Graphics Reference
In-Depth Information
3
y-=
0
has a horizontal tangent at (0,0) but still can also be solved for both x and y in a neigh-
borhood of that point.
4.4.7. Theorem.
(The Implicit Function Theorem) Let
n
m m
f
:
RR R
¥Æ
be a continuously differentiable function in an open set about a point (
a
,
b
) and
assume that
(
)
=
f
ab
,
0
.
If the m ¥ m matrix
(
(
)
)
MDf
=
ab
,
nij
+
1
££
ij m
,
is nonsingular, then there exists an open neighborhood
A
about
a
in
R
n
, an open neigh-
borhood
B
about
b
in
R
m
, and a differentiable function
g:
AB
Æ
with the property that
(
,
()
)
=
f
xx 0
for all
x
in
A
.
Proof.
Define a function
n
m n
m
F
:
RR RR
¥Æ¥
by
(
)
=
(
(
)
)
F
xy
,
x xy
,
f
,
.
Our hypotheses imply that F is a continuously differentiable function whose deriva-
tive DF is nonsingular at (
a
,
b
) because det DF(
a
,
b
) = det M. The inverse function
theorem (Theorem 4.4.2) now implies that F has an inverse G in an open neighbor-
hood of (
a
,
0
) = F(
a
,
b
), which we may assume to have the form
A
¥
B
. It is easy
to check that G has the form G(
x
,
y
) = (
x
,k(
x
,
y
)) for some differentiable function k.
Let
n
m m
p:
RR R
¥Æ
be the natural projection. Then p
∞
F = f and
