Math 317, Section 202, Homework no. 5
(due Friday February 12, 2010)
[7 problems, 26 points + 4 bonus points]
Problem 1
(12 points)
.
Compute the gradient of the following potentials.
We use the
abbreviation
r
=
h
x, y, z
i
and
r
=

r

.
(a) (4 points)
f
(
x, y, z
) =
c r
k
where
k
= 0
,
1
,

1
,
2
,

2
, . . .
and
c
is a constant.
(b) (4 points)
g
(
x, y, z
) =
c
r
e

mr
, where
c
and
m >
0 are constants.
(c) (4 points)
h
(
x, y, z
) =
c
ln
r
r
0
, where
c
and
r
0
>
0 are constants.
[Hint: Try to be as economical as possible and present the results in a concise way.
You
will need them again.
Background information:
These gradients appear in the following
applications.
∇
g
is a good approximation to the strong nuclear force for scattering processes
of not very high energy.
∇
f
for
k
=

1 is the form of the electrostatic and the gravitational
force.
∇
f
for
k
= 2 is the force that acts on an object attached to the origin by an elastic
spring.]
Problem 2
(2 points)
.
In your garden, there is a wooden fence whose base is the curve
x
2
+
y
2
= 2,
y
≥
0, and whose height is given by
h
(
x, y
) = 1 +
x
2
+
xy
2
. You plan to paint
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 Spring '10
 phfiier
 2 K, Strong interaction, Fundamental physics concepts, Gradient

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