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824
CHAPTER 15
Vector Fields
Exercises 15.3
1.
Show that the curve
C
given by
r
=
a
cos
t
sin
t
i
+
a
sin
2
t
j
+
a
cos
t
k
,(
0
≤
t
≤
π
2
),
lies on a sphere centred at the origin. Find
Z
C
zds
.
2.
Let
C
be the conical helix with parametric equations
x
=
t
cos
t
,
y
=
t
sin
t
,
z
=
t
,
(
0
≤
t
≤
2
π)
.F
ind
Z
C
.
3.
Find the mass of a wire along the curve
r
=
3
t
i
+
3
t
2
j
+
2
t
3
k
0
≤
t
≤
1
),
if the density at
r
(
t
)
is 1
+
t
g/unit length.
4.
Show that the curve
C
in Example 3 also has parametrization
x
=
cos
t
,
y
=
sin
t
,
z
=
cos
2
t
,
(
0
≤
t
≤
π/
2
)
,and
recalculate the mass of the wire in that example using this
parametrization.
5.
Find the moment of inertia about the
z
axis (i.e., the value of
δ
Z
C
(
x
2
+
y
2
)
ds
), for a wire of constant density
δ
lying
along the curve
C
:
r
=
e
t
cos
t
i
+
e
t
sin
t
j
+
t
k
, from
t
=
0
to
t
=
2
π
.
6.
Evaluate
Z
C
e
z
,where
C
is the curve in Exercise 5.
7.
Find
Z
C
x
2
along the line of intersection of the two planes
x
−
y
+
z
=
0and
x
+
y
+
2
z
=
0, from the origin to the
point
(
3
,
1
,
−
2
)
.
8.
Find
Z
C
p
1
+
4
x
2
z
2
C
is the curve of intersection
of the surfaces
x
2
+
z
2
=
1and
y
=
x
2
.
9.
Find the mass and centre of mass of a wire bent in the shape
of the circular helix
x
=
cos
t
,
y
=
sin
t
,
z
=
t
,
(
0
≤
t
≤
2
, if the wire has line density given by
δ(
x
,
y
,
z
)
=
z
.
10.
Repeat Exercise 9 for the part of the wire corresponding to
0
≤
t
≤
π
.
11.
Find the moment of inertia about the
y
axis, that is,
Z
C
(
x
2
+
z
2
)
,
of the curve
x
=
e
t
,
y
=
√
2
t
,
z
=
e
−
t
,
(
0
≤
t
≤
1
)
.
12.
Find the centroid of the curve in Exercise 11.
13.
3
Find
Z
C
xds
along the ±rst octant part of the curve of
intersection of the cylinder
x
2
+
y
2
=
a
2
and the plane
z
=
x
.
14.
3
Find
Z
C
along the part of the curve
x
2
+
y
2
+
z
2
=
1,
x
+
y
=
1, where
z
≥
0.
15.
3
Find
Z
C
(
2
y
2
+
1
)
3
/
2
C
is the parabola
z
2
=
x
2
+
y
2
,
x
+
z
=
1.
Hint:
Use
y
=
t
as parameter.
16.
Express as a de±nite integral, but do not try to evaluate, the
value of
Z
C
xyzds
C
is the curve
y
=
x
2
,
z
=
y
2
from
(
0
,
0
,
0
)
to
(
2
,
4
,
16
)
.
17.
3
The function
E
(
k
,φ)
=
Z
φ
0
p
1
−
k
2
sin
2
tdt
is called the
elliptic integral function of the second
kind
.The
complete elliptic integral
of the second kind is
the function
E
(
k
)
=
E
(
k
,π/
2
)
. In terms of these functions,
express the length of one complete revolution of the elliptic
helix
x
=
a
cos
t
,
y
=
b
sin
t
,
z
=
ct
,
where 0
<
a
<
b
. What is the length of that part of the helix
lying between
t
=
t
=
T
,where0
<
T
<π/
2?
18.
3
Evaluate
Z
L
x
2
+
y
2
L
is the entire straight line with
equation
Ax
+
By
=
C
,
(
C
6=
0
)
.
Hint:
Use the symmetry
of the integrand to replace the line with a line having a
simpler equation but giving the same value to the integral.
15.4
Line Integrals of Vector Fields
In elementary physics the
work
done by a constant force of magnitude
F
in moving
an object a distance
d
is de±ned to be the product of
F
and
d
:
W
=
Fd
.T
h
e
r
e
is, however, a catch to this; it is understood that the force is exerted in the direction
of motion of the object. If the object moves in a direction different from that of the
force (because of some other forces acting on it), then the work done by the particular
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This note was uploaded on 03/21/2012 for the course STAT 101 taught by Professor Graham during the Spring '08 term at Iowa State.
 Spring '08
 Graham

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