Section 15.6
Directional Derivatives and the Gradient Vector
“Finding rates of change in different directions”
Recall that when we first started considering derivatives of functions
of more than one variable, we had to introduce the concept of a partial
derivative since there is more than one variable we can differentiate
with respect to. For a function of two variables, the partial derivatives
f
x
and
f
y
had a geometric interpretation  specifically, they represented
the rate of change of
f
in the
vector
i
direction and
vector
j
directions respectively.
In this section we consider the more general case of determining the
rate of change of a function
f
in the direction of any arbitrary vector
vectorv
by using partial derivaitives.
1.
Directional Derivatives
As with single variable calculus, by “the rate of change of
f
in the
direction of the vector
vectorv
”, we mean the slope of a tangent line to
f
at the point (
a,b
) in the direction of
vectorv
. In order to calculate this, we
simply generalize the ideas from single variable calculus.
•
Suppose that we want to find the rate of change of
f
(
x,y
) in
the direction of some unit vector
vectoru
=
u
1
vector
i
+
u
2
vector
j
at the point
P
(
a,b
) (it is important that
vectoru
is a unit vector or this will not
work).
•
Let
Q
= (
a
+
u
1
h,b
+
u
2
h
) (so the point in the
xy
plane a
distance
h
from
P
along the vector
vectoru
).
h
P
Q
•
The average rate of change from
P
to
Q
can be calculated
using a difference quotient.
Specifically, the average rate of
change will equal
f
(
Q
)
−
f
(
P
)

vector
PQ

=
f
(
a
+
u
1
h,b
+
u
2
h
)
−
f
(
a,b
)
h
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
(observe that since
vectoru
is a unit vector, the length of
vector
PQ
will be
h
(WHY?)).
•
As with single variable, we can take the limit
h
→
0, and as
h
gets smaller, the difference quotient approximates the slope
of
f
at
P
in the direction of
vectoru
more accurately.
This motivates the following definition.
Definition 1.1.
The directional derivative of
f
(
x,y
) at (
a,b
) in the
direction of a unit vector
vectoru
is
D
vectoru
f
(
a,b
) = lim
h
→
0
f
(
a
+
u
1
h,b
+
u
2
h
)
−
f
(
a,b
)
h
if this limit exists.
Example 1.2.
For an arbitrary function
f
(
x,y
), determine the direc
tional derivatives
D
vector
i
f
(
x,y
) and
D
vector
j
f
(
x,y
).
By definition,
D
vector
i
f
(
x,y
) = lim
h
→
0
f
(
x
+
h,y
)
−
f
(
x,y
)
h
=
f
x
and
D
vector
j
f
(
x,y
) = lim
h
→
0
f
(
x
+
u
1
h,y
)
−
f
(
x,y
)
h
=
f
y
.
Example 1.3.
If the following is a contour diagram for
f
(
x,y
) with
the
z
= 0 contour at the origin, going up by 1 for each concentric circle,
approximate the rate of change of
f
(
x,y
) at (1
,
1) in the direction of
vectoru
=
vector
i
+
vector
j
.
Drawing a vector out from the point (1
,
1) in the direction of
vectoru
=
vector
i
+
vector
j
,
we can use a difference quotient to approximate the rate of change.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Stefanov
 Calculus, Derivative, Du f

Click to edit the document details