Download EXAMPLE 6 Find the gradient vector field of . Plot the gradient vector

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CHAPTER 13 VECTOR CALCULUS
EXAMPLE 6 Find the gradient vector field of f 共x, y兲 苷 x 2 y y 3. Plot the gradient
vector field together with a contour map of f. How are they related?
4
SOLUTION The gradient vector field is given by
f
f
i
j 苷 2xy i 共x 2 3y 2 兲 j
x
y
∇f 共x, y兲 苷
_4
4
Figure 15 shows a contour map of f with the gradient vector field. Notice that the
gradient vectors are perpendicular to the level curves, as we would expect from
Section 11.6. Notice also that the gradient vectors are long where the level curves
are close to each other and short where they are farther apart. That’s because the
length of the gradient vector is the value of the directional derivative of f and close
level curves indicate a steep graph.
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FIGURE 15
A vector field F is called a conservative vector field if it is the gradient of some
scalar function, that is, if there exists a function f such that F 苷 ∇f . In this situation
f is called a potential function for F.
Not all vector fields are conservative, but such fields do arise frequently in physics.
For example, the gravitational field F in Example 4 is conservative because if we
define
f 共x, y, z兲 苷
mMG
sx y 2 z 2
2
then
∇f 共x, y, z兲 苷
苷
f
f
f
i
j
k
x
y
z
mMGx
mMGy
mMGz
i 2
j 2
k
共x 2 y 2 z 2 兲3兾2
共x y 2 z 2 兲3兾2
共x y 2 z 2 兲3兾2
苷 F共x, y, z兲
In Sections 13.3 and 13.5 we will learn how to tell whether or not a given vector field
is conservative.
13.1
Exercises
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1. F共x, y兲 苷 共i j兲
2. F共x, y兲 苷 i x j
3. F共x, y兲 苷 x i y j
4. F共x, y兲 苷 x i y j
1
2
5. F共x, y兲 苷
yixj
sx 2 y 2
6. F共x, y兲 苷
yixj
sx 2 y 2
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9. F共x, y, z兲 苷 y j
1–10
■ Sketch the vector field F by drawing a diagram like
Figure 5 or Figure 9.
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10. F共x, y, z兲 苷 j i
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11–14
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■ Match the vector fields F with the plots labeled I –IV.
Give reasons for your choices.
11. F共x, y兲 苷 具 y, x典
12. F共x, y兲 苷 具2x 3y, 2x 3y典
7. F共x, y, z兲 苷 j
13. F共x, y兲 苷 具sin x, sin y典
8. F共x, y, z兲 苷 z j
14. F共x, y兲 苷 具ln共1 x 2 y 2 兲, x典
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◆
SECTION 13.1 VECTOR FIELDS
I
II
5
923
use it to plot
6
F共x, y兲 苷 共 y 2 2xy兲 i 共3xy 6x 2 兲 j
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5
Explain the appearance by finding the set of points 共x, y兲
such that F共x, y兲 苷 0.
6
CAS
a CAS to plot this vector field in various domains until you
can see what is happening. Describe the appearance of the
plot and explain it by finding the points where F共x兲 苷 0.
_6
_5
III
ⱍ ⱍ
20. Let F共x兲 苷 共r 2 2r兲x, where x 苷 具x, y典 and r 苷 x . Use
IV
5
5
21–24
■
Find the gradient vector field of f .
22. f 共x, y兲 苷 x e x
21. f 共x, y兲 苷 ln共x 2y兲
23. f 共x, y, z兲 苷 sx 2 y 2 z 2
_5
5
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5
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25–26
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24. f 共x, y, z兲 苷 x cos共 y兾z兲
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■ Match the vector fields F on ⺢ with the plots labeled
I–IV. Give reasons for your choices.
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1
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I
II
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1
z 0
z 0
_1
_1
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28. f 共x, y兲 苷 sin共x y兲
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30. f 共x, y兲 苷 x 2 y 2
31. f 共x, y兲 苷 x 2 y 2
32. f 共x, y兲 苷 sx 2 y 2
II
4
_4
_1 0
1
y
_1
1 0x
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1
0
4
4
_4
4
_1
x
_4
III
_4
IV
III
IV
4
4
1
1
z 0
z 0
_1
_1
_4
_1
0
1 x
_1 0
1
y
CAS
■
29. f 共x, y兲 苷 xy
I
1
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■ Match the functions f with the plots of their gradient
vector fields (labeled I –IV). Give reasons for your choices.
18. F共x, y, z兲 苷 x i y j z k
y
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29–32
17. F共x, y, z兲 苷 x i y j 3 k
1
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26. f 共x, y兲 苷 4 共x y兲2
27. f 共x, y兲 苷 sin x sin y
16. F共x, y, z兲 苷 i 2 j z k
0
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■ Plot the gradient vector field of f together with a contour map of f . Explain how they are related to each other.
15. F共x, y, z兲 苷 i 2 j 3 k
_1
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27–28
CAS
3
15–18
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Find the gradient vector field ∇ f of f and sketch it.
25. f 共x, y兲 苷 x y 2x
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y
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0
4
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4
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1 0x
1
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19. If you have a CAS that plots vector fields (the command is
fieldplot in Maple and PlotVectorField in Mathematica),
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CHAPTER 13 VECTOR CALCULUS
solve the differential equations to find an equation of
the flow line that passes through the point (1, 1).
33. The flow lines (or streamlines) of a vector field are the
paths followed by a particle whose velocity field is the
given vector field. Thus, the vectors in a vector field are
tangent to the flow lines.
(a) Use a sketch of the vector field F共x, y兲 苷 x i y j to
draw some flow lines. From your sketches, can you
guess the equations of the flow lines?
(b) If parametric equations of a flow line are x 苷 x共t兲,
y 苷 y共t兲, explain why these functions satisfy the differential equations dx兾dt 苷 x and dy兾dt 苷 y. Then
13.2
Line Integrals
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34. (a) Sketch the vector field F共x, y兲 苷 i x j and then
sketch some flow lines. What shape do these flow lines
appear to have?
(b) If parametric equations of the flow lines are x 苷 x共t兲,
y 苷 y共t兲, what differential equations do these functions
satisfy? Deduce that dy兾dx 苷 x.
(c) If a particle starts at the origin in the velocity field
given by F, find an equation of the path it follows.
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In this section we define an integral that is similar to a single integral except that
instead of integrating over an interval 关a, b兴, we integrate over a curve C. Such integrals are called line integrals, although “curve integrals” would be better terminology.
They were invented in the early 19th century to solve problems involving fluid flow,
forces, electricity, and magnetism.
We start with a plane curve C given by the parametric equations
1
y
Pi-1
Pi
Pn
P™
y 苷 y共t兲
P¡
P¸
x
0
n
兺 f 共x *, y*兲 s
t i*
a
FIGURE 1
t i-1
atb
or, equivalently, by the vector equation r共t兲 苷 x共t兲 i y共t兲 j, and we assume that C
is a smooth curve. [This means that r is continuous and r共t兲 苷 0. See Section 10.2.]
If we divide the parameter interval 关a, b兴 into n subintervals 关ti1, ti 兴 of equal width
and we let x i 苷 x共ti 兲 and yi 苷 y共ti 兲, then the corresponding points Pi 共x i, yi 兲 divide C
into n subarcs with lengths s1, s2, . . . , sn . (See Figure 1.) We choose any point
Pi*共x i*, yi*兲 in the i th subarc. (This corresponds to a point t*i in 关ti1, ti兴.) Now if f is
any function of two variables whose domain includes the curve C, we evaluate f at
the point 共x i*, yi*兲, multiply by the length si of the subarc, and form the sum
P *i (x *i , y *i )
C
x 苷 x共t兲
i
i
i
i苷1
ti
b t
which is similar to a Riemann sum. Then we take the limit of these sums and make
the following definition by analogy with a single integral.
2 Definition If f is defined on a smooth curve C given by Equations 1, then
the line integral of f along C is
n
y
C
f 共x, y兲 ds 苷 lim
兺 f 共x *, y*兲 s
n l i苷1
i
i
i
if this limit exists.
