Calculo 2 De dos variables_9na Edición - Ron Larson

053714_1501.qxp 10/27/08 1:43 PM Page 1067
053714_1501.qxp 10/27/08 1:43 PM Page 1067
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15.1 Exercises Ejercicios See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
15.1 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
15.1 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
15.1 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
15.1 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
En In los Exercises ejercicios 1–6, 1 a match 6, asociar the vector el campo field vectorial with its con graph. su gráfica. [The
[Las In graphs Exercises gráficas are labeled se 1–6, marcan match (a), (b), a), the b), (c), vector c), (d), d), (e), field y and ƒ).]
with (f).]
In Exercises 1–6, match the vector field with
its
its
graph.
graph.
[The
[The
graphs are labeled (a), (b), (c), (d), (e), and (f).]
a) (a) graphs In Exercises are labeled y1–6, y match (a), (b), the (c), vector (d), b) (b)(e), field and with (f).] y yits graph. [The
(a) graphs are labeled 44
(a), (b), (c), (d), (b)(e), and (f).]
In Exercises 1–6, match the vector field with 6 6
(a)
y
(b)
yits graph. [The
In graphs (a) Exercises are labeled 1–6, y match (a), (b), the (c), vector (d), (b)(e), field and with (f).]
yits graph. [The
4
6
graphs are labeled (a), (b), (c), (d), (e), and (f).]
(a)
4 y
(b)
6 y
x
x x
(a)
y
(b)
y
4 4 4
−6 −6
4
4
−6
−6
4
−6
c)
(c)
y
y
d)
(d)
4
(c)
5
5
(c)
y
(d)
(d)
(c)
5 y
(d)
(c)
(c)
5 y
y
x
5
5
5
5
5
5
(e)
y
(f)
e) y
5 ƒ)
(e)
(e)
y
(f)
(f)
(e)
y
(f)
(e)
(e)
5
4
5
y
y
5 x
5
5
−5
5
−5
−5
5
1. F x, y −5 yi
2. F x, y −1 xj
1. Fx, y −5
yi 5 2. Fx, −3 −2 y xj
−1
−3 2 3
1. 3. 3.
1. Fx,
F x,
x, y
y
yi
yi
yi
xj
2. 4.
xj
4.
2.
Fx,
F x,
x, y
y
xj xi
xi
xj
3yj
1 3yj
3. 5. −5
−3
5. Fx, y x, sin y
6. Fx, y 1 2 xy, 3. 1. F x, x, y yi x, sen xjy
4. 6. 2.
x, xi xj 2xy,
sen
1 3yj
1 4x 2
−5 yi xj
4. F x, y −3 xi 3yj
1 4
5. x, sen 6.
2xy, 1x2 In 1. 3.
Exercises x, 7–16, yi xj
compute F and 2. 4.
sketch x, several xj xi 1
5. F x, y x, sen y
6. F x, y
4x 3yj
2xy, representative
1 4x 2 2
En In vectors 1.
3. 5. F
1
Exercises los x, ejercicios in y
the yi
7–16, yi vector x, 7 sen compute a xjy16, field. 2.
calcular and 4.
6. F
F sketch x,
y y
dibujar xj
several xi 2xy, varios representative 3yj
1 4x 2
vectores
In
representativos vectors 3. Exercises in the vector del campo field. vectorial.
7. 5.
F 1
x, x,
y 7–16, yi compute
ix, sen
xj F and
j y
4. sketch
8. 6.
F x, x,
yseveral xi representative
2i 2xy, 1 3yj
vectors In Exercises in the 7–16, vector compute field. F and sketch several 4x 2
1 representative
vectors
5. F x, 7. 9. x, in
y
the yi vector
x, sen y
xjfield.
6. F x, y
8.
2xy, 1 4x 2
In 7. Exercises F x, y 7–16, i compute j F and 10.
8. sketch F x,
x, yseveral 2i yi
2i representative 2xj
11. In vectors 9. 7. Exercises x, in y, the z 7–16, yi ivector 3yj compute j
xj field. F and 10. 12. 8. sketch x, several yi xi 2i representative
9. F x, y yi xj
10. F x, y yi
2xj
2xj
vectors
11. 13. in y, the vector field.
3yj
12. 14.
11. 7. 9. F x, x, y, y z
4xi iyi 3yjj
xj yj
12. 10. 8. F x, x, y
xi
xi 2i yi x 2 2xj y 2 i j
13. 15. 11. 7.
9. F x, y y, z
4xi i
yii3yj
j
xj
yj j k
14. 16. 10. 12. 8.
F x, y, y
z 2i
2 xi 2 yj zk
13. x, 4xi yj 14.
x, yi xi x 2 2xj y 2 i j
15. y, 16.
y, xi yj zk
In 11. 13. 9. F
Exercises x, y yi 4xi
17–20, 3yj
xjyj
10.
use a computer 12. 14. F x,
algebra y yi
xi x
system 2 2xj
15. x, y, z i j k 16.
x, y, z xi yyj 2 i
to graph zk j
En In several
11.
