1.
(a)
[7]
Find the
constant
such that
, where
0
≠
k
0
)
(
=
∇
⋅
∇
k
r
r
=
r
,
r
.
0
≠
=
)
,
,
(
z
y
x
=
⋅
∇
=
⋅
∇
=
∇
⋅
−
−
)
(
)
(
)
(
2
1
r
r
k
k
k
r
k
r
r
k
r
∇
2
3
2
)
1
(
]
)
2
(
)
3
(
[
−
−
−
+
=
⋅
−
+
k
k
k
r
k
k
r
r
k
r
k
r
r
= 0
for
. Here the identity 5.28 from the text was used to shorten calculations.
0
1
≠
−
=
k
(b)
[8]
Evaluate
, where
∫
⋅
C
d
curl
x
F
)
(
)
,
,
(
)
,
,
(
z
yz
y
xz
z
y
x
−
+
=
F
and
C
is the segment of the
curve of intersection of the plane
x
y
=
and the paraboloid
from (0,0,0) to (1,1,2).
2
2
y
x
z
+
=
Direct calculation shows that
)
1
,
,
(
−
=
x
y
curl
F
. Setting
x
=
t
, we get
y
=
t
and
,
2
2
t
z
=
hence the parametrization of
C
is
g
and we get
1
0
),
2
,
,
(
)
(
2
≤
≤
=
t
t
t
t
t
.
1

)
2
(
)
4
,
1
,
1
(
)
1
,
,
(
)
(
1
0
1
0
2
1
0
−
=
−
=
−
=
⋅
−
=
⋅
∫
∫
∫
t
dt
t
dt
t
t
t
d
curl
C
x
F
2