In Section 6.3 we found that the length of C is
L苷
y
b
a
冑冉 冊 冉 冊
dx
dt
2
dy
dt
2
dt
SECTION 13.2 LINE INTEGRALS
▲ Figure 13 shows the twisted cubic C
EXAMPLE 8 Evaluate
in Example 8 and some typical vectors
acting at three points on C.
xC F ⴢ dr, where F共x, y, z兲 苷 xy i yz j zx k and C is the
x苷t
y 苷 t2
F { r(3/4)}
0t1
r共t兲 苷 t i t 2 j t 3 k
r共t兲 苷 i 2t j 3t 2 k
(1, 1, 1)
0.5
z 苷 t3
SOLUTION We have
F { r(1)}
z 1
933
twisted cubic given by
2
1.5
◆
C
F共r共t兲兲 苷 t 3 i t 5 j t 4 k
F { r(1/ 2)}
0
0
y1 2
2
y
Thus
0
1
x
C
1
F ⴢ dr 苷 y F共r共t兲兲 ⴢ r共t兲 dt
0
苷y
FIGURE 13
1
0
t4
5t 7
共t 5t 兲 dt 苷
4
7
3
6
册
1
苷
0
27
28
Finally, we note the connection between line integrals of vector fields and line integrals of scalar fields. Suppose the vector field F on ⺢ 3 is given in component form by
the equation F 苷 P i Q j R k. We use Definition 13 to compute its line integral
along C:
y
C
b
F ⴢ dr 苷 y F共r共t兲兲 ⴢ r共t兲 dt
a
b
苷 y 共P i Q j R k兲 ⴢ 共x共t兲 i y共t兲 j z共t兲 k兲 dt
a
b
苷 y 关P共x共t兲, y共t兲, z共t兲兲x共t兲 Q共x共t兲, y共t兲, z共t兲兲y共t兲 R共x共t兲, y共t兲, z共t兲兲z共t兲兴 dt
a
But this last integral is precisely the line integral in (10). Therefore, we have
y
C
where F 苷 P i Q j R k
F ⴢ dr 苷 y P dx Q dy R dz
C
For example, the integral
as xC F ⴢ dr where
xC y dx z dy x dz in Example 6 could be expressed
F共x, y, z兲 苷 y i z j x k
13.2
1–12
1.
2.
3.
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Exercises
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Evaluate the line integral, where C is the given curve.
xC y ds, C: x 苷 t , y 苷 t, 0 t 2
xC 共 y兾x兲 ds, C: x 苷 t 4, y 苷 t 3, 12 t 1
xC xy 4 ds, C is the right half of the circle x 2 y 2 苷 16
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4.
xC
5.
xC xy dx 共x y兲 dy,
2
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sin x dx,
C is the arc of the curve x 苷 y 4 from 共1, 1兲 to 共1, 1兲
C consists of line segments from
共0, 0兲 to 共2, 0兲 and from 共2, 0兲 to 共3, 2兲
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6.
CHAPTER 13 VECTOR CALCULUS
xC x sy dx 2y sx dy,
15–18 ■ Evaluate the line integral xC F ⴢ dr, where C is given
by the vector function r共t兲.
C consists of the shortest arc of the circle x y 苷 1 from
共1, 0兲 to 共0, 1兲 and the line segment from 共0, 1兲 to 共4, 3兲
2
2
r共t兲 苷 t 2 i t 3 j,
7.
xC xy
8.
xC x 2z ds, C is the line segment from (0, 6, 1) to (4, 1, 5)
xC xe yz ds, C is the line segment from (0, 0, 0) to (1, 2, 3)
xC yz dy xy dz, C: x 苷 st, y 苷 t, z 苷 t 2, 0 t 1
xC z 2 dx z dy 2y dz,
9.
10.
11.
15. F共x, y兲 苷 x 2y 3 i y sx j,
3
ds,
C: x 苷 4 sin t, y 苷 4 cos t, z 苷 3t, 0 t 兾2
16. F共x, y, z兲 苷 yz i xz j xy k,
r共t兲 苷 t i t 2 j t 3 k,
r共t兲 苷 t 3 i t 2 j t k,
18. F共x, y, z兲 苷 x i xy j z 2 k,
r共t兲 苷 sin t i cos t j t 2 k,
CAS
C consists of line segments from 共0, 0, 0兲 to 共2, 0, 0兲, from
共2, 0, 0兲 to 共1, 3, 1兲, and from 共1, 3, 1兲 to 共1, 3, 0兲
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_3
CAS
Are the line integrals of F over C1 and C2 positive, negative,
or zero? Explain.
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F共x, y兲 苷 e x1 i xy j and C is given by
r共t兲 苷 t 2 i t 3 j, 0 t 1.
(b) Illustrate part (a) by using a graphing calculator or computer to graph C and the vectors from the vector field
corresponding to t 苷 0, 1兾s2, and 1 (as in Figure 13).
F共x, y, z兲 苷 x i z j y k and C is given by
r共t兲 苷 2t i 3t j t 2 k, 1 t 1.
(b) Illustrate part (a) by using a computer to graph C and
the vectors from the vector field corresponding to
t 苷 1 and 12 (as in Figure 13).
23. Find the exact value of xC x 3 y 5 ds, where C is the part of the
24. (a) Find the work done by the force field
y
CAS
x
■
astroid x 苷 cos 3t, y 苷 sin 3t in the first quadrant.
14. The figure shows a vector field F and two curves C1 and C2.
C™
■
22. (a) Evaluate the line integral xC F ⴢ dr, where
3x
C¡
0 t 兾2
21. (a) Evaluate the line integral xC F ⴢ dr, where
1
_2
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■ Use a graph of the vector field F and the curve C to
guess whether the line integral of F over C is positive, negative,
or zero. Then evaluate the line integral.
■
;
2
■
y
x
i
j,
sx 2 y 2
sx 2 y 2
2
C is the parabola y 苷 1 x from 共1, 2兲 to (1, 2)
y
3
1
■
20. F共x, y兲 苷
2
_1 0
_1
■
C is the arc of the circle x 2 y 2 苷 4 traversed counterclockwise from (2, 0) to 共0, 2兲
1
_2
■
19. F共x, y兲 苷 共x y兲 i xy j,
■
(a) If C1 is the vertical line segment from 共3, 3兲 to
共3, 3兲, determine whether xC F ⴢ dr is positive, negative, or zero.
(b) If C2 is the counterclockwise-oriented circle with radius
3 and center the origin, determine whether xC F ⴢ dr is
positive, negative, or zero.
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19–20
13. Let F be the vector field shown in the figure.
2
0t1
2
■
xC yz dx xz dy xy dz,
0t2
17. F共x, y, z兲 苷 sin x i cos y j xz k,
C consists of line segments from 共0, 0, 0兲 to 共0, 1, 1兲, from
共0, 1, 1兲 to 共1, 2, 3兲, and from 共1, 2, 3兲 to 共1, 2, 4兲
12.
0t1
F共x, y兲 苷 x 2 i xy j on a particle that moves once
around the circle x 2 y 2 苷 4 oriented in the
counterclockwise direction.
(b) Use a computer algebra system to graph the force field
and circle on the same screen. Use the graph to explain
your answer to part (a).
25. A thin wire is bent into the shape of a semicircle
x 2 y 2 苷 4, x 0. If the linear density is a constant k,
find the mass and center of mass of the wire.
26. Find the mass and center of mass of a thin wire in the shape
of a quarter-circle x 2 y 2 苷 r 2, x 0, y 0, if the density function is 共x, y兲 苷 x y.
SECTION 13.2 LINE INTEGRALS
27. (a) Write the formulas similar to Equations 4 for the center
of mass 共 x, y, z 兲 of a thin wire with density function
共x, y, z兲 in the shape of a space curve C.
(b) Find the center of mass of a wire in the shape of the
helix x 苷 2 sin t, y 苷 2 cos t, z 苷 3t, 0 t 2, if the
density is a constant k.
28. Find the mass and center of mass of a wire in the shape of
the helix x 苷 t, y 苷 cos t, z 苷 sin t, 0 t 2, if the
density at any point is equal to the square of the distance
from the origin.
29. If a wire with linear density 共x, y兲 lies along a plane curve
935
revolutions, how much work is done by the man against
gravity in climbing to the top?
36. Suppose there is a hole in the can of paint in Exercise 35
and 9 lb of paint leak steadily out of the can during the
man’s ascent. How much work is done?
37. An object moves along the curve C shown in the figure
from (1, 2) to (9, 8). The lengths of the vectors in the force
field F are measured in newtons by the scales on the axes.
Estimate the work done by F on the object.
y
(meters)
C, its moments of inertia about the x- and y-axes are
defined as
Ix 苷 y y 2 共x, y兲 ds
◆
C
Iy 苷 y x 2 共x, y兲 ds
C
C
Find the moments of inertia for the wire in Example 3.
