929 Chapter 15: Vector Analysis
15.1 Vector Fields
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Sketch the vector field .
a. d.
b. e.
15.1 Vector Fields
930
c.
____ 2.
Sketch several representative vectors in the vector field given by
.
a.
d.
b. e.
931 Chapter 15: Vector Analysis
c.
____ 3. Compute for the vector field given by .
a.
b.
c.
d.
e.
____ 4. Find the gradient vector for the scalar function. (That is, find the conservative vector field
for the potential function.)
a.
b.
c.
d.
e.
15.1 Vector Fields
932
____ 5.
Find the gradient vector for the scalar function. (That is, find the conservative vector
field for the potential function.)
a.
b.
c.
d.
e.
____ 6. Find the conservative vector field for the potential function by
finding its gradient.
a.
b.
c.
d.
e.
933 Chapter 15: Vector Analysis
____ 7. Determine whether the vector field is conservative.
conservative
not conservative
____ 8. Determine whether the vector field is conservative. If it is, find a potential function for the
vector field.
conservative with potential function
conservative with potential function
conservative with potential function
conservative with potential function
not conservative
____ 9. Determine whether the vector field is conservative. If it is, find a potential function for the
vector field.
a.
conservative with potential function
b.
conservative with potential function
c.
conservative with potential function
d.
conservative with potential function
e. not conservative
15.1 Vector Fields
934
____ 10. Find the curl for the vector field at the given point.
a.
b.
c.
d.
e.
____ 11. Determine whether the vector field is conservative. If it is, find a potential function for
the vector field.
conservative with potential function
conservative with potential function
conservative with potential function
conservative with potential function
not conservative
____ 12. Find the divergence of the vector field.
a.
b.
935 Chapter 15: Vector Analysis
c.
d.
e.
____ 13. Find the divergence of the vector field F given by
.
a.
b.
c.
d.
e.
____ 14. Find the divergence of the vector field at the given point.
a.
b.
c.
d.
e.
____ 15. Find the divergence at for the vector field
a.
b.
c.
d.
e.
15.1 Vector Fields
936
____ 16. Find .
a.
b.
c.
d.
e.
____ 17. Find .
a.
b.
c.
d.
937 Chapter 15: Vector Analysis
____ 18. Find .
a.
b.
c.
d.
e.
____ 19. Find for the vector field given by
.
a.
b.
c.
d.
e.
15.1 Vector Fields 938
15.1 Vector Fields
Answer Section
939 Chapter 15: Vector Analysis
15.2 Line Integrals
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find a piecewise smooth parametrization of the path C given in the following graph.
a.
b.
c.
d.
15.2 Line Integrals
940
e.
____ 2. Find a piecewise smooth parametrization of the path C given in the following graph.
a.
b.
c.
d.
e.
____ 3. Evaluate the line integral along the given path.
a.
b.
c.
d.
e.
941
Chapter 15: Vector Analysis
____
4.
Evaluate
along the path C, defined as counterclockwise along the
circle
from
to
.
a.
b.
c.
d.
e.
____ 5. Evaluate along the path C, defined as y-axis from .
a.
b.
c.
d.
e.
____ 6. Find the total mass of the wire with density .
a.
b.
c.
15.2 Line Integrals
942
d.
e.
____ 7. Evaluate where C is represented by .
a.
b.
c.
d.
e.
____ 8. Evaluate where C is represented by .
a.
b.
c.
d.
e.
943 Chapter 15: Vector Analysis
____ 9. Find the work done by the force field on a particle moving along the given path.
from to .
a.
b.
c.
d.
e.
____
10.
Evaluate
where
and
.
a.
b.
c.
d.
e.
____
11.
Evaluate the integral
along the path C, defined as y-axis
from
.
a.
b.
c.
d.
e.
15.2 Line Integrals
944
____ 12.
Evaluate the integral
along the path C, defined as
from
a.
b.
c.
d.
e.
____ 13. Find the area of the lateral surface over the curve C in the xy-plane and under the
surface where Lateral surface .
line from .
a.
b.
c.
d.
e.
____ 14. A tractor engine has a steel component with a circular base modeled by the
vector-valued function . Its height is given by . (All measurements of
the component are given in centimeters.) Find the lateral surface area of the component. Round your
answer to two decimal places.
a.
b.
c.
d.
e.
945
Chapter 15: Vector Analysis
____
15.
Find the moments of inertia for a wire that lies along
with density
.
a.
b.
c.
d.
e.
____
16.
Find the work done by a person weighing
pounds walking exactly one
revolution up a circular helical staircase of radius feet if the person rises
feet.
a.
b.
c.
d.
e.
15.2 Line Integrals
946
15.2 Line Integrals
Answer Section
947 Chapter 15: Vector Analysis
15.3 Conservative Vector Fields and Independence of Path
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1.
Calculate the line integral along
for
and C is any
path starting at the point
and ending at
.
a.
b.
c.
d.
e.
____ 2. Calculate the line integral along for and C is any path starting
at the point and ending at .
a.
b.
c.
d.
e.
15.3 Conservative Vector Fields and Independence of Path
948
____ 3.
For the vector field
, find the value of A for which the
field is conservative.
5
6
4
25
is not conservative for any value of A.
____ 4. Determine whether or not the vector field is conservative.
conservative
not conservative
____ 5. Find the value of the line integral , where and
.
(Hint: If F is conservative, the integration may be easier on an alternate path.)
3,072
192
6,144
6,144
6,912
____ 6.
Find the value of the line integral
on the closed path consisting of line
segments from
to
from
to
and then from
to
where
.
(Hint: If F is conservative, the integration may be easier on an alternate path.)
99
12
36
30
0