Lecture 3
2
Differentiation
2.1
Differentiation in
n
dimensions
We
are
setting out
to generalize
to
n
dimensions the notion
of differentiation
in
one
dimensional
calculus. We
begin with
a
review
of elementary
onedimensional calculus.
Let
I
⊆
R
be
an open interval, let
f
:
I
→
R
be a
map, and
let
a
∈
I
.
Definition 2.1.
The
derivative
of
f
at
a
is
f
�
(
a
)
=
lim
f
(
a
+
t
)
−
f
(
a
)
,
(2.1)
t
0
t
→
provided
that
the
limit
exists.
If
the limit
exists, then
f
is differentiable at
a
.
There
are
half
a dozen or
so
possible reasonable generalizations
of the notion
of derivative
to higher
dimensions.
One candidate generalization
which
you
have
probably already encountered is
the directional derivative.
Definition 2.2.
Given an open set
U
in
R
n
, a
map
f
:
U
→
R
m
, a
point
a
∈
U
, and
a
point
u
∈
R
n
,
the
directional derivative of
f
in the direction of
u
at
a
is
D
u
f
(
a
) =
lim
f
(
a
+
tu
)
−
f
(
a
)
,
(2.2)
t
0
t
→
provided that
the
limit
exists.
In
particular,
we
can calculate the directional derivatives
in
the direction
of the
standard basis
vectors
e
1
, . . . , e
n
of
R
n
, where
e
1
=
(1
,
0
, . . . ,
0)
,
(2.3)
e
2
=
(0
,
1
,
0
, . . . ,
0)
,
(2.4)
.
.
.
(2.5)
e
n
=
(0
, . . . ,
0
,
1)
.
(2.6)
Notation.
The
directional
derivative in
the direction
of a
standard
basis
vector
e
i
of
R
n
is denoted by
∂
D
i
f
(
a
) =
D
e
i
f
(
a
) =
f
(
a
)
.
(2.7)
∂x
i
We
now
try to answer
the
following
question: What is
an
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 Fall '04
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 Calculus, Linear Algebra, Derivative, basis, Standard basis

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