30. If a wire with linear density 共x, y, z兲 lies along a space
curve C, its moments of inertia about the x-, y-, and z -axes
are defined as
Ix 苷 y 共 y 2 z 2 兲 共x, y, z兲 ds
C
Iy 苷 y 共x 2 z 2 兲 共x, y, z兲 ds
C
Iz 苷 y 共x 2 y 2 兲 共x, y, z兲 ds
C
C
1
0
x
(meters)
1
38. Experiments show that a steady current I in a long wire pro-
duces a magnetic field B that is tangent to any circle that
lies in the plane perpendicular to the wire and whose center is the axis of the wire (as in the figure). Ampère’s Law
relates the electric current to its magnetic effects and states
that
Find the moments of inertia for the wire in Exercise 27.
y
C
B ⴢ dr 苷 0 I
31. Find the work done by the force field
F共x, y兲 苷 x i 共 y 2兲 j in moving an object along an arch
of the cycloid r共t兲 苷 共t sin t兲 i 共1 cos t兲 j,
0 t 2.
32. Find the work done by the force field
F共x, y兲 苷 x sin y i y j on a particle that moves along the
parabola y 苷 x 2 from 共1, 1兲 to 共2, 4兲.
where I is the net current that passes through any surface
bounded by a closed curve C and 0 is a constant called the
permeability of free space. By taking C to be a circle with
radius r , show that the magnitude B 苷 B of the magnetic
field at a distance r from the center of the wire is
ⱍ ⱍ
B苷
33. Find the work done by the force field
F共x, y, z兲 苷 xz i yx j zy k on a particle that moves
along the curve r共t兲 苷 t 2 i t 3 j t 4 k, 0 t 1.
I
34. The force exerted by an electric charge at the origin on a
charged particle at a point 共x, y, z兲 with position vector
r 苷 具x, y, z典 is F共r兲 苷 Kr兾 r 3 where K is a constant. (See
Example 5 in Section 13.1.) Find the work done as the particle moves along a straight line from 共2, 0, 0兲 to 共2, 1, 5兲.
ⱍ ⱍ
35. A 160-lb man carries a 25-lb can of paint up a helical stair-
case that encircles a silo with a radius of 20 ft. If the silo
is 90 ft high and the man makes exactly three complete
0 I
2 r
B
SECTION 13.3 THE FUNDAMENTAL THEOREM FOR LINE INTEGRALS
◆
943
Therefore
ⱍ
ⱍ
ⱍ
W 苷 12 m v共b兲 2 12 m v共a兲
15
ⱍ
2
where v 苷 r is the velocity.
The quantity 12 m v共t兲 2, that is, half the mass times the square of the speed, is
called the kinetic energy of the object. Therefore, we can rewrite Equation 15 as
ⱍ
ⱍ
W 苷 K共B兲 K共A兲
16
which says that the work done by the force field along C is equal to the change in
kinetic energy at the endpoints of C.
Now let’s further assume that F is a conservative force field; that is, we can write
F 苷 ∇f . In physics, the potential energy of an object at the point 共x, y, z兲 is defined
as P共x, y, z兲 苷 f 共x, y, z兲, so we have F 苷 ∇P. Then by Theorem 2 we have
W 苷 y F ⴢ dr 苷 y ∇P ⴢ dr
C
C
苷 关P共r共b兲兲 P共r共a兲兲兴
苷 P共A兲 P共B兲
Comparing this equation with Equation 16, we see that
P共A兲 K共A兲 苷 P共B兲 K共B兲
which says that if an object moves from one point A to another point B under the influence of a conservative force field, then the sum of its potential energy and its kinetic
energy remains constant. This is called the Law of Conservation of Energy and it is
the reason the vector field is called conservative.
13.3
Exercises
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1. The figure shows a curve C and a contour map of a function
f whose gradient is continuous. Find xC f ⴢ dr.
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y
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0
1
2
0
1
6
4
1
3
5
7
2
8
2
9
x
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y
60
40
C
50
30
3–10
■ Determine whether or not F is a conservative vector
field. If it is, find a function f such that F 苷 f .
20
10
3. F共x, y兲 苷 共6x 5y兲 i 共5x 4y兲 j
4. F共x, y兲 苷 共x 3 4xy兲 i 共4xy y 3 兲 j
0
x
5. F共x, y兲 苷 xe y i ye x j
6. F共x, y兲 苷 e y i xe y j
2. A table of values of a function f with continuous gradient is
given. Find xC f ⴢ dr, where C has parametric equations
x 苷 t 2 1, y 苷 t 3 t, 0 t 1.
7. F共x, y兲 苷 共2x cos y y cos x兲 i 共x 2 sin y sin x兲 j
8. F共x, y兲 苷 共1 2xy ln x兲 i x 2 j
9. F共x, y兲 苷 共 ye x sin y兲 i 共e x x cos y兲 j
●
■
944
CHAPTER 13 VECTOR CALCULUS
10. F共x, y兲 苷 共 ye xy 4x 3 y兲 i 共xe xy x 4 兲 j
■
■
■
■
■
■
■
■
■
■
■
■
■
11. The figure shows the vector field F共x, y兲 苷 具2xy, x 典 and
2
22. F共x, y兲 苷 共 y 2兾x 2 兲 i 共2y兾x兲 j;
P共1, 1兲, Q共4, 2兲
■
■
■
■
■
■
■
■
■
■
■
■
■
23. Is the vector field shown in the figure conservative?
three curves that start at (1, 2) and end at (3, 2).
(a) Explain why xC F ⴢ dr has the same value for all three
curves.
(b) What is this common value?
Explain.
y
y
3
x
2
1
CAS
0
1
2
x
3
■ From a plot of F guess whether it is conservative.
Then determine whether your guess is correct.
24. F共x, y兲 苷 共2xy sin y兲 i 共x 2 x cos y兲 j
12–18 ■ (a) Find a function f such that F 苷 ∇ f and (b) use
part (a) to evaluate xC F ⴢ dr along the given curve C.
25. F共x, y兲 苷
12. F共x, y兲 苷 y i 共x 2y兲 j,
■
13. F共x, y兲 苷 x y i x y j,
(a)
0t1
15. F共x, y, z兲 苷 yz i xz j 共xy 2z兲 k,
16. F共x, y, z兲 苷 共2xz y 兲 i 2x y j 共x 3z 兲 k,
2
2
C: x 苷 t 2, y 苷 t 1, z 苷 2t 1,
2
0t1
0t
19–20
■
■
■
■
■
■
■
■
■
■
■
xC 2x sin y dx 共x
20.
xC 共2y
■
32.
cos y 3y 兲 dy,
C is any path from 共1, 0兲 to 共5, 1兲
2
■
■
3
■
■
3
4 2
■
■
■
■
■
■
21–22
■
■
■ Find the work done by the force field F in moving an
object from P to Q.
21. F共x, y兲 苷 x 2 y 3 i x 3 y 2 j;
y
C1
F ⴢ dr 苷 0
y
(b)
C2
■
■
F ⴢ dr 苷 1
P
R
苷
z
x
R
Q
苷
z
y
30. 兵共x, y兲 ⱍ x 苷 0其
ⱍ
兵共x, y兲 ⱍ 1 x y 4其
兵共x, y兲 ⱍ x y 1 or 4 x y 9其
2
2
■
■
2
2
■
2
■
■
■
2
■
■
P共0, 0兲, Q共2, 1兲
■
■
■
■
y i x j
.
x2 y2
(a) Show that P兾y 苷 Q兾x.
(b) Show that xC F ⴢ dr is not independent of path.
[Hint: Compute xC F ⴢ dr and xC F ⴢ dr, where C1 and
C2 are the upper and lower halves of the circle
x 2 y 2 苷 1 from 共1, 0兲 to 共1, 0兲.] Does this
contradict Theorem 6?
33. Let F共x, y兲 苷
12x y 兲 dx 共4xy 9x y 兲 dy,
C is any path from 共1, 1兲 to 共3, 2兲
2
■
29. 兵共x, y兲 x 0, y 0其
31.
Show that the line integral is independent of path and
evaluate the integral.
19.
■
■ Determine whether or not the given set is (a) open,
(b) connected, and (c) simply-connected.
■
2
■
29–32
y
C: r共t兲 苷 t i t 2 j t 3 k, 0 t 1
■
■
xC y dx x dy xyz dz is not independent of path.
18. F共x, y, z兲 苷 e i xe j 共z 1兲e z k,
■
■
28. Use Exercise 27 to show that the line integral
17. F共x, y, z兲 苷 y cos z i 2xy cos z j xy 2 sin z k,
2
y
■
P
Q
苷
y
x
C is the line segment from 共1, 0, 2兲 to 共4, 6, 3兲
C: r共t兲 苷 t i sin t j t k,
■
servative and P, Q, R have continuous first-order partial
derivatives, then
0t1
2
■
27. Show that if the vector field F 苷 P i Q j R k is con-
14. F共x, y兲 苷 e 2y i 共1 2 xe 2y 兲 j,
C: r共t兲 苷 te t i 共1 t兲 j,
■
and C2 that are not closed and satisfy the equation.
