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Derivatives

of vector functions.
Recall that if x

is a point of
R"
and if f
(5)
is
a scalar function of

x, then the derivative of f (if it
exists) is the vector
For some purposes, it will be convenient to denote the derivative
of
f
by a

row matrix rather than by a vector. When we do this,
we usually denote the derivative by
Df
rather than ?f. Thus
If we use this notation,.the definition of the derivative
takes the following form:
where
(
)

0
as

h
>

0.
Here the dot denotes matrix
multiplication, so we must write
h

a
column matrix in
order for the formula to work;
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is the formula that will generalize to vector functions
i
Definition. Let S be a sybset of
R~.
If

k
f
:
S
>
R
,
then

f(5) is called
a
vector function

of a
vector variable.
In
scalar form, we can write

f(5) out in the
form
Said differently,

f consists of
"k
realvalued functions of
n variables." Suppose now that

f is defined in an open ball
about the point

a. We say that

f is differentiable at

a
if
each of the functions fl(x), .
..,fk(~) is differentiable at

a
!
(in the sense already defined)
.
Furthermore, we define the
derivative of

f
at

a to be the matrix
That is,
Df(=)
is the matrix whose
1
Oth
row is the derivative
th
Dfi
(=)
of
the i
coordinate function of

f.

the' derivative 'Df (5)

of'.
.

f at


a
is. the
k
by n matrix whose entry in row i and column
j
is
it
is
often called the Jacobian matrix of

f
(5).
Another
notation for this matrix
the notation
With this notation, many of the formulas we proved for
a
scalar function
f
(5)
hold without change for a vector function

f(5).
We
consider some of them here:
Theorem
1.

The
function

f

differentiable

at

a

if and only

if
where

g(&)
>
0 as h
0.
_I


\
(Here

f,
5,
and

E
are written as column matrices.)
Proof:
Both sides of this equation represent column
th
matrices.
If we consider the
I
entries of these
matrices,
we have the following equation:
NOW

f
differentiable at

a if and only if each function
is.
And
fi

a if and only
if
Ei
(&)
0 as

h

0.
But
Ei
(h)

0
as h


0,
for
each
i,

E
(h)


0
as
h


0.
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2.

If
1

f(x)

is
differentiable

at

a,

then

f

continuous

at a.
Proof.

f
differentiable at

then so
each
function
fi.
Then in particular,
fi
continuous at

a,
whence

f

a.

The general chain

rule.
Before considering the general chain rule,
let
us take the
chain rule we have already proved and reformulate
it
in terms of
matrices.
Assume
that
f
(5)
=
f(xl,.
.
.
,x
)
a scalar function
n
defined in an open ball about

and that

x
(t)
=
(xl
,
...
,
xn(t))
a parametrized curve passing through

Let

x(tO)
=

f (x)


and if

x(t)
)
to,
and we have shown that the composite
f(~(t))
and
its
derivative
given
by the equation
when
t=tO.
We
can
rewrite this formula in scalar form as follows:
or we can rewrite
in the following matrix form:
Recalling the definition of the Jacobian matrix
Df,

we see
that the latter formula can be written in the form
(Note that the matrix
Df
is
a row matrix, while the matrix
DX
d
by
its
definition
a
column matrix.)
This
the form of the chain rule that we find especially
useful, for
it
the formula that generalizes to higher dimen
sions.
Let us now consider a composite of vector functions of
vector variables.
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This note was uploaded on 01/19/2012 for the course MATH 18.024 taught by Professor Christinebreiner during the Spring '11 term at MIT.
 Spring '11
 ChristineBreiner
 Derivative, Multivariable Calculus, Scalar

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