Notes by David Groisser for MAS 4105, Copyright c 1998
Jacobians, Directional Derivatives, and the Chain Rule.
Suppose
f
1
, f
2
, ...f
q
are functions of
p
variables
x
1
, ..., x
p
; thus for 1
≤
i
≤
q
,
f
i
(
x
1
, x
2
, ..., x
p
) is some real number. For a given point
x
= (
x
1
, x
2
, ..., x
p
)
∈
R
p
, we can
assemble the numbers
f
1
(
x
)
, f
2
(
x
)
, ..., f
q
(
x
) into a
q
component column vector
f(x)
. (We
will also write
x
as a column vector below.) Thus we obtain a map
f
:
R
p
→
R
q
.
(If one or more of the
f
i
’s is not defined at every point of
R
p
, we actually get a function
whose domain is just a subset of
R
p
, not all of
R
p
.) This is an important and sophisticated
perspective on which much of advanced calculus is based. For what we do below, it is
best to write points in
R
p
and
R
q
as column vectors, rather than row vectors.
We call
f
:
R
p
→
R
q
differentiable
at a point
a
∈
R
p
if all the partial derivatives
∂f
i
/∂x
j
(1
≤
i
≤
q,
1
≤
j
≤
p
) exist at
x
=
a
and are continuous there. (Technically this
definition is not quite right, but it will suffice for us.) At each such
a
, we define a
q
×
p
matrix
J
f
(
a
), called the
Jacobian
of
f
at
a
; it is defined by
(
J
f
(
a
))
ij
=
∂f
i
∂x
j
(
a
)
.
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 Spring '09
 RUDYAK
 Chain Rule, Derivative, rp, Jf ◦g

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