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MAT 473
1
1.
Euclidean Space, Linear Maps.
Recall that a function
f
:
R
→
R
is
diferentiable
at
x
∈
R
if
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
exists. When it does, we call the value of the limit the
derivative
of
f
at
x
, and denote it
by
f
°
(
x
). Equivalently,
f
is diFerentiable at
x
if there exists
c
∈
R
such that
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
−
ch
h
=0
.
(Take
c
=
f
°
(
x
).) Equivalently,
f
is diFerentiable at
x
if there exists a linear function
T
:
R
→
R
such that
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
−
T
(
h
)
h
=0
.
(Take
T
(
h
)=
ch
.) Equivalently,
f
is diFerentiable at
x
if there exists a linear function
T
:
R
→
R
such that
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
−
T
(
h
)

h

=0
.
Since we can talk about limits and linear functions and 0 and we can divide by

h

in higher
dimensional spaces, this reformulation suggests a way to de±ne derivatives of vectorvalued
functions of several variables. ²irst we need some notation and basic facts concerning linear
maps.
I’ll use the symbol
R
n
to denote the set of all real
n
×
1 column vectors
x
=
x
1
.
.
.
x
n
.T
o
save space, column vectors are often written as
n
tuples
x
=(
x
1
,...,x
n
). (The commas
distinguish an
n
tuple from a 1
×
n
row vector.)
R
n
is a vector space over
R
under the
componentwise operations
(
x
1
,...,x
n
)+(
y
1
,...,y
n
)=(
x
1
+
y
1
,...,x
n
+
y
n
)
and
c
(
x
1
,...,x
n
)=(
cx
1
,...,cx
n
)
.
In fact,
R
n
is a
normed vector space
with the norm de±ned by
°
x
°
=
°
x
2
1
+
···
x
2
n
,
which means that
(i)
°
x
°≥
0 for all
x
,and
°
x
°
= 0 if and only if
x
=
0
=(0
,
0
,...,
0);
(ii)
°
c
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This document was uploaded on 03/11/2012.
 Spring '09
 Derivative

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