Graphics Reference
In-Depth Information
Then there exist C
s
diffeomorphisms h and g of neighborhoods of the origins in
R
n
and
R
m
, respectively, such that
(
(
(
)
)
)
=
(
)
gfhx
,...,
x
x
,...,
x
, ,...,
00
1
n
1
k
in some neighborhood of the origin in
R
n
.
Proof.
The proof divides into two cases.
Case 1.
n ≥ m.
Let p :
R
m
Æ
R
k
be the natural projection. We may assume without loss of gener-
ality that the matrix
∂
∂
f
x
Ê
Á
ˆ
˜
i
j
1,
££
ij k
is nonsingular on
U
. It follows from Lemma 4.4.4 (applied to p of) that there is a C
s
diffeomorphism h such that
(
(
)
(
(
)
(
)
)
f h x
,...,
x
=
x
,...,
x
,
f
x
,...,
x
,...,
f
x
,...,
x
.
1
n
1
k
1
1
n
m
-
k
1
n
o
Since the rank of D(f h) is k, we must have ∂f
i
/∂x
j
= 0 for j > k. It follows that the f
i
are independent of x
k+1
,..., x
n
. Now define a map f
1
:
R
k
Æ
R
m
by
(
)
=
(
(
)
(
)
)
fx
,...,
x
x
,...,
x
,
f
x
,...,
x
, ,...,
00
,...,
f
x
,...,
x
, ,...,
00
11
k
1
k
11
k
m
-
k
1
k
and apply Lemma 4.4.3.
Case 2.
n £ m.
This is proved similarly to Case 1 and is left as an exercise.
One aspect worth noting about the hypotheses of Theorem 4.4.5 is that it is not
enough to simply assume that the map f has rank k at the origin. One needs to know
that this holds in a neighborhood of the origin. In the special case where Df has
maximal rank, then the assumption of rank k at the origin
is
enough because this by
itself implies that Df will have maximal rank in a neighborhood. This explains the
slight difference in hypotheses between Lemmas 4.4.3 and 4.4.4 and Theorem 4.4.5.
It is worth summarizing an aspect of these observations.
4.4.6. Theorem.
Let
U
be an open subset of
R
n
. Let f :
U
Æ
R
m
, n £ m, be a C
k
map,
k ≥ 1. Let
p
be a point of
U
and assume that Df(
p
) has rank n. Then there is a neigh-
borhood of
V
of
p
in
U
so that f |
V
is one-to-one.
Implicit function theorems are another major application of the inverse function
theorem. They have to do with solving equations of the form
(
)
= 0
fxy
,
(4.9)
