x x x x x x x x x x x fy dx x x x x x

CHAPTER 16 REVIEW
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1107
EXERCISES
1. A vector field F, a curve C, and a point P are shown.
13–14 Show that F is conservative and use this fact to evaluate
(a) Is xC F ⴢ dr positive, negative, or zero? Explain.
(b) Is div F共P兲 positive, negative, or zero? Explain.
xC F ⴢ dr along the given curve.
13. F共x, y兲苷共4 x 3 y 2 ⫺ 2 x y 3兲 i ⫹ 共2 x 4 y ⫺ 3 x 2 y 2 ⫹ 4y 3 兲 j,
C: r共t兲苷共t ⫹ sin ␲ t兲 i ⫹ 共2t ⫹ cos ␲ t兲 j, 0 艋 t 艋 1
y
14. F共x, y, z兲苷 e y i ⫹ 共xe y ⫹ e z 兲 j ⫹ ye z k,
C is the line segment from 共0, 2, 0兲 to 共4, 0, 3兲
C
15. Verify that Green’s Theorem is true for the line integral
xC xy 2 dx ⫺ x 2 y dy, where C consists of the parabola y 苷 x 2
x
from 共⫺1, 1兲 to 共1, 1兲 and the line segment from 共1, 1兲 to
共⫺1, 1兲.
P
16. Use Green’s Theorem to evaluate xC s1 ⫹ x 3 dx ⫹ 2 xy dy ,
where C is the triangle with vertices 共0, 0兲, 共1, 0兲, and 共1, 3兲.
17. Use Green’s Theorem to evaluate xC x 2 y dx ⫺ x y 2 dy,
2–9 Evaluate the line integral.
2.
where C is the circle x 2 ⫹ y 2 苷 4 with counterclockwise
orientation.
xC x ds,
C is the arc of the parabola y 苷 x 2 from (0, 0) to (1, 1)
3.
18. Find curl F and div F if
xC yz cos x ds ,
C: x 苷 t, y 苷 3 cos t, z 苷 3 sin t, 0 艋 t 艋 ␲
F共x, y, z兲苷 e⫺x sin y i ⫹ e⫺y sin z j ⫹ e⫺z sin x k
4.
xC y dx ⫹ 共x ⫹ y 兲 dy, C is the ellipse 4x ⫹ 9y 苷 36 with
counterclockwise orientation
19. Show that there is no vector field G such that
5.
xC y dx ⫹ x dy , C is the arc of the parabola x 苷 1 ⫺ y
from 共0, ⫺1兲 to 共0, 1兲
20. Show that, under conditions to be stated on the vector fields F
6.
xC sxy dx ⫹ e y dy ⫹ xz dz,
2
3
2
2
2
C is given by r共t兲苷 t 4 i ⫹ t 2 j ⫹ t 3 k, 0 艋 t 艋 1
7.
C is the line segment from 共1, 0, ⫺1兲, to 共3, 4, 2兲
8.
xC F ⴢ dr,
where F共x, y兲苷 x y i ⫹ x 2 j and C is given by
r共t兲苷 sin t i ⫹ 共1 ⫹ t兲 j, 0 艋 t 艋 ␲
9.
where F共x, y, z兲苷 e z i ⫹ xz j ⫹ 共x ⫹ y兲 k and
C is given by r共t兲苷 t 2 i ⫹ t 3 j ⫺ t k, 0 艋 t 艋 1
xC F ⴢ dr,
10. Find the work done by the force field
F共x, y, z兲苷 z i ⫹ x j ⫹ y k in moving a particle from the
point 共3, 0, 0兲 to the point 共0, ␲ 兾2, 3兲 along
(a) a straight line
(b) the helix x 苷 3 cos t, y 苷 t, z 苷 3 sin t
11–12 Show that F is a conservative vector field. Then find a func-
tion f such that F 苷 ∇ f .
11. F共x, y兲苷共1 ⫹ x y兲e
and G,
curl共F ⫻ G兲苷 F div G ⫺ G div F ⫹ 共G ⴢ ⵜ 兲F ⫺ 共F ⴢ ⵜ 兲G
21. If C is any piecewise-smooth simple closed plane curve
xC x y dx ⫹ y 2 dy ⫹ yz dz,
xy
curl G 苷 2 x i ⫹ 3yz j ⫺ xz 2 k.
2
i ⫹ 共e ⫹ x e 兲 j
y
2 xy
12. F共x, y, z兲苷 sin y i ⫹ x cos y j ⫺ sin z k
and f and t are differentiable functions, show that
xC f 共x兲 dx ⫹ t共 y兲 dy 苷 0 .
22. If f and t are twice differentiable functions, show that
ⵜ 2共 ft兲苷 f ⵜ 2t ⫹ tⵜ 2 f ⫹ 2ⵜ f ⴢ ⵜt
23. If f is a harmonic function, that is, ⵜ 2 f 苷 0, show that the line
integral x fy dx ⫺ fx dy is independent of path in any simple
region D.
