(8/17/08)
Math 20C. Lecture Examples.
Section 14.5, Part 1. Directional derivatives and gradient vectors in the plane
†
The
x
derivative
f
x
(
a, b
) is the derivative of
f
at (
a, b
) in the direction of the unit vector
u
, and the
y
derivative
f
y
(
a, b
) is the derivative in the direction of the unit vector
j
. To find the derivative of
z
=
f
(
x, y
) at (
a, b
) in the direction of an arbitrary unit vector
u
=
(
u
1
, u
2
)
, we introduce an
t
axis, as
in Figure 1, with its origin at (
a, b
), with its positive direction in the direction of
u
, and with the scale
used on the
x
 and
y
axes. Then the point at
t
on the
t
axis has
xy
coordinates
x
=
a
+
tu
1
, y
=
b
+
tu
2
,
and the value of
z
=
f
(
x, y
) at the point
t
on the
t
axis is
F
(
t
) =
f
(
a
+
tu
1
, b
+
tu
2
)
.
(1)
We call
z
=
F
(
t
) the
cross section
through (
a, b
) of
z
=
f
(
x, y
) in the direction of
u
. Its
t
derivative
at
t
= 0 is the directional derivative of
f
at (
a, b
).
braceleftbigg
x
=
a
+
tu
1
y
=
b
+
tu
2
Tangent line of slope
F
prime
(0) =
D
u
f
(
a, b
)
FIGURE 1
FIGURE 2
Definition 1
The directional derivative of
z
=
f
(
x, y
)
at
(
a, b
)
in the direction of the unit vector
u
=
(
u
1
, u
2
)
is the derivative of the cross section function
(1)
at
t
= 0
:
D
u
f
(
a, b
) =
bracketleftbigg
d
dt
f
(
a
+
tu
1
, b
+
tu
2
)
bracketrightbigg
t
=0
.
(2)
The directional derivative
(2)
is the rate of change of
f
at (
a, b
) in the direction of
u
. Its geometric
meaning is shown in Figure 2. We introduce a second vertical
z
axis with its origin at (
a, b
) as in Figure 2.
Then the graph of
z
=
F
(
t
) is the intersection of the surface
z
=
f
(
x, y
) with the
tz
plane and the
directional derivative of
z
=
f
(
x, y
) is the slope of the tangent line to this curve in the positive
t
direction
at
t
= 0.
†
Lecture notes to accompany Section 14.5, Part 1 of
Calculus, Early Transcendentals
by Rogawski.
1
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Math 20C. Lecture Examples. (8/17/08)
Section 14.5, Part 1, p. 2
The next theorem is used to calculate directional derivatives from partial derivatives.
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 Spring '08
 Helton
 Calculus, Derivative, Vectors, Dot Product, Gradient

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