4 3
C: r共t兲 苷 st i 共1 t 3 兲 j,
■
共x 2y兲 i 共x 2兲 j
s1 x 2 y 2
26. Let F 苷 f , where f 共x, y兲 苷 sin共x 2y兲. Find curves C1
C is the upper semicircle that starts at (0, 1) and ends
at (2, 1)
3 4
24–25
1
2
■
◆
SECTION 13.4 GREEN’S THEOREM
(at a maximum distance of 1.52 10 8 km from the
Sun) to perihelion (at a minimum distance of
1.47 10 8 km). (Use the values m 苷 5.97 10 24 kg,
M 苷 1.99 10 30 kg, and G 苷 6.67 10 11 Nm 2兾kg2.兲
(c) Another example of an inverse square field is the electric field E 苷 qQr兾 r 3 discussed in Example 5 in
Section 13.1. Suppose that an electron with a charge of
1.6 10 19 C is located at the origin. A positive unit
charge is positioned a distance 10 12 m from the electron and moves to a position half that distance from
the electron. Use part (a) to find the work done by the
electric field. (Use the value 苷 8.985 10 10.)
34. (a) Suppose that F is an inverse square force field, that is,
F共r兲 苷
cr
r 3
ⱍ ⱍ
for some constant c, where r 苷 x i y j z k. Find
the work done by F in moving an object from a point P1
along a path to a point P2 in terms of the distances d1
and d2 from these points to the origin.
(b) An example of an inverse square field is the gravitational field F 苷 共mMG 兲r兾 r 3 discussed in Example 4
in Section 13.1. Use part (a) to find the work done by
the gravitational field when Earth moves from aphelion
ⱍ ⱍ
ⱍ ⱍ
13.4
Green’s Theorem
y
D
C
0
x
●
●
●
945
●
●
●
●
●
●
●
●
●
●
●
●
●
Green’s Theorem gives the relationship between a line integral around a simple closed
curve C and a double integral over the plane region D bounded by C. (See Figure 1.
We assume that D consists of all points inside C as well as all points on C.) In stating
Green’s Theorem we use the convention that the positive orientation of a simple
closed curve C refers to a single counterclockwise traversal of C. Thus, if C is given
by the vector function r共t兲, a t b, then the region D is always on the left as the
point r共t兲 traverses C. (See Figure 2.)
y
FIGURE 1
y
C
D
D
C
0
FIGURE 2
x
0
(a) Positive orientation
x
(b) Negative orientation
Green’s Theorem Let C be a positively oriented, piecewise-smooth, simple
closed curve in the plane and let D be the region bounded by C. If P and Q
have continuous partial derivatives on an open region that contains D, then
y
C
P dx Q dy 苷
yy
D
NOTE
●
冉
P
Q
x
y
冊
dA
The notation
y
䊊
C
P dx Q dy
or
gC P dx Q dy
is sometimes used to indicate that the line integral is calculated using the positive orientation of the closed curve C. Another notation for the positively oriented boundary
13.4
Exercises
●
●
●
●
●
●
●
●
●
●
1–4
■ Evaluate the line integral by two methods: (a) directly
and (b) using Green’s Theorem.
1.
●
16.
xy 2 dx x 3 dy,
C is the rectangle with vertices (0, 0), (2, 0), (2, 3), and (0, 3)
x
■
3.
xy dx x 2 y 3 dy,
C is the triangle with vertices (0, 0), (1, 0), and (1, 2)
4.
共x 2 y 2 兲 dx 2xy dy, C consists of the arc of the
parabola y 苷 x 2 from 共0, 0兲 to 共2, 4兲 and the line segments
from 共2, 4兲 to 共0, 4兲 and from 共0, 4兲 to 共0, 0兲
x
䊊
C
■
■
■
■
■
■
■
■
5–6
■ Verify Green’s Theorem by using a computer algebra
system to evaluate both the line integral and the double integral.
5. P共x, y兲 苷 x y ,
6. P共x, y兲 苷 y sin x,
■
■
■
■
■
■
■
■
■
■
■
■
■
■
13.
14.
xC e y dx 2xe y dy,
xC x 2 y 2 dx 4xy 3 dy,
xC ( y e sx ) dx 共2x cos y 2 兲 dy,
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
y
C
x dy y dx 苷 x 1 y2 x 2 y1
(b) If the vertices of a polygon, in counterclockwise order,
are 共x 1, y1 兲, 共x 2 , y2 兲, . . . , 共x n, yn 兲, show that the area of
the polygon is
A 苷 12 关共x 1 y2 x 2 y1 兲 共x 2 y3 x 3 y2 兲 共x n1 yn x n yn1 兲 共x n y1 x 1 yn 兲兴
A苷
(c) Find the area of the pentagon with vertices 共0, 0兲, 共2, 1兲,
共1, 3兲, 共0, 2兲, and 共1, 1兲.
22. Let D be a region bounded by a simple closed path C in the
xC 共 y 2 tan1x兲 dx 共3x sin y兲 dy,
xC y 3 dx x 3 dy, C is the circle x 2 y 2 苷 4
xC sin y dx x cos y dy, C is the ellipse x 2 xy y 2 苷 1
xC xy dx 2x 2 dy,
xy-plane. Use Green’s Theorem to prove that the coordinates of the centroid 共 x, y 兲 of D are
x苷
1
2A
y
䊊
C
x 2 dy
y苷
1
2A
y
䊊
C
y 2 dx
where A is the area of D.
C consists of the line segment from 共2, 0兲 to 共2, 0兲 and the
top half of the circle x 2 y 2 苷 4
23. Use Exercise 22 to find the centroid of the triangle with
xC 共x 3 y 3 兲 dx 共x 3 y 3 兲 dy,
24. Use Exercise 22 to find the centroid of a semicircular region
C is the boundary of the region between the circles
x 2 y 2 苷 1 and x 2 y 2 苷 9
15.
■
the point 共x 2 , y2兲, show that
C is the boundary of the region enclosed by the parabola
y 苷 x 2 and the line y 苷 4
12.
■
■
C is the boundary of the region enclosed by the parabolas
y 苷 x 2 and x 苷 y 2
11.
●
21. (a) If C is the line segment connecting the point 共x 1, y1兲 to
C is the triangle with vertices (0, 0), (1, 3), and (0, 3)
10.
●
r共t兲 苷 cos t i sin 3t j, 0 t 2
C is the square with sides x 苷 0, x 苷 1, y 苷 0, and y 苷 1
9.
●
20. The region bounded by the curve with vector equation
■ Use Green’s Theorem to evaluate the line integral along
the given positively oriented curve.
8.
●
xC F ⴢ dr, where F共x, y兲 苷 y 6 i xy 5 j,
7–16
7.
●
■ Find the area of the given region using one of the
formulas in Equations 5.
2
■
●
equation r共t兲 苷 cos 3t i sin 3t j, 0 t 2
Q共x, y兲 苷 x sin y,
C consists of the arc of the parabola y 苷 x 2 from (0, 0) to
(1, 1) followed by the line segment from (1, 1) to (0, 0)
■
●
19–20
7 6
2
●
19. The region bounded by the hypocycloid with vector
Q共x, y兲 苷 x y ,
C is the circle x 2 y 2 苷 1
4 5
●
●
to 共2, 0兲, and then along the semicircle y 苷 s4 x 2 to the
starting point. Use Green’s Theorem to find the work done
on this particle by the force field F共x, y兲 苷 具x, x 3 3xy 2 典 .
x
■
●
18. A particle starts at the point 共2, 0兲, moves along the x-axis
䊊
C
■
●
F共x, y兲 苷 x共x y兲 i xy 2 j in moving a particle from the
origin along the x-axis to 共1, 0兲, then along the line segment
to 共0, 1兲, and then back to the origin along the y-axis.
y dx x dy,
C is the circle with center the origin and radius 1
䊊
C
■
●
951
17. Use Green’s Theorem to find the work done by the force
x
2.
■
●
◆
C is the ellipse 4x 2 y 2 苷 1
䊊
C
■
CAS
●
SECTION 13.4 GREEN’S THEOREM
xC F ⴢ dr, where F共x, y兲 苷 共 y
x y兲 i xy j,
C consists of the circle x y 苷 4 from 共2, 0兲 to (s2, s2 )
and the line segments from (s2, s2 ) to 共0, 0兲 and from
共0, 0兲 to 共2, 0兲
2
2
2
2
2
vertices 共0, 0兲, 共1, 0兲, and 共0, 1兲.
of radius a.
25. A plane lamina with constant density 共x, y兲 苷 occupies a
region in the xy-plane bounded by a simple closed path C.
Show that its moments of inertia about the axes are
Ix 苷 3
y
䊊
C
y 3 dx
Iy 苷
3
y
䊊
C
x 3 dy
952
■
CHAPTER 13 VECTOR CALCULUS
where f 共x, y兲 苷 1:
26. Use Exercise 25 to find the moment of inertia of a circular
disk of radius a with constant density about a diameter.
