1
Mathematics 1c: Homework Set 2
Due: Monday, April 12th by 10am.
1. (10 Points)
Section 2.5, Exercise 8
Suppose that a function is given in
terms of rectangular coordinates by
u
=
f
(
x,y,z
)
. If
x
=
ρ
cos
θ
sin
φ,y
=
ρ
sin
θ
sin
φ,z
=
ρ
cos
φ,
express the partial derivatives
∂u/∂ρ,∂u/∂θ,
and
∂u/∂φ
in terms of
∂u/∂x,∂u/∂y
, and
∂u/∂z
.
2. (10 Points)
Section 2.5, Exercise 12
Suppose that the temperature at the
point
(
x,y,z
)
in space is
T
(
x,y,z
) =
x
2
+
y
2
+
z
2
. Let a particle follow the
right circular helix
σ
(
t
) = (cos
t,
sin
t,t
)
and let
T
(
t
)
be its temperature at time
t
.
(a)
What is
T
0
(
t
)
?
(b)
Find an approximate value for the temperature at
t
= (
π/
2) + 0
.
01
.
3. (10 Points)
Section 2.6, Exercise 3(c)
Compute the directional derivative
of the function
f
(
x,y,z
) =
xyz
at the point
(
x
0
,y
0
,z
0
) = (1
,
0
,
1)
in the direction of the unit vector parallel to
the vector
d
= (1
,
0
,

1)
.
4. (20 Points)