Lecture 13  Wednesday April 29th
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Key words: Systems of equations, Implicit differentiation
Know how to do implicit differentiation, how to use implicit and inverse
function theorems
13.1 Examples
Recall the two main theorems of the last section:
Implicit Function Theorem
Let
f
(
x,y
) :
R
n
+
m
→
R
m
where
x
∈
R
n
and
y
∈
R
m
and let
U
be an
open ball containing a point
a
∈
R
n
. Suppose that det(
∇
f
)
6
= 0 on
U
when
f
is treated only as a function of
y
. Then there is a diﬀerentiable
function
g
:
R
n
→
R
m
such that
f
(
x,g
(
x
)) = 0 in some open ball
containing
a
.
Inverse Function Theorem
Let
f
:
R
n
→
R
n
be deﬁned on an open ball around a point
a
∈
R
n
and
suppose
f
has continuous partial derivatives on this ball. If
∇
f
is non
singular on this ball, then there exists a unique function
g
:
R
n
→
R
n
such that
f
(
g
(
x
)) =
x
in some open ball containing
a
.
We now show how to use these theorems. In the ﬁrst example, we can explicitly ﬁnd the
inverse of the function, without using the implicit function theorem.
Example 1.
The function
f
(
x,y
) = (
x
+
y,x

y
) has an inverse on
U
=
R
: we have to
solve
u
+
v
=
x
and
u

v
=
y
. Clearly this means
u
=
x
+
y
2
and
v
=
x

y
2
. In other words,
the function
g
(
x,y
) = ((
x
+
y
)
/
2
,
(
x

y
)
/
2) is an inverse of
f
(
x,y
). What this means is
that the equations
u
+
v
=
x
and
u

v
=
y
are solvable for
x
and
y
in terms of
u
and
v
.
However for most functions
f
(
x,y
) there is no hope of computing an inverse. The inverse
function theorem nevertheless tells us when an inverse exists, even though it may be
impossible to ﬁnd explicitly. Let’s use the inverse function theorem to show that the
function
f
in the last example has an inverse.
1
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View Full DocumentExample 2.
In the last example the Jacobian matrix is
∇
f
(
x,y
) =
±
1 1
1

1
¶
.
The determinant is

2, so this function
f
is invertible (i.e. the equations
u
+
v
=
x
and
u

v
=
y
are solvable for
x
and
y
in terms of
u
and
v
. Next we consider
f
(
x,y
) =
(
x
2
+
y
2
,xy
). Then
∇
f
(
x,y
) =
±
2
x
2
y
y
x
¶
.
This is nonsingular everywhere except at (0
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 Spring '07
 Enright
 Math, Systems Of Equations, Equations, Derivative, Implicit Differentiation, Continuous function, Inverse function, Function composition, implicit function theorem, Inverse Function Theorem

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