(Compare with Example 4 in Section 12.5.)
yy dx dy 苷 yy
R
27. If F is the vector field of Example 5, show that
xC F ⴢ dr 苷 0 for every simple closed path that does not
pass through or enclose the origin.
by proving Equation 3.
29. Use Green’s Theorem to prove the change of variables for-
mula for a double integral (Formula 12.9.9) for the case
Curl and Divergence
●
●
冟
共x, y兲
du dv
共u, v兲
Here R is the region in the xy-plane that corresponds to the
region S in the uv-plane under the transformation given by
x 苷 t共u, v兲, y 苷 h共u, v兲.
[Hint: Note that the left side is A共R兲 and apply the first
part of Equation 5. Convert the line integral over R to a
line integral over S and apply Green’s Theorem in the
uv-plane.]
28. Complete the proof of the special case of Green’s Theorem
13.5
S
冟
●
●
●
●
●
●
●
●
●
●
●
●
●
In this section we define two operations that can be performed on vector fields and that
play a basic role in the applications of vector calculus to fluid flow and electricity and
magnetism. Each operation resembles differentiation, but one produces a vector field
whereas the other produces a scalar field.
Curl
If F 苷 P i Q j R k is a vector field on ⺢ 3 and the partial derivatives of P, Q, and
R all exist, then the curl of F is the vector field on ⺢ 3 defined by
1
curl F 苷
冉
R
Q
y
z
冊 冉
i
P
R
z
x
冊 冉
j
Q
P
x
y
冊
k
As an aid to our memory, let’s rewrite Equation 1 using operator notation. We introduce the vector differential operator ∇ (“del”) as
j
k
x
y
z
∇ 苷i
It has meaning when it operates on a scalar function to produce the gradient of f :
∇f 苷 i
f
f
f
f
f
f
j
k
苷
i
j
k
x
y
z
x
y
z
If we think of ∇ as a vector with components 兾x, 兾y, and 兾z, we can also consider the formal cross product of ∇ with the vector field F as follows:
ⱍ ⱍ
i
∇F苷
x
P
苷
冉
j
y
Q
R
Q
y
z
苷 curl F
k
z
R
冊 冉
i
P
R
z
x
冊 冉
j
Q
P
x
y
冊
k
■
958
CHAPTER 13 VECTOR CALCULUS
by Green’s Theorem. But the integrand in this double integral is just the divergence
of F. So we have a second vector form of Green’s Theorem.
y
13
䊊
C
F ⴢ n ds 苷 yy div F共x, y兲 dA
D
This version says that the line integral of the normal component of F along C is equal
to the double integral of the divergence of F over the region D enclosed by C.
13.5
1–6
■
Exercises
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
y
9.
Find (a) the curl and (b) the divergence of the vector
●
field.
1. F共x, y, z兲 苷 xy i yz j zx k
2. F共x, y, z兲 苷 共x 2z兲 i 共x y z兲 j 共x 2y兲 k
3. F共x, y, z兲 苷 xyz i x 2y k
4. F共x, y, z兲 苷 xe y j ye z k
x
0
5. F共x, y, z兲 苷 e x sin y i e x cos y j z k
■
x
y
z
i 2
j 2
k
x y2 z2
x y2 z2
x y2 z2
6. F共x, y, z兲 苷
■
■
■
2
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
10. Let f be a scalar field and F a vector field. State whether
each expression is meaningful. If not, explain why. If so,
state whether it is a scalar field or a vector field.
(a) curl f
(b) grad f
(c) div F
(d) curl共grad f 兲
(e) grad F
(f) grad共div F兲
(g) div共grad f 兲
(h) grad共div f 兲
(i) curl共curl F兲
( j) div共div F兲
(k) 共grad f 兲 共div F兲
(l) div共curl共grad f 兲兲
■
7–9
■ The vector field F is shown in the xy-plane and looks the
same in all other horizontal planes. (In other words, F is independent of z and its z-component is 0.)
(a) Is div F positive, negative, or zero? Explain.
(b) Determine whether curl F 苷 0. If not, in which direction
does curl F point?
7.
■
11–16 ■ Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F 苷 ∇ f .
y
11. F共x, y, z兲 苷 yz i xz j xy k
12. F共x, y, z兲 苷 x i y j z k
13. F共x, y, z兲 苷 2xy i 共x 2 2yz兲 j y 2 k
14. F共x, y, z兲 苷 xy 2z 3 i 2x 2yz 3 j 3x 2y 2z 2 k
0
x
15. F共x, y, z兲 苷 e x i e z j e y k
16. F共x, y, z兲 苷 yze xz i e xz j xye xz k
8.
■
y
■
■
■
■
■
■
■
■
■
■
■
17. Is there a vector field G on ⺢ such that
3
curl G 苷 xy 2 i yz 2 j zx 2 k? Explain.
18. Is there a vector field G on ⺢ 3 such that
curl G 苷 yz i xyz j xy k? Explain.
19. Show that any vector field of the form
0
x
F共x, y, z兲 苷 f 共x兲 i t共 y兲 j h共z兲 k
where f , t, h are differentiable functions, is irrotational.
■
◆
SECTION 13.5 CURL AND DIVERGENCE
20. Show that any vector field of the form
where D and C satisfy the hypotheses of Green’s Theorem
and the appropriate partial derivatives of f and t exist and
are continuous.
F共x, y, z兲 苷 f 共 y, z兲 i t共x, z兲 j h共x, y兲 k
is incompressible.
21–27
959
33. This exercise demonstrates a connection between the curl
vector and rotations. Let B be a rigid body rotating about
the z-axis. The rotation can be described by the vector
w 苷 k, where is the angular speed of B, that is, the tangential speed of any point P in B divided by the distance d
from the axis of rotation. Let r 苷 具x, y, z典 be the position
vector of P.
(a) By considering the angle in the figure, show that the
velocity field of B is given by v 苷 w r.
(b) Show that v 苷 y i x j.
(c) Show that curl v 苷 2w.
■
Prove the identity, assuming that the appropriate
partial derivatives exist and are continuous. If f is a scalar field
and F, G are vector fields, then f F, F ⴢ G, and F G are
defined by
共 f F兲共x, y, z兲 苷 f 共x, y, z兲F共x, y, z兲
共F ⴢ G兲共x, y, z兲 苷 F共x, y, z兲 ⴢ G共x, y, z兲
共F G兲共x, y, z兲 苷 F共x, y, z兲 G共x, y, z兲
21. div共F G兲 苷 div F div G
22. curl共F G兲 苷 curl F curl G
z
23. div共 f F兲 苷 f div F F ⴢ f
w
24. curl共 f F兲 苷 f curl F 共 f 兲 F
25. div共F G兲 苷 G ⴢ curl F F ⴢ curl G
26. div共 f t兲 苷 0
27. curl curl F 苷 grad div F 2 F
■
■
28–30
■
■
■
■
■
■
d
B
■
■
■
■
■
v
P
■
ⱍ ⱍ
Let r 苷 x i y j z k and r 苷 r .
28. Verify each identity.
¨
(a) ⴢ r 苷 3
(c) 2r 3 苷 12r
(b) ⴢ 共rr兲 苷 4r
0
29. Verify each identity.
(a) r 苷 r兾r
(c) 共1兾r兲 苷 r兾r 3
y
(b) r 苷 0
(d) ln r 苷 r兾r 2
x
30. If F 苷 r兾r , find div F. Is there a value of p for which
p
34. Maxwell’s equations relating the electric field E and mag-
div F 苷 0?
■
■
■
■
■
■
■
■
■
■
■
■
■
31. Use Green’s Theorem in the form of Equation 13 to prove
div E 苷 0
Green’s first identity:
yy f t dA 苷 y
2
䊊
C
f 共t兲 ⴢ n ds yy f ⴢ t dA
D
D
where D and C satisfy the hypotheses of Green’s Theorem
and the appropriate partial derivatives of f and t exist and
are continuous. (The quantity t ⴢ n 苷 Dn t occurs in the
line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of t.)
32. Use Green’s first identity (Exercise 31) to prove Green’s
second identity:
yy 共 f t t f 兲 dA 苷 y
2
D
2
netic field H as they vary with time in a region containing
no charge and no current can be stated as follows:
䊊
C
共 f t t f 兲 ⴢ n ds
curl E 苷 div H 苷 0
1 H
c t
curl H 苷
1 E
c t
where c is the speed of light. Use these equations to prove
the following:
1 2 E
c 2 t 2
1 2 H
(b) 共 H兲 苷 2
c t 2
2
1
E
(c) 2 E 苷 2
[Hint: Use Exercise 27.]
c t 2
2
1 H
(d) 2 H 苷 2
c t 2
(a) 共 E兲 苷 ■
970
CHAPTER 13 VECTOR CALCULUS
13.6
Exercises
●
●
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●
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●
1. Let S be the cube with vertices 共1, 1, 1兲. Approximate
xxS sx 2 2y 2 3z 2 dS by using a Riemann sum as in Definition 1, taking the patches Sij to be the squares that are the
faces of the cube and the points Pij* to be the centers of the
squares.
