Graphics Reference
In-Depth Information
as “one-sided” limits, extending the idea of the derivative at the endpoints of a
function defined on a closed interval [a,b]. All the theorems and definitions in this
section will be applicable.
4.4
The Inverse and Implicit Function Theorem
Definition.
Let
U
and
V
be open subsets of
R
n
.A C
k
map f :
U
Æ
V
, k =•or k ≥ 1,
that has a C
k
inverse is called a C
k
diffeomorphism
of
U
onto
V
. A C
•
diffeomorphism
will be called simply a
diffeomorphism
.
Because diffeomorphisms are one-to-one and onto maps, one can think of them
as defining a change of coordinates. Another definition that often comes in handy is
the following:
Definition.
Let
U
be an open subset of
R
n
and let f :
U
Æ
R
n
. If
p
Œ
U
, then f is called
a
local (C
k
) diffeomorphism at
p
if f is a (C
k
) diffeomorphism of an open neighborhood
of
p
onto an open neighborhood of f(
p
).
If a differentiable map f :
B
n
(r) Æ
R
n
satisfies
4.4.1. Lemma.
∂
∂
f
x
i
£
b
,
for all i and j
,
j
then it satisfies the Lipschitz condition
()
-
()
£
n
f
xy
f
bn
xy
-
,
for all
xyB
,
Œ
.
Proof.
Use Taylor series and the mean-value theorem.
4.4.2. Theorem.
(The Inverse Function Theorem) Let
U
be an open subset of
R
n
. Let f :
U
Æ
R
n
be a C
k
function, k ≥ 1, and assume that Df(
x
0
) is nonsingular at
x
0
Œ
U
. Then f is a local C
k
diffeomorphism.
Outline of proof.
By composing f with linear maps if necessary one may assume
that
x
0
= f(
x
0
) =
0
and that Df(
x
0
) is the identity map. Next, let g(
x
) = f(
x
) -
x
. It follows
that Dg(
0
) is the zero map and since f is at least C
1
, there is a small neighborhood
B
n
(r) about the origin so that
∂
∂
g
x
1
2
i
j
£
.
n
For each
y
Œ
B
n
(r/2) there is a unique
x
Œ
B
n
(r) such that f(
x
) =
y
.
Claim.
See Figure 4.10. To prove the existence of
x
note that Lemma 4.4.1 implies
that
