22. The cylinder
x
2
+ 4
y
2
= 4 is intersected with the plane
x
+ 3
y
+ 2
z
= 1
.
This
yields a closed curve,
C.
Orient this curve in the counter clockwise direction when
viewed from a point high on the
z
axis. Let
F
=
(
y, z
+
y, x
2
)
. Find
R
C
F
·
d
R
.
23. The cylinder
x
2
+
y
2
= 4 is intersected with the plane
x
+ 3
y
+ 2
z
= 1
.
This yields
a closed curve,
C.
Orient this curve in the clockwise direction when viewed from
a point high on the
z
axis. Let
F
= (
y, z
+
y, x
). Find
R
C
F
·
d
R
.
24. Let
F
=
(
xz, z
2
(
y
+ sin
x
)
, z
3
y
)
.
Find the surface integral,
R
S
curl (
F
)
·
n
dA
where
S
is the surface,
z
= 4

(
x
2
+
y
2
)
,
z
≥
0
.
25. Let
F
=
(
xz,
(
y
3
+
x
)
, z
3
y
)
.
Find the surface integral,
R
S
curl (
F
)
·
n
dA
where
S
is the surface,
z
= 16

(
x
2
+
y
2
)
,
z
≥
0
.
410
STOKES AND GREEN’S THEOREMS
26. The cylinder
z
=
y
2
intersects the surface
z
= 8

x
2

4
y
2
in a curve,
C
which is
oriented in the counter clockwise direction when viewed high on the
z
axis. Find
R
C
F
·
d
R
if
F
=
‡
z
2
2
, xy, xz
·
.
Hint:
This is not too hard if you show you can use
Stokes theorem on a domain in the
xy
plane.
27. Suppose solutions have been found to 18.17, 18.16, and 18.12. Then define
E
and
B
using 18.14 and 18.13. Verify Maxwell’s equations hold for
E
and
B
.
28. Suppose now you have found solutions to 18.17 and 18.16,
ψ
1
and
A
1
.
Then go
show again that if
φ
satisfies 18.10 and
ψ
≡
ψ
1
+
1
c
∂φ
∂t
,
while
A
≡
A
1
+
∇
φ,
then
18.12 holds for
A
and
ψ.
29. Why consider Maxwell’s equations?
Why not just consider 18.17, 18.16, and
18.12?
30. Tell which open sets are simply connected.
(a) The inside of a car radiator.
(b) A donut.
(c) The solid part of a cannon ball which contains a void on the interior.
(d) The inside of a donut which has had a large bite taken out of it.
(e) All of
R
3
except the
z
axis.
(f) All of
R
3
except the
xy
plane.
31. Let
P
be a polygon with vertices (
x
1
, y
1
)
,
(
x
2
, y
2
)
,
· · ·
,
(
x
n
, y
n
)
,
(
x
1
, y
1
) encoun
tered as you move over the boundary of the polygon in the counter clockwise
direction. Using Problem 13, find a nice formula for the area of the polygon in
terms of the vertices.
The Mathematical Theory Of
Determinants
*
A.1
The Function
sgn
n
It is easiest to give a different definition of the determinant which is clearly well defined
and then prove the earlier one in terms of Laplace expansion.
Let (
i
1
,
· · ·
, i
n
) be an
ordered list of numbers from
{
1
,
· · ·
, n
}
.
This means the order is important so (1
,
2
,
3)
and (2
,
1
,
3) are different. There will be some repetition between this section and the
earlier section on determinants.
The main purpose is to give all the missing proofs.
Two books which give a good introduction to determinants are Apostol [2] and Rudin
[23]. A recent book which also has a good introduction is Baker [4].
The following Lemma will be essential in the definition of the determinant.
Lemma A.1.1
There exists a unique function,
sgn
n
which maps each list of numbers
from
{
1
,
· · ·
, n
}
to one of the three numbers,
0
,
1
,
or

1
which also has the following
properties.