1032
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CHAPTER 16 VECTOR CALCULUS
Notice also that the gradient vectors are long where the level curves are close to each
other and short where the curves are farther apart. That’s because the length of the gradient vector is the value of the directional derivative of f and closely spaced level curves
M
indicate a steep graph.
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
mMG
f 共x, y, z兲 苷
2 ⫹ y2 ⫹ z2
sx
then
ⵜf 共x, y, z兲 苷
苷
⭸f
⭸f
⭸f
i⫹
j⫹
k
⭸x
⭸y
⭸z
⫺mMGy
⫺mMGz
⫺mMGx
k
2
2 3兾2 i ⫹
2
2
2 3兾2 j ⫹
2
共x ⫹ y ⫹ z 兲
共x ⫹ y ⫹ z 兲
共x ⫹ y 2 ⫹ z 2 兲3兾2
2
苷 F共x, y, z兲
In Sections 16.3 and 16.5 we will learn how to tell whether or not a given vector field is
conservative.
16.1
EXERCISES
1–10 Sketch the vector field F by drawing a diagram like
I
II
3
5
Figure 5 or Figure 9.
1. F共x, y兲 苷 2 共i ⫹ j兲
2. F共x, y兲 苷 i ⫹ x j
3. F共x, y兲 苷 y i ⫹ j
4. F共x, y兲 苷 共x ⫺ y兲 i ⫹ x j
1
1
2
5. F共x, y兲 苷
yi⫹xj
sx 2 ⫹ y 2
6. F共x, y兲 苷
_3
3
_5
5
yi⫺xj
sx 2 ⫹ y 2
7. F共x, y, z兲 苷 k
8. F共x, y, z兲 苷 ⫺y k
_3
III
_5
IV
3
5
9. F共x, y, z兲 苷 x k
10. F共x, y, z兲 苷 j ⫺ i
_3
3
_5
5
11–14 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
_3
_5
12. F共x, y兲 苷 具1, sin y典
13. F共x, y兲 苷 具x ⫺ 2, x ⫹ 1 典
14. F共x, y兲 苷 具 y, 1兾x典
15–18 Match the vector fields F on ⺢3 with the plots labeled I–IV.
Give reasons for your choices.
15. F共x, y, z兲 苷 i ⫹ 2 j ⫹ 3 k
16. F共x, y, z兲 苷 i ⫹ 2 j ⫹ z k
SECTION 16.1 VECTOR FIELDS
31. f 共x, y兲 苷 共x ⫹ y兲2
32. f 共x, y兲 苷 sin sx 2 ⫹ y 2
18. F共x, y, z兲 苷 x i ⫹ y j ⫹ z k
I
II
1
1
z 0
z 0
_1
_1
y
0
_1 0
1
y
_1
1 0x
1
_4
1
0
_1
x
III
IV
1
z 0
z 0
_1
_1
4
4
_4
_4
III
4
_4
IV
4
4
1
_1 0
1
y
1
0
_1
x
_4
_1
y
0
1
_1
1 0x
19. If you have a CAS that plots vector fields (the command
is fieldplot in Maple and PlotVectorField in
Mathematica), use it to plot
F共x, y兲 苷 共 y 2 ⫺ 2 x y兲 i ⫹ 共3x y ⫺ 6 x 2 兲 j
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 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.
21–24 Find the gradient vector field of f .
21. f 共x, y兲 苷 xe xy
22. f 共x, y兲 苷 tan共3x ⫺ 4y兲
23. f 共x, y, z兲 苷 sx ⫹ y ⫹ z
2
2
2
24. f 共x, y, z兲 苷 x cos共 y兾z兲
25–26 Find the gradient vector field ∇ f of f and sketch it.
25. f 共x, y兲 苷 x 2 ⫺ y
26. f 共x, y兲 苷 sx 2 ⫹ y 2
27–28 Plot the gradient vector field of f together with a contour
map of f . Explain how they are related to each other.
27. f 共x, y兲 苷 sin x ⫹ sin y
28. f 共x, y兲 苷 sin共x ⫹ y兲
29–32 Match the functions f with the plots of their gradient
vector fields (labeled I– IV). Give reasons for your choices.
29. f 共x, y兲 苷 x 2 ⫹ y 2
4
_4
_4
4
_4
33. A particle moves in a velocity field V共x, y兲 苷 具x 2, x ⫹ y 2 典 .
If it is at position 共2, 1兲 at time t 苷 3, estimate its location at
time t 苷 3.01.
34. At time t 苷 1 , a particle is located at position 共1, 3兲. If it
Explain the appearance by finding the set of points 共x, y兲
such that F共x, y兲 苷 0.
CAS
4
II
_1
CAS
1033
17. F共x, y, z兲 苷 x i ⫹ y j ⫹ 3 k
I
CAS
||||
30. f 共x, y兲 苷 x共x ⫹ y兲
moves in a velocity field
F共x, y兲 苷 具xy ⫺ 2, y 2 ⫺ 10 典
find its approximate location at time t 苷 1.05 .
35. 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 solve
the differential equations to find an equation of the flow
line that passes through the point (1, 1).
36. (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.
APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES
CHAPTER 16
EXERCISES 16.1
27.
A125
6
PAGE 1032
N
1.
||||
y
6
6
1
_2
0
_1
6
x
1
29. III
35. (a)
_1
33. 共2.04, 1.03兲
31. II
(b) y 苷 1兾x, x 0
y
_2
3.
5.
y
y
x
0
2
0
x
2
x
0
y 苷 C兾x
_2
EXERCISES 16.2
7.
9.
z
x
11. II
19.
z
x
y
13. I
y
15. IV
N
PAGE 1043
243
17
3. 1638.4
5. 8
7. 3
共145 1兲
97
1
13. 5
15. 3
s14 共e 6 1兲
17. (a) Positive
(b) Negative
6
19. 45
21. 5 cos 1 sin 1
23. 1.9633
1
1. 54
1
11. 12
3兾2
27. 3 2
3
25. 15.0074
2.5
2.5
17. III
9. s5 2
.5
The line y 苷 2x
4.5
2.5
4.5
4.5
29. (a)
11
8
1兾e
(b)
1.6
F(r(1))
4.5
1
F ” r ” ’’
2
œ„
21. f 共x, y兲 苷 共xy 1兲e xy i x 2e xy j
x
i
sx 2 y 2 z 2
y
z
j
k
sx 2 y 2 z 2
sx 2 y 2 z 2
23. f 共x, y, z兲 苷
25. f 共x, y兲 苷 2x i j
y
2
_6
_4
_2
0
_2
4
6
x
0
F(r(0))
1
1.6
0.2
33. 2k, 共4兾, 0兲
s2 共1 e14 兲
35. (a) x 苷 共1兾m兲 xC x 共x, y, z兲 ds,
y 苷 共1兾m兲 xC y 共x, y, z兲 ds,
z 苷 共1兾m兲 xC z 共x, y, z兲 ds, where m 苷 xC 共x, y, z兲 ds
(b) 共0, 0, 3兲
1
4
1
2
37. Ix 苷 k ( 2 3 ), Iy 苷 k ( 2 3 )
2
39. 2
41. 26
43. 1.67 10 4 ft-lb
45. (b) Yes
47. ⬇22 J
31.
172,704
5,632,705