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
.
Defnition 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.
Defnition 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:
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 Fall '04
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 Calculus, Linear Algebra, Derivative, basis, Standard basis

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