Math 317, Section 202, Homework no. 8
(due Friday March 19, 2010)
[8 problems, 40 points + 5 bonus points]
Problem 1
(4 points)
.
Suppose
F
(
x,y,z
) =
g
(
r
)
r
with
r
=
h
x,y,z
i
,
r
=

r

and a continu
ously diﬀerentiable function
g
(
r
),
r >
0. Find all such functions
g
(
r
) for which
∇ ·
F
= 0.
Problem 2
(4 points)
.
Suppose
f
(
x,y,z
) =
h
(
r
) with
r
=

r

,
r
=
h
x,y,z
i
and some twice
continuously diﬀerentiable function
h
(
r
),
r >
0. Find all such functions
h
(
r
) for which
f
satisﬁes Laplace’s equation.
Problem 3
(6 points, 2 points each)
.
Suppose
F
(
x,y,z
) is a vector ﬁeld, and
f
(
x,y,z
) and
g
(
x,y,z
) are real valued functions such that all second order partial derivatives exist. Show
that
(a)
∇ ×
(
f
F
) =
f
(
∇ ×
F
) + (
∇
f
)
×
F
,
(b)
∇ ·
(
f
F
) =
f
(
∇ ·
F
) + (
∇
f
)
·
F
,
(c)
∇ ·
(
f
∇
g

g
∇
f
) =
f
∇
2
g

g
∇
2
f.
Problem 4