MATH 237
Tutorial 5
Wednesday June 7th, 2006
The Chain Rule, Directional Derivative and the Gradient Vector
1) Atmospheric temperature depends on position and time.
If we denote position by
three coordinates
x, y, z
(in km) and time by
t
(in hours), then the temperature
T
is a
function of four variables
T
(
x, y, z, t
).
a) If a thermometer is attached to a weather balloon that moves through the atmo
sphere on a path given by
x
(
t
)
, y
(
t
) and
z
(
t
), what is the rate of change of
T
with
respect to time recorded by the thermometer?
b) Find the rate of change of
T
with time at
t
= 1 if
T
(
x, y, z, t
) =
100
5 +
x
2
+
y
2
(1 + sin[
πt/
12])

20(1 +
z
2
) [
◦
C
]
and if the balloon moves along the curve
x
(
t
) =
t
,
y
(
t
) = 2
t
,
z
(
t
) =
t

t
4
4
.
2) Let
u
(
x, y, z
) =
F
1
x

1
y
,
1
y

1
z
and assume that
x
6
= 0
, y
6
= 0 and
z
6
= 0 and
F
is
differentiable. Show that
x
2
∂u
∂x
+
y
2
∂u
∂y
+
z
2
∂u
∂z
= 0.
3) Let
f
(
x, y
) =
H
(
xy
2
) where
H
is differentiable.
Given that
H
0
(3) = 2, calculate
∇
f
(3
,
1).
4) Show that if
u
=
u
(
x, y
) and
x
= e
s
, y
= e
t
then
∂
2
u
∂s
2
+
∂
2
u
∂t
2
=
x
2
∂
2
u
∂x
2
+
y
2
∂
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 Spring '08
 WOLCZUK
 Calculus, Chain Rule, Derivative, The Chain Rule, Continuous function, directional derivative equals

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