24. (a) Sketch the curve C with parametric equations
x 苷 cos t
y 苷 sin t
z 苷 sin t
0 艋 t 艋 2␲
(b) Find xC 2 xe 2y dx ⫹ 共2 x 2e 2y ⫹ 2y cot z兲 dy ⫺ y 2 csc 2z dz.
25. Find the area of the part of the surface z 苷 x 2 ⫹ 2y that lies
above the triangle with vertices 共0, 0兲, 共1, 0兲, and 共1, 2兲.
26. (a) Find an equation of the tangent plane at the point 共4, ⫺2, 1兲
to the parametric surface S given by
r共u, v兲苷 v 2 i ⫺ u v j ⫹ u 2 k
0 艋 u 艋 3, ⫺3 艋 v 艋 3
1108
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;
(b) Use a computer to graph the surface S and the tangent
plane found in part (a).
(c) Set up, but do not evaluate, an integral for the surface area
of S.
(d) If
x2
y2
z2
k
F共x, y, z兲苷
2 i ⫹
2 j ⫹
1⫹x
1⫹y
1 ⫹ z2
CAS
CHAPTER 16 VECTOR CALCULUS
Evaluate xC F ⴢ dr, where C is the curve with initial point
共0, 0, 2兲 and terminal point 共0, 3, 0兲 shown in the figure.
z
(0, 0, 2)
0
find xxS F ⴢ dS correct to four decimal places.
(0, 3, 0)
(1, 1, 0)
27–30 Evaluate the surface integral.
y
(3, 0, 0)
27.
xxS z dS, where S is the part of the paraboloid z 苷 x 2 ⫹ y 2
that lies under the plane z 苷 4
28.
xxS 共x 2 z ⫹ y 2 z兲 dS,
where S is the part of the plane
z 苷 4 ⫹ x ⫹ y that lies inside the cylinder x 2 ⫹ y 2 苷 4
29.
xxS F ⴢ dS, where F共x, y, z兲苷 x z i ⫺ 2y j ⫹ 3x k and S is
the sphere x 2 ⫹ y 2 ⫹ z 2 苷 4 with outward orientation
30.
xxS F ⴢ dS,
x
38. Let
F共x, y兲苷
共2 x 3 ⫹ 2 x y 2 ⫺ 2y兲 i ⫹ 共2y 3 ⫹ 2 x 2 y ⫹ 2 x兲 j
x2 ⫹ y2
Evaluate x䊊C F ⴢ dr, where C is shown in the figure.
where F共x, y, z兲苷 x 2 i ⫹ x y j ⫹ z k and S is the
part of the paraboloid z 苷 x 2 ⫹ y 2 below the plane z 苷 1
with upward orientation
y
C
x
0
31. Verify that Stokes’ Theorem is true for the vector field
F共x, y, z兲苷 x i ⫹ y j ⫹ z k, where S is the part of the
paraboloid z 苷 1 ⫺ x 2 ⫺ y 2 that lies above the xy-plane and
S has upward orientation.
2
2
2
32. Use Stokes’ Theorem to evaluate xxS curl F ⴢ dS, where
F共x, y, z兲苷 x 2 yz i ⫹ yz 2 j ⫹ z 3e xy k, S is the part of the
sphere x 2 ⫹ y 2 ⫹ z 2 苷 5 that lies above the plane z 苷 1, and
S is oriented upward.
39. Find xxS F ⴢ n dS, where F共x, y, z兲苷 x i ⫹ y j ⫹ z k and S is
the outwardly oriented surface shown in the figure (the
boundary surface of a cube with a unit corner cube removed).
z
33. Use Stokes’ Theorem to evaluate xC F ⴢ dr, where
F共x, y, z兲苷 x y i ⫹ yz j ⫹ z x k, and C is the triangle with
vertices 共1, 0, 0兲, 共0, 1, 0兲, and 共0, 0, 1兲, oriented counterclockwise as viewed from above.
(0, 2, 2)
(2, 0, 2)
1
34. Use the Divergence Theorem to calculate the surface integral
xxS F ⴢ dS, where F共x, y, z兲苷 x i ⫹ y j ⫹ z k and S is the
surface of the solid bounded by the cylinder x 2 ⫹ y 2 苷 1 and
the planes z 苷 0 and z 苷 2.
3
3
3
35. Verify that the Divergence Theorem is true for the vector
field F共x, y, z兲苷 x i ⫹ y j ⫹ z k, where E is the unit ball
x 2 ⫹ y 2 ⫹ z 2 艋 1.
36. Compute the outward flux of
F共x, y, z兲苷
xi⫹yj⫹zk
共x 2 ⫹ y 2 ⫹ z 2 兲 3兾2
through the ellipsoid 4 x 2 ⫹ 9y 2 ⫹ 6z 2 苷 36.
37. Let
F共x, y, z兲苷共3x 2 yz ⫺ 3y兲 i ⫹ 共x 3 z ⫺ 3x兲 j ⫹ 共x 3 y ⫹ 2z兲 k
1
1
y
S
x
(2, 2, 0)
40. If the components of F have continuous second partial deriva-
tives and S is the boundary surface of a simple solid region,
show that xxS curl F ⴢ dS 苷 0.
