of the form w on
R
m
.
Using equation (4.25f), the action of the induced map f* on differential k-forms
can be described in more detail as follows: Let
v
i
Œ
R
n
. Then
(
)
=
(
(
(
)
(
)
(
)
)
(
)( ) (
)
(
)
(
)
()
( )
()
()
f
*
w
pv
,
v
,...,
v
w
f
p
f
v
,
f
v
,...,
*
f
v
.
(4.34)
1
2
k
*
1
*
2
k
p
p
p
p
p
p
The next theorem lists the main properties of the induced map f* that enables us
to compute the map easily.
If f :
R
n
Æ
R
m
is a differentiable function, then the induced map
4.9.3. Theorem.
(
)
Æ
(
)
km
kn
f
*: W
R
W
R
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*w Ÿ f*h
n
n
∂
∂
f
x
Â
Â
i
*
(
)
=
(4)
fdx
Dfdx
=
dx
i
j i
j
j
j
j
=
1
j
=
1
(5) f*(g dx
1
Ÿ dx
2
Ÿ ...Ÿ dx
n
) = (g
∞
f)(det f¢) dx
1
Ÿ dx
2
Ÿ ...Ÿ dx
n
Proof.
The proofs of (1)-(4) are simply a case of expanding all the expressions using
their definitions. As an example, (4) is proved by the following equalities:
(
)( )(
)
=
(
()
)
(
(
)
)
fdx
*
pv
dxf
p
f
v
i
p
i
p
*
n
n
n
Ê
Á
ˆ
˜
Â
Â
Â
(
()
)
()
()
()
=
dx f
p
v Df
p
,
v Df
p
,...,
v Df
p
i
j
j
1
j
j
2
j
j m
j
=
1
j
=
1
j
=
1
n
Â
=
vDf
()
p
j
j i
j
=
1
n
Â
1
()
( ( )
=
Df
p
dx
pv
p
.
ji
j
j
=
For a proof of (5), see [Spiv65].