●
●
●
13.
3. Let H be the hemisphere x 2 y 2 z 2 苷 50, z 0, and
suppose f is a continuous function with f 共3, 4, 5兲 苷 7,
f 共3, 4, 5兲 苷 8, f 共3, 4, 5兲 苷 9, and f 共3, 4, 5兲 苷 12.
By dividing H into four patches, estimate the value of
xxH f 共x, y, z兲 dS.
4. Suppose that f 共x, y, z兲 苷 t(sx 2 y 2 z 2 ), where t is a
function of one variable such that t共2兲 苷 5. Evaluate
xxS f 共x, y, z兲 dS, where S is the sphere x 2 y 2 z 2 苷 4.
5–18
5.
■
Evaluate the surface integral.
xxS yz dS,
S is the surface with parametric equations x 苷 uv,
y 苷 u v, z 苷 u v, u 2 v 2 1
6.
7.
xxS s1 x 2 y 2 dS,
S is the helicoid with vector equation
r共u, v兲 苷 u cos v i u sin v j v k, 0 u 1,
0v
14.
8.
9.
xxS yz dS,
S is the part of the plane x y z 苷 1 that lies in the
first octant
10.
xxS y dS,
S is the surface z 苷 共x
11.
2
3
3兾2
y
3兾2
xxS x dS,
xxS 共 y
z 兲 dS,
S is the part of the paraboloid x 苷 4 y 2 z 2 that lies in
front of the plane x 苷 0
2
2
●
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●
●
●
●
●
●
●
xxS yz dS,
xxS xy dS,
xxS 共x 2z y 2z兲 dS,
S is the hemisphere x 2 y 2 z 2 苷 4, z 0
16.
xxS xyz dS,
S is the part of the sphere x 2 y 2 z 2 苷 1 that lies above
the cone z 苷 sx 2 y 2
17.
xxS 共x 2 y z 2 兲 dS,
S is the part of the cylinder x 2 y 2 苷 9 between the planes
z 苷 0 and z 苷 2
18.
xxS 共x 2 y 2 z 2 兲 dS,
S consists of the cylinder in Exercise 17 together with its
top and bottom disks
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■
■
Evaluate the surface integral xxS F ⴢ dS for the given
vector field F and the oriented surface S. In other words, find
the flux of F across S. For closed surfaces, use the positive
(outward) orientation.
19–27
■
19. F共x, y, z兲 苷 xy i yz j zx k,
S is the part of the
paraboloid z 苷 4 x 2 y 2 that lies above the square
0 x 1, 0 y 1, and has upward orientation
20. F共x, y, z兲 苷 y i x j z 2 k,
S is the helicoid of Exercise 6 with upward orientation
21. F共x, y, z兲 苷 xze y i xze y j z k,
S is the part of the plane x y z 苷 1 in the first octant
and has downward orientation
22. F共x, y, z兲 苷 x i y j z 4 k,
S is the part of the cone z 苷 sx 2 y 2 beneath the plane
z 苷 1 with downward orientation
23. F共x, y, z兲 苷 x i y j z k,
S is the sphere x 2 y 2 z 2 苷 9
24. F共x, y, z兲 苷 y i x j 3z k,
S is the hemisphere
z 苷 s16 x 2 y 2 with upward orientation
25. F共x, y, z兲 苷 y j z k,
S consists of the paraboloid y 苷 x 2 z 2, 0 y 1, and
the disk x 2 z 2 1, y 苷 1
兲, 0 x 1, 0 y 1
S is the surface y 苷 x 2 4z, 0 x 2, 0 z 2
12.
15.
xxS xy dS,
S is the triangular region with vertices (1, 0, 0), (0, 2, 0),
and (0, 0, 2)
●
S is the boundary of the region enclosed by the cylinder
x 2 z 2 苷 1 and the planes y 苷 0 and x y 苷 2
xxS x 2yz dS,
S is the part of the plane z 苷 1 2x 3y that lies above
the rectangle 关0, 3兴 关0, 2兴
●
S is the part of the plane z 苷 y 3 that lies inside the
cylinder x 2 y 2 苷 1
2. A surface S consists of the cylinder x 2 y 2 苷 1,
1 z 1, together with its top and bottom disks.
Suppose you know that f is a continuous function with
f 共1, 0, 0兲 苷 2, f 共0, 1, 0兲 苷 3, and f 共0, 0, 1兲 苷 4.
Estimate the value of xxS f 共x, y, z兲 dS by using a Riemann
sum, taking the patches Sij to be four quarter-cylinders and
the top and bottom disks.
●
26. F共x, y, z兲 苷 x i y j 5 k,
S is the surface of Exercise 14
27. F共x, y, z兲 苷 x i 2y j 3z k,
S is the cube with vertices 共1, 1, 1兲
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◆
SECTION 13.7 STOKES’ THEOREM
CAS
28. Let S be the surface z 苷 xy, 0 x 1, 0 y 1.
(b) Find the moment of inertia about the z -axis of the
funnel in Exercise 34.
(a) Evaluate xxS xyz dS correct to four decimal places.
(b) Find the exact value of xxS x 2 yz dS.
CAS
36. The conical surface z 2 苷 x 2 y 2, 0 z a, has constant
29. Find the value of xxS x 2 y 2z 2 dS correct to four decimal
density k. Find (a) the center of mass and (b) the moment of
inertia about the z -axis.
places, where S is the part of the paraboloid
z 苷 3 2x 2 y 2 that lies above the xy-plane.
CAS
37. A fluid with density 1200 flows with velocity
30. Find the flux of F共x, y, z兲 苷 sin共xyz兲 i x 2 y j z 2e x兾5 k
v 苷 y i j z k. Find the rate of flow upward through the
paraboloid z 苷 9 14 共x 2 y 2 兲, x 2 y 2 36.
across the part of the cylinder 4y z 苷 4 that lies above
the xy-plane and between the planes x 苷 2 and x 苷 2
with upward orientation. Illustrate by using a computer
algebra system to draw the cylinder and the vector field on
the same screen.
2
2
38. A fluid has density 1500 and velocity field
v 苷 y i x j 2z k. Find the rate of flow outward
through the sphere x 2 y 2 z 2 苷 25.
31. Find a formula for xxS F ⴢ dS similar to Formula 10 for the
39. Use Gauss’s Law to find the charge contained in the solid
case where S is given by y 苷 h共x, z兲 and n is the unit
normal that points toward the left.
hemisphere x 2 y 2 z 2 a 2, z 0, if the electric field is
E共x, y, z兲 苷 x i y j 2z k.
32. Find a formula for xxS F ⴢ dS similar to Formula 10 for the
40. Use Gauss’s Law to find the charge enclosed by the cube
case where S is given by x 苷 k共 y, z兲 and n is the unit normal that points forward (that is, toward the viewer when the
axes are drawn in the usual way).
with vertices 共1, 1, 1兲 if the electric field is
E共x, y, z兲 苷 x i y j z k.
41. The temperature at the point 共x, y, z兲 in a substance with
33. Find the center of mass of the hemisphere
x 2 y 2 z 2 苷 a 2, z 0, if it has constant density.
conductivity K 苷 6.5 is u共x, y, z兲 苷 2y 2 2z 2. Find the
rate of heat flow inward across the cylindrical surface
y 2 z 2 苷 6, 0 x 4.
34. Find the mass of a thin funnel in the shape of a cone
z 苷 sx 2 y 2, 1 z 4, if its density function is
共x, y, z兲 苷 10 z.
42. The temperature at a point in a ball with conductivity K is
35. (a) Give an integral expression for the moment of inertia Iz
inversely proportional to the distance from the center of the
ball. Find the rate of heat flow across a sphere S of radius a
with center at the center of the ball.
about the z -axis of a thin sheet in the shape of a surface
S if the density function is .
13.7
Stokes’ Theorem
z
n
n
S
C
0
x
FIGURE 1
y
●
●
971
●
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●
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●
●
●
●
●
●
Stokes’ Theorem can be regarded as a higher-dimensional version of Green’s Theorem. Whereas Green’s Theorem relates a double integral over a plane region D to a
line integral around its plane boundary curve, Stokes’ Theorem relates a surface integral over a surface S to a line integral around the boundary curve of S (which is a space
curve). Figure 1 shows an oriented surface with unit normal vector n. The orientation
of S induces the positive orientation of the boundary curve C shown in the figure.