41. If a is a constant vector, r 苷 x i ⫹ y j ⫹ z k, and S is an ori-
ented, smooth surface with a simple, closed, smooth,
positively oriented boundary curve C, show that
yy 2a ⴢ dS 苷 y
C
S
共a ⫻ r兲 ⴢ dr
APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES
23. 3 21. 6
713
180
19.
1
29. 2 8
3
4
31. 0.1642
25. 0
33. 3.4895
EXERCISES 17.1
[
1
All solutions approach either
0 or as x l .
10
g
9. 80
f
_3
3
(b)
5
_10
z 0
5
2
0
(c) x 苷 3 cos t, y 苷 3 sin t,
z 苷 1 3共cos t sin t兲,
0 t 2
0
2
2
y
17. y 苷 2e3x兾2 ex
19. y 苷 e x/2 2xe x兾2
21. y 苷 3 cos 4x sin 4x
23. y 苷 ex共2 cos x 3 sin x兲
2
25. y 苷 3 cos( 2 x) 4 sin( 2 x)
1
x
0
EXERCISES 17.2
y
0
2
2
0
_2
x
PAGE 1124
3. y 苷 c1 c2 e 2x 3
7
4
cos 4x 20 sin 4x
1
5. y 苷 e 共c1 cos x c2 sin x兲 10 ex
3
11
1 x
7. y 苷 2 cos x 2 sin x 2 e x 3 6x
1
9. y 苷 e x ( 2 x 2 x 2)
3
11.
The solutions are all asymptotic
to yp 苷 101 cos x 103 sin x as
x l . Except for yp , all
_3
8 solutions approach either yp
or as x l .
1
40
1
2x
17. 3
PAGE 1103
7. 9兾2
11. 32兾3
13. 0
arcsin(s3兾3)
17. 13兾20
19. Negative at P1 , positive at P2
21. div F 0 in quadrants I, II; div F 0 in quadrants III, IV
15. 341 s2兾60 N
1. y 苷 c1 e2x c2 ex 2 x 2 2 x 1
_2
N
e 2x
e x3
e 1
1 e3
3
2
_2
EXERCISES 16.9
27. y 苷
1
29. No solution
31. y 苷 e2x 共2 cos 3x e sin 3x兲
33. (b) 苷 n 2 2兾L2, n a positive integer; y 苷 C sin共n x兾L兲
4
z
5. 2
9. 0
]
1
PAGE 1097
7. 1
3. 0
5. 0
11. (a) 81兾2
PAGE 1117
13. P 苷 et c1 cos (10 t) c 2 sin (10 t)
15.
EXERCISES 16.8
N
1. y 苷 c1 e 3x c 2 e2x
3. y 苷 c1 cos 4x c 2 sin 4x
5. y 苷 c1 e 2x兾3 c 2 xe 2x兾3
7. y 苷 c1 c 2 e x兾2
2x
9. y 苷 e 共c1 cos 3x c 2 sin 3x兲
11. y 苷 c1 e (s31) t兾2 c 2 e (s31) t兾2
where D 苷 projection of S on xz-plane
37. 共0, 0, a兾2兲
39. (a) Iz 苷 xxS 共x 2 y 2 兲共x, y, z兲 dS
(b) 4329 s2 兾5
8
41. 0 kg兾s
43. 3 a 30
45. 1248
N
A127
CHAPTER 17
27. 48
xxS F ⴢ dS 苷 xxD 关P共h兾x兲 Q R共h兾z兲兴 dA,
35.
||||
81
20
_3
13. yp 苷 Ae 共Bx 2 Cx D兲 cos x 共Ex 2 Fx G兲 sin x
15. yp 苷 Ax 共Bx C 兲e 9x
17. yp 苷 xex 关共Ax 2 Bx C 兲 cos 3x 共Dx 2 Ex F兲 sin 3x兴
2x
CHAPTER 16 REVIEW
N
PAGE 1106
True-False Quiz
1. False
19. y 苷 c1 cos ( 2 x) c 2 sin ( 2 x) 3 cos x
1
3. True
5. False
7. True
7.
110
3
9.
4兾e
33. 2
1
25.
3. 6 s10
5.
11. f 共x, y兲苷 e y xe xy
(27 5 s5 )
29.
共兾60兲(391 s17 1)
17. 8
27.
11
12
(b) Positive
1
21. y 苷 c1e x c2 xe x e 2x
23. y 苷 c1 sin x c 2 cos x sin x ln共sec x tan x兲 1
25. y 苷关c1 ln共1 ex 兲兴e x 关c2 ex ln共1 ex 兲兴e 2x
Exercises
1. (a) Negative
1
4
15
13. 0
[
]
27. y 苷 e x c1 c 2 x 2 ln共1 x 2 兲 x tan1 x
1
1
6
37. 4
39. 21
64兾3
EXERCISES 17.3
N
PAGE 1132
1. x 苷 0.35 cos (2 s5 t)
3. x 苷 5 e6t 5 et
1
6
5.
49
12
kg

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