Professor John Nachbar
Econ 4111
January 29, 2008
Multivariate Differentiation
1
Preliminaries
These notes provide an introduction to multivariate calculus. I assume that you are
already familiar with standard concepts and results from univariate calculus.
To avoid extra notational complication, I will, with a few exceptions, take the
domain of functions to be all of
R
N
. Everything generalizes immediately to functions
whose domain is some open subset of
R
N
.
In my notation, a point in
R
N
, which I also refer to as a vector (
vector
and
point
mean exactly the same thing), is written
x
= (
x
1
, . . . , x
N
) =
x
1
.
.
.
x
N
.
Thus, a vector in
R
N
always
corresponds to an
N
×
1 (column) matrix. This ensures
that the matrix multiplication below makes sense (the matrices conform).
If
f
:
R
N
→
R
M
then
f
can be written in terms of
M
coordinate functions
f
(
x
) = (
f
1
(
x
)
, . . . , f
M
(
x
))
.
Again,
f
(
x
), being a point in
R
M
, can be written as an
M
×
1 matrix.
2
Partial Derivatives and the Jacobian.
Given a function
f
:
R
N
→
R
M
, define the
partial derivative
of
f
m
with respect to
the
n
th coordinate,
x
n
, evaluated at the point
x
*
, by
D
n
f
m
(
x
*
) = lim
t
→
0
f
m
(
x
*
1
, . . . , x
*
n
+
t, . . . , x
*
N
)

f
m
(
x
*
)
t
,
assuming that this limit exists. One frequently sees the alternate notation,
∂f
m
(
x
*
)
∂x
n
=
D
n
f
m
(
x
*
)
.
If you can take derivatives in the onedimensional case, you can just as easily
take partial derivatives in the multivariate case. When taking the derivative with
respect to
x
n
, just treat the other variables like constants.
1
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Example
1
.
f
:
R
2
+
\
0
→
R
is defined by
f
(
x
1
, x
2
) =
√
x
1
+ 3
x
2
. (
R
2
+
\
0
is
R
2
+
with
the point zero removed.) Then at the point
x
*
= (1
,
1),
D
1
f
(
x
*
) =
1
2
1
p
x
*
1
+ 3
x
*
2
=
1
4
,
D
2
f
(
x
*
) =
1
2
1
p
x
*
1
+ 3
x
*
2
3 =
3
4
.
The
M
×
N
matrix of partial derivatives is called the
Jacobian
of
f
at
x
*
, denoted
Jf
(
x
*
).
Jf
(
x
*
) =
D
1
f
1
(
x
*
)
. . .
D
N
f
1
(
x
*
)
.
.
.
.
.
.
.
.
.
D
1
f
M
(
x
*
)
. . .
D
N
f
M
(
x
*
)
.
Example
2
.
In the previous example,
Jf
(
x
*
) =
h
1
/
4
3
/
4
i
.
If the partial derivatives
D
n
f
m
(
x
*
) are defined for all
x
*
in the domain of
f
then
one can define a function
D
n
f
m
.
Example
3
.
If, as above,
f
:
R
2
+
\
0
→
R
is defined by
f
(
x
1
, x
2
) =
√
x
1
+ 3
x
2
then
the functions
D
n
f
are defined by
D
1
f
(
x
) =
1
2
√
x
1
+ 3
x
2
,
D
2
f
(
x
*
) =
3
2
√
x
1
+ 3
x
2
.
And one can ask whether the
D
n
f
m
are continuous and one can compute partial
derivatives of the
D
n
f
m
, which would be second order partials of
f
. And so on.
3
Directional Derivatives.
D
n
f
m
(
x
*
) measures movement of
f
m
in response to changes in
x
n
.
One can also
measure the movement of
f
m
when more than one variable is changing at the same
time.
Formally, by a
direction
in
R
N
I simply mean any vector
v
in
R
N
such that
k
v
k
= 1.
The restriction
k
v
k
= 1 makes it easier to compare different direction
vectors; requiring
k
v
k
=
c
, for some positive integer
c
6
= 1, would work just as well
as long as it were consistently applied.
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 Fall '08
 JohnNachbar
 Derivative, x∗, jf

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