This means that if you walk in the positive direction around C with your head pointing in the direction of n, then the surface will always be on your left.
Stokes’ Theorem Let S be an oriented piecewise-smooth surface that is bounded
by a simple, closed, piecewise-smooth boundary curve C with positive orientation. Let F be a vector field whose components have continuous partial derivatives on an open region in ⺢ 3 that contains S. Then
y
C
F ⴢ dr 苷 yy curl F ⴢ dS
S
■
976
13.7
CHAPTER 13 VECTOR CALCULUS
Exercises
●
●
●
●
●
●
●
●
●
●
●
●
●
Suppose F is a vector field on ⺢3 whose components have
continuous partial derivatives. Explain why
z
z
4
4
■
■
2
2
y
;
■
Use Stokes’ Theorem to evaluate xxS curl F ⴢ dS.
2
3. F共x, y, z兲 苷 x 2e yz i y 2e xz j z 2e xy k,
;
4. F共x, y, z兲 苷 共x tan1yz兲 i y 2z j z k,
;
S is the hemisphere x 2 y 2 z 2 苷 4, z 0,
oriented upward
S is the part of the hemisphere x 苷 s9 y 2 z 2 that lies
inside the cylinder y 2 z 2 苷 4, oriented in the direction of
the positive x-axis
■
■
■
■
■
■
■
■
■
■
14. F共x, y, z兲 苷 x i y j xyz k,
■
■
■
■
■
■ Use Stokes’ Theorem to evaluate x F ⴢ dr. In each case
C
C is oriented counterclockwise as viewed from above.
7–10
7. F共x, y, z兲 苷 共x y 2 兲 i 共 y z 2 兲 j 共z x 2 兲 k,
C is the triangle with vertices (1, 0, 0), (0, 1, 0),
and (0, 0, 1)
8. F共x, y, z兲 苷 ex i e x j e z k,
■
S is the part of the paraboloid z 苷 x 2 y 2 that lies below
the plane z 苷 1, oriented upward
S is the part of the plane 2x y z 苷 2 that lies in the
first octant, oriented upward
S consists of the four sides of the pyramid with vertices
共0, 0, 0兲, 共1, 0, 0兲, 共0, 0, 1兲, 共1, 0, 1兲, and 共0, 1, 0兲 that lie to
the right of the xz-plane, oriented in the direction of the
positive y-axis [Hint: Use Equation 3.]
■
●
13. F共x, y, z兲 苷 y 2 i x j z 2 k,
6. F共x, y, z兲 苷 xy i e z j xy 2 k,
■
●
■ Verify that Stokes’ Theorem is true for the given vector
field F and surface S.
S consists of the top and the four sides (but not the bottom)
of the cube with vertices 共1, 1, 1兲, oriented outward
[Hint: Use Equation 3.]
■
●
13–15
5. F共x, y, z兲 苷 xyz i xy j x 2 yz k,
■
●
F共x, y, z兲 苷 x 2 y i 13 x 3 j xy k and C is the curve of
intersection of the hyperbolic paraboloid z 苷 y 2 x 2
and the cylinder x 2 y 2 苷 1 oriented counterclockwise
as viewed from above.
(b) Graph both the hyperbolic paraboloid and the cylinder
with domains chosen so that you can see the curve C
and the surface that you used in part (a).
(c) Find parametric equations for C and use them to
graph C.
S is the part of the paraboloid z 苷 9 x y that lies
above the plane z 苷 5, oriented upward
2
■
●
12. (a) Use Stokes’ Theorem to evaluate xC F ⴢ dr, where
2. F共x, y, z兲 苷 yz i xz j xy k,
■
●
and C is the curve of intersection of the plane
x y z 苷 1 and the cylinder x 2 y 2 苷 9 oriented
counterclockwise as viewed from above.
(b) Graph both the plane and the cylinder with domains
chosen so that you can see the curve C and the surface
that you used in part (a).
(c) Find parametric equations for C and use them to graph C.
;
2–6
●
F共x, y, z兲 苷 x 2z i xy 2 j z 2 k
H
x
●
11. (a) Use Stokes’ Theorem to evaluate xC F ⴢ dr, where
P
y
●
C is the boundary of the part of the paraboloid
z 苷 1 x 2 y 2 in the first octant
■
2
●
10. F共x, y, z兲 苷 x i y j 共x 2 y 2 兲 k,
P
2
●
C is the curve of intersection of the plane z 苷 x 4 and
the cylinder x 2 y 2 苷 4
yy curl F ⴢ dS 苷 yy curl F ⴢ dS
x
●
9. F共x, y, z兲 苷 2z i 4x j 5y k,
1. A hemisphere H and a portion P of a paraboloid are shown.
H
●
C is the boundary of the part of the plane 2x y 2z 苷 2
in the first octant
15. F共x, y, z兲 苷 y i z j x k,
S is the hemisphere x 2 y 2 z 2 苷 1, y 0, oriented in
the direction of the positive y-axis
■
■
■
■
■
■
■
■
■
■
■
■
■
■
16. Let
F共x, y, z兲 苷 具ax 3 3xz 2, x 2 y by 3, cz 3 典
Let C be the curve in Exercise 12 and consider all possible
smooth surfaces S whose boundary curve is C. Find the values of a, b, and c for which xxS F ⴢ dS is independent of the
choice of S.
WRITING PROJECT THREE MEN AND TWO THEOREMS
17. Calculate the work done by the force field
2
y
2
z
2
when a particle moves under its influence around the edge
of the part of the sphere x 2 y 2 z 2 苷 4 that lies in the
first octant, in a counterclockwise direction as viewed from
above.
18. Evaluate xC 共 y sin x兲 dx 共z 2 cos y兲 dy x 3 dz,
where C is the curve r共t兲 苷 具sin t, cos t, sin 2t典 ,
0 t 2. [Hint: Observe that C lies on the surface
z 苷 2xy.]
Writing
Project
▲ The photograph shows a stained-
glass window at Cambridge University
in honor of George Green.
977
19. If S is a sphere and F satisfies the hypotheses of Stokes’
F共x, y, z兲 苷 共x z 兲 i 共 y x 兲 j 共z y 兲 k
x
◆
Theorem, show that xxS curl F ⴢ dS 苷 0.
20. Suppose S and C satisfy the hypotheses of Stokes’ Theorem
and f , t have continuous second-order partial derivatives.
Use Exercises 22 and 24 in Section 13.5 to show the
following.
(a) xC 共 f t兲 ⴢ dr 苷 xxS 共 f t兲 ⴢ dS
(b)
(c)
xC 共 f f 兲 ⴢ dr 苷 0
xC 共 f t t f 兲 ⴢ dr 苷 0
Three Men and Two Theorems
Although two of the most important theorems in vector calculus are named after George
Green and George Stokes, a third man, William Thomson (also known as Lord Kelvin),
played a large role in the formulation, dissemination, and application of both of these
results. All three men were interested in how the two theorems could help to explain and
predict physical phenomena in electricity and magnetism and fluid flow. The basic facts of
the story are given in the margin notes on pages 946 and 972.
Write a report on the historical origins of Green’s Theorem and Stokes’ Theorem. Explain
the similarities and relationship between the theorems. Discuss the roles that Green, Thomson, and Stokes played in discovering these theorems and making them widely known.
Show how both theorems arose from the investigation of electricity and magnetism and
were later used to study a variety of physical problems.
The dictionary edited by Gillispie [2] is a good source for both biographical and scientific
information. The book by Hutchinson [5] gives an account of Stokes’ life and the book by
Thompson [8] is a biography of Lord Kelvin. The articles by Grattan-Guinness [3] and
Gray [4] and the book by Cannell [1] give background on the extraordinary life and works
of Green. Additional historical and mathematical information is found in the books by
Katz [6] and Kline [7].
1. D. M. Cannell, George Green, Mathematician and Physicist 1793–1841: The Back-
ground to his Life and Work (London: Athlone Press, 1993).
2. C. C. Gillispie, ed., Dictionary of Scientific Biography (New York: Scribner’s, 1974).
See the article on Green by P. J. Wallis in Volume XV and the articles on Thomson by
Jed Buchwald and on Stokes by E. M. Parkinson in Volume XIII.
3. I. Grattan-Guinness, “Why did George Green write his essay of 1828 on electricity and
magnetism?” Amer. Math. Monthly, Vol. 102 (1995), pp. 387–396.
4. J. Gray, “There was a jolly miller.” The New Scientist, Vol. 139 (1993), pp. 24–27.
5. G. E. Hutchinson, The Enchanted Voyage (New Haven: Yale University Press, 1962).
6. Victor Katz, A History of Mathematics: An Introduction (New York: HarperCollins,
1993), pp. 678–680.
7. Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oxford
University Press, 1972), pp. 683–685.
8. Sylvanus P. Thompson, The Life of Lord Kelvin (New York: Chelsea, 1976).
◆
SECTION 13.8 THE DIVERGENCE THEOREM
983
For the vector field in Figure 4, it appears that the vectors that end near P1 are
shorter than the vectors that start near P1. Thus, the net flow is outward near P1, so
div F共P1兲 0 and P1 is a source. Near P2 , on the other hand, the incoming arrows
are longer than the outgoing arrows. Here the net flow is inward, so div F共P2 兲 0
and P2 is a sink. We can use the formula for F to confirm this impression. Since
F 苷 x 2 i y 2 j, we have div F 苷 2x 2y, which is positive when y x. So the
points above the line y 苷 x are sources and those below are sinks.
y
P¡
x
P™
FIGURE 4
The vector field F=≈ i+¥ j
13.8
Exercises
●
●
●
●
●
●
●
●
●
1. A vector field F is shown. Use the interpretation of
divergence derived in this section to determine whether
div F is positive or negative at P1 and at P2.
●
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●
●
●
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●
●
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●
●
●
●
●
●
3–6
■ Verify that the Divergence Theorem is true for the vector
field F on the region E.
3. F共x, y, z兲 苷 3x i xy j 2xz k,
E is the cube bounded by the planes x 苷 0, x 苷 1, y 苷 0,
y 苷 1, z 苷 0, and z 苷 1
2
P¡
4. F共x, y, z兲 苷 xz i yz j 3z 2 k,
_2
E is the solid bounded by the paraboloid z 苷 x 2 y 2 and
the plane z 苷 1
2
P™
5. F共x, y, z兲 苷 xy i yz j zx k,
E is the solid cylinder x 2 y 2 1, 0 z 1
_2
6. F共x, y, z兲 苷 x i y j z k,
E is the unit ball x 2 y 2 z 2 1
2. (a) Are the points P1 and P2 sources or sinks for the vector
field F shown in the figure? Give an explanation based
solely on the picture.
(b) Given that F共x, y兲 苷 具x, y 2 典 , use the definition of divergence to verify your answer to part (a).
2
■
■
■
■
■
■
■
■
■
■
■
■
7–15
■ Use the Divergence Theorem to calculate the surface
integral xxS F ⴢ dS; that is, calculate the flux of F across S.
7. F共x, y, z兲 苷 e x sin y i e x cos y j yz 2 k,
S is the surface of the box bounded by the planes x 苷 0,
x 苷 1, y 苷 0, y 苷 1, z 苷 0, and z 苷 2
P¡
_2
2
P™
8. F共x, y, z兲 苷 x 2z 3 i 2xyz 3 j xz 4 k,
S is the surface of the box with vertices 共1, 2, 3兲
9. F共x, y, z兲 苷 3xy 2 i xe z j z 3 k,
_2
S is the surface of the solid bounded by the cylinder
y 2 z 2 苷 1 and the planes x 苷 1 and x 苷 2
■
■
984
CHAPTER 13 VECTOR CALCULUS
10. F共x, y, z兲 苷 x 3y i x 2y 2 j x 2yz k,
and components of the vector fields have continuous secondorder partial derivatives.
S is the surface of the solid bounded by the hyperboloid
x 2 y 2 z 2 苷 1 and the planes z 苷 2 and z 苷 2
21.
11. F共x, y, z兲 苷 xy sin z i cos共xz兲 j y cos z k,
2
2
2
2
yy a ⴢ n dS 苷 0,
2
22. V共E 兲 苷
12. F共x, y, z兲 苷 x 3 i 2xz 2 j 3y 2z k,
S is the surface of the solid bounded by the paraboloid
z 苷 4 x 2 y 2 and the xy-plane
23.
24.
25.
15. F共x, y, z兲 苷 e y tan z i y s3 x 2 j x sin y k,
CAS
■
■
■
■
■
■
■
■
16. Use a computer algebra system to plot the vector field
F共x, y, z兲 苷 sin x cos 2 y i sin 3 y cos 4z j sin 5z cos 6x k
in the cube cut from the first octant by the planes x 苷 兾2,
y 苷 兾2, and z 苷 兾2. Then compute the flux across the
surface of the cube.
17. Use the Divergence Theorem to evaluate xxS F ⴢ dS, where
F共x, y, z兲 苷 z 2x i ( 13 y 3 tan z) j 共x 2z y 2 兲 k
and S is the top half of the sphere x 2 y 2 z 2 苷 1.
[Hint: Note that S is not a closed surface. First compute
integrals over S1 and S2, where S1 is the disk x 2 y 2 1,
oriented downward, and S2 苷 S 傼 S1.]
18. Let F共x, y, z兲 苷 z tan1共 y 2 兲 i z 3 ln共x 2 1兲 j z k.
Find the flux of F across the part of the paraboloid
x 2 y 2 z 苷 2 that lies above the plane z 苷 1 and is
oriented upward.
19. Verify that div E 苷 0 for the electric field
E共x兲 苷
Q
x
x 3
ⱍ ⱍ
20. Use the Divergence Theorem to evaluate
yy 共2x 2y z
2
兲 dS
S
where S is the sphere x 2 y 2 z 2 苷 1.
21–26
n
f dS 苷 yyy 2 f dV
E
yy 共 f t兲 ⴢ n dS 苷 yyy 共 f t f ⴢ t兲 dV
2
S
S is the surface of the solid that lies above the xy-plane
and below the surface z 苷 2 x 4 y 4, 1 x 1,
1 y 1
■
yy D
S
S is the surface of the solid bounded by the hemispheres
z 苷 s4 x 2 y 2, z 苷 s1 x 2 y 2 and the plane z 苷 0
■
where F共x, y, z兲 苷 x i y j z k
S
14. F共x, y, z兲 苷 共x 3 y sin z兲 i 共 y 3 z sin x兲 j 3z k,
■
yy F ⴢ dS,
yy curl F ⴢ dS 苷 0
S is the sphere x 2 y 2 z 2 苷 1
■
1
3
S
13. F共x, y, z兲 苷 x 3 i y 3 j z 3 k,
CAS
where a is a constant vector
S
S is the ellipsoid x 兾a y 兾b z 兾c 苷 1
2
■ Prove each identity, assuming that S and E satisfy the
conditions of the Divergence Theorem and the scalar functions
26.
E
yy 共 f t t f 兲 ⴢ n dS 苷 yyy 共 f t t
2
S
■
■
■
2
f 兲 dV
E
■
■
■
■
■
■
■
■
■
■
■
27. Suppose S and E satisfy the conditions of the Divergence
Theorem and f is a scalar function with continuous partial
derivatives. Prove that
yy f n dS 苷 yyy f dV
S
E
These surface and triple integrals of vector functions are
vectors defined by integrating each component function.
[Hint: Start by applying the Divergence Theorem to F 苷 f c,
where c is an arbitrary constant vector.]
28. A solid occupies a region E with surface S and is immersed
in a liquid with constant density . We set up a coordinate
system so that the xy-plane coincides with the surface of the
liquid and positive values of z are measured downward into
the liquid. Then the pressure at depth z is p 苷 tz, where t
is the acceleration due to gravity (see Section 6.5). The total
buoyant force on the solid due to the pressure distribution is
given by the surface integral
F 苷 yy pn dS
S
where n is the outer unit normal. Use the result of Exercise 27 to show that F 苷 W k, where W is the weight of
the liquid displaced by the solid. (Note that F is directed
upward because z is directed downward.) The result is
Archimedes’ principle: The buoyant force on an object
equals the weight of the displaced liquid.
◆
SECTION 13.9 SUMMARY
13.9
Summary
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985
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The main results of this chapter are all higher-dimensional versions of the Fundamental Theorem of Calculus. To help you remember them, we collect them together
here (without hypotheses) so that you can see more easily their essential similarity.
Notice that in each case we have an integral of a “derivative” over a region on the left
side, and the right side involves the values of the original function only on the boundary of the region.
y
Fundamental Theorem of Calculus
b
a
F共x兲 dx 苷 F共b兲 F共a兲
a
b
r(b)
y
Fundamental Theorem for Line Integrals
C
∇f ⴢ dr 苷 f 共r共b兲兲 f 共r共a兲兲
C
r(a)
Green’s Theorem
yy
D
冉
Q
P
x
y
冊
C
dA 苷 y P dx Q dy
D
C
n
Stokes’ Theorem
yy curl F ⴢ dS 苷 y
C
F ⴢ dr
S
S
C
n
S
Divergence Theorem
yyy div F dV 苷 yy F ⴢ dS
E
S
E
n