Math 255
Winter 2010
Review problems for Final Exam
1. The electrostatic potential at (0
,
0
,

a
) of a charge of constant density
σ
on the hemisphere
S
:
x
2
+
y
2
+
z
2
=
a
2
,
z
≥
0 is
U
=
S
σ
x
2
+
y
2
+ (
z
+
a
)
2
dS.
Show that
U
= 2
πσ
a
(2

√
2).
Solution
The surface is a graph
z
=
a
2

x
2

y
2
.
We therefore project the integral onto the
xy
plane and convert to polar coordinates to find that we need to calculate
2
π
0
a
0
σ
2
a
2
+ 2
a
√
a
2

r
2
a
√
a
2

r
2
r
d
r
d
θ
.
After integrating with respect to
θ
we make the substitution
w
=
a
2

r
2
to find that
√
2
a
πσ
a
2
0
d(
a
+
√
w
)
a
+
√
w
= 2
πσ
a
(2

√
2)
.
1
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2. Find the value of the line integral
C
yzdx
+
xzdy
+
xydz
where
C
is the curve with initial point
P
0
= (1
,
0
,
0) given by.
a)
x
= cos
t
,
y
= sin
t
,
z
=
t
2
with
t
∈
[0
,
π
].
b) The line segment between
P
0
and (

1
,
0
,

π
).
c) The ellipse
x
2
+ 4
y
2
= 1 and
z
= 0 parametrized clockwise.
d) The circle
x
2
+
y
2
= 9 and
z
= 0 parametrized counterclockwise as viewed from above. Use the
fact that the path C is the boundary of the region
x
2
+
y
2
≤
9 with
z
= 0. Use Green’s Theorem
to compute value of the line integral.
e) The same path as in part
d
), but now using Stokes Theorem. Use the fact that the path C in this
case is the boundary of the hemisphere
x
2
+
y
2
+
z
2
= 9 with
z
≤
0.
f) Can you use the fundamental theorem of line integrals to compute the value of any of the above
line integrals? Explain.
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 Fall '08
 JackWaddell
 Math, Vector Calculus, Cos, Line integral, Stokes' theorem

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