984
Chapter 15: Vector Analysis
____
3.
Let
and let S be the surface bounded by the
plane
and the coordinates planes. Verify the Divergence Theorem by evaluating
as a surface integral and as a triple integral.
a.
b.
c.
d.
e.
15.7 Divergence Theorem
985
____ 4.
Let
and let S be the surface bounded by the planes
and the coordinate planes. Verify the Divergence Theorem by evaluating
as a surface integral and as a triple integral. Round your answer to two decimal places
wherever applicable.
a.
b.
c.
d.
e.
____
5.
Let
and let S be the surface bounded by
and
. Verify the Divergence Theorem by evaluating
as a surface integral
and as a triple integral. Round your answer to two decimal places.
a.
b.
c.
d.
e.
____ 6. Let and let S be the surface bounded by
and . Verify the Divergence Theorem by evaluating as a surface
integral and as a triple integral. Round your answer to two decimal places.
986 Chapter 15: Vector Analysis
a.
b.
c.
d.
e.
____ 7. Use Divergence Theorem to evaluate and find the outward flux of
through the surface S of the solid bounded by the planes
.
a.
b.
c.
d.
e.
____ 8. Use the Divergence Theorem to evaluate and find the outward flux of
through the surface of the solid bounded by the graphs of the equations. Use a computer
algebra system to verify your results.
a.
b.
c.
d.
e.
____ 9. Use Divergence Theorem to evaluate and find the outward flux of
through the surface S of the solid bounded by the sphere .
a.
b.
c.
15.7 Divergence Theorem
987
d.
e.
____ 10. Use the Divergence Theorem to evaluate and find the outward flux of
through the surface of the solid bounded by the graphs of the equations. Use a computer
algebra system to verify your results.
a.
b.
c.
d.
e.
____ 11. Use Divergence Theorem to evaluate and find the outward flux of through the
surface S of the solid bounded by
a.
b.
c.
d.
e. 0
____ 12. Use Divergence Theorem to evaluate and find the outward flux of
through the surface S of the solid bounded by the planes
.
a.
b.
c.
988 Chapter 15: Vector Analysis
d.
e.
____ 13. Use the Divergence Theorem to evaluate and find the outward flux of
through the surface of the solid bounded by the graphs of the equations. Use a computer
algebra system to verify your results.
a.
b.
c.
d.
e.
____ 14. Evaluate , where
and S is the closed surface of the solid bounded by the graphs , , and
the coordinate planes.
a.
b.
c.
d.
e.
15.7 Divergence Theorem
989
____ 15. Evaluate where S is the closed surface of the solid bounded by the
graphs of
a.
b.
c.
d.
e.
____ 16. Let and let S be the cube bounded by the planes
. Verify the Divergence Theorem by evaluating as a
surface integral and as a triple integral.
a.
b.
c.
d.
e.
990 Chapter 15: Vector Analysis
15.7 Divergence Theorem
Answer Section
15.8 Stokes’s Theorem
991
15.8 Stokes’s Theorem
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Find the curl of the vector field
.
a.
b.
c.
d.
e.
____
2.
Find the curl of the vector field
a.
b.
c.
d.
e.
____
3.
Find the curl of the vector field
a.
b.
c.
992 Chapter 15: Vector Analysis
d.
e.
____ 4. Verify Stokes’s Theorem by evaluating
as a line integral and as a double integral .
a.
b.
c.
d.
15.8 Stokes’s Theorem
993
e.
____ 5.
Let
and let S be the plane
in the first
octant. Verify Stokes’s Theorem by evaluating
as a line integral and as a
double integral.
a.
b.
c.
d.
0
e.
1
____ 6.
Let
and let S be the graph of
.
Verify Stokes’s Theorem by evaluating
as a line integral and as a double
integral.
a.
b.
c.
d.
e.
____ 7. Use Stokes’s Theorem to evaluate Use a computer algebra system to
verify your results. Note: C is oriented counterclockwise as viewed from above.
994 Chapter 15: Vector Analysis
C: triangle with vertices
a.
b.
c.
d.
e.
____ 8.
143. Let
and let C be the triangle
with vertices of
oriented counterclockwise. Use Stokes’s
Theorem to evaluate
a.
b.
c.
d.
e.
____ 9. Let and let S be the graph of
oriented counterclockwise. Use Stokes’s Theorem to evaluate .
a.
b.
c.
d.
e.
____ 10. Use Stokes’s Theorem to evaluate .Use a computer algebra system to
verify your results. Note: C is oriented counterclockwise as viewed from above.
a.
15.8 Stokes’s Theorem
995
b.
c.
d.
e.
____ 11. Use Stokes’s Theorem to evaluate where and S is
.Use a computer algebra system to verify your result.
a.
b.
c.
d.
e.
____ 12. Use Stokes’s Theorem to evaluate where
and S is over in the first
octant. Use a computer algebra system to verify your result.
0
____ 13.
Use Stokes’s Theorem to evaluate
where
and S is the first-octant portion of
over
Use a computer algebra system to verify your result.
a.
b.
996 Chapter 15: Vector Analysis
c.
d.
e.
____ 14. Use Stokes’s Theorem to evaluate Use a computer algebra system to
verify your results. Note: C is oriented counterclockwise as viewed from above.
a.
b.
c.
d.
e.
____ 15. Use Stokes’s Theorem to evaluate where
and S is the first-octant portion of over .
Use a computer algebra system to verify your result.
a.
b.
15.8 Stokes’s Theorem
997
c.
d.
e.
____ 16.
The motion of a liquid in a cylindrical container of radius 1 is described by the
velocity field
Find
where S is the upper surface of
the cylindrical container.
a.
b.
c.
d. 0
e.
____ 17. The motion of a liquid in a cylindrical container of radius 1 is described by the velocity
field Find where S is the upper surface of the
cylindrical container.
a.
b.
c.
d.
e.
998 Chapter 15: Vector Analysis
15.8 Stokes’s Theorem
Answer Section