13. 15. F
los Exercises x,
ejercicios representative
y,
y z
17–20, 4xi
3yj i j a 20, yj
use vectors k
utilizar computer in
12.
un the 14. 16.
sistema vector
F x, algebra y
y,
algebraico field. z xi
system x 2 xi
por y to 2 yji zk
compu-
graph j
In
tadora several
13. Exercises
y representative representar 1 gráficamente vectors in the varios vector vectores 17.
15.
F
In Exercises x,
x,
x,
y 17–20,
y
y, z
4xi
17–20,
field. representativos
15. Fdel x, campo y, z 1 vectorial. i j k 16. F x, y, z xi yj zk
17. 18. several
8 2xyi
i j
yj use a computer
use y 2 k
14.
ja computer 16.
Falgebra x,
algebra x,
y system
y, z
x 2
system xi
y
yj 2 to i graph
to graph zk
j
several representative vectors in the vector field.
In Exercises x, representative 17.
Fx, y 1 817–20, 12y 2xyi 3x use vectors i 2 a computer 2y in the 3x jvector field.
17. F x, y
algebra system to graph
8 2xyi y 2 j
18. In several 17. Exercises x, representative 82xyi 17–20, 1
2y xi
3x
y
yj vectors 2 j
2y zk in the 3x vector field.
18.
19. 18. F
Fx, x, y
y
y,
z
82y 2xyi use
3x yi 2 ja computer algebra system to graph
2y 3x j
several representative 2y 1
xi
x 2 3xi vectors
yjy 2 2y zkz 2 in 3xj the vector field.
19. 17. 18.
F x, x, y, y 2y 8 2xyi xi 3xyjy i 2 j zk 2y 3x j
xi yj zk
19.
20. 19.
Fx, y, y,
z z 1
17.
xi 2 yj zk
18.
F x,
x,
y
2y xi 3xyj i zk 2y 8 2xyi 2
3x j
19.
20.
x, y,
y, z
x x 2 y
y 2 j 2 z 2
2 yj y 2 zkz 2
20. 18.
F x, y, y z 2yxi xi
x 2 3x yj yj
iy 2 zk zk
2yz 2 3x j
20. 19. 20.
Fx, x, x,
y, y, y,
z xi
xi
yj zk
xi x 2 yj yjy 2 zk zk
19. F x, y, z
z 2
20. F x, y, z xi x 2 yj y 2 zk z 2
20. F x, y, z xi yj zk
(d)
(d)
(f)
(f)
−6
−6
6 6
6
x
6
6 x
−6 −6
6
x
−6
−6 y 6 x
y
−6
6
5
5 y
−6
y
−6 5
5 y
y
x
5
5
5
5
5
−5
−5
5
−5
−5 y 5
y
−5
5
3
y
3
−5
2 y
2−5
3
1
123
y
−3 −2 12
y
2 3
−3 −2
−13
−11
2 3
−3
−2
3
−12
−3 −2
−1
2 3
−3 −2 −32
1 2 3
−3 −1
1
−3
−3 −2 −3 2 3
x
SECCIÓN 15.1 Campos vectoriales
15.1 Vector Fields
1067
1067
15.1
15.1 Vector Vector
Fields
Fields 1067
1067
15.1 Vector Fields 1067
15.1 Vector Fields 1067
15.1 Vector Fields 1067
In En Exercises los ejercicios 21–30, a find 30, hallar the conservative el campo vectorial field conservativo for the
In potential para Exercises la función function 21–30, potencial, by find finding the encontrando its conservative gradient. su gradiente.
In Exercises 21–30, find the conservative
vector
vector
field
field
for
for
the
the
potential function by finding its gradient.
21. potential In Exercises 22. f x, y x 2 1
f x, y function x21–30, 2 2y by 2
finding the its conservative gradient. vector field
4 y2 for the
potential function 21. 22. fx, 2 1
23. fx, g x, 5x 2 by
2y 2 finding
3xy y 2 its gradient.
21.
In Exercises 21–30, find the conservative 24.
22. fx,
g x,
y vector
x
sen 3x 4
1
fx, y x 2 2y
field cos 2 4y for the
23. 5x 2 3xy 2 4 y
In
25.
potential Exercises y,
function 21–30, 2
21. z 6xyz
by finding its 24. 26. 22.
gradient.
gx, f y, z sen x 2 1
f x, x 2 2y 2
the conservative vector
3x x 2 field 2
4 ycos 2
for
4y 2 the
23. z 2
potential g x, yfunction 5x 2 by 3xy finding y 2 its 24. gradient. gx, y sen 3x cos 4y
25. y, 6xyz
26.
y, 2 4y 2 27. 21. y z xz
z ye 28. 22. g x 2 1
25.
23.
fx, g x,
y, y
z x5x 2 6xyz
2 2y3xy 2 y 2 2
26.
24.
fx,
gx,
y,
y
z
sen 3x
x4 2 ycos 2 4y
4y z 2
21. z
x
xz y
27. 23. 25. gx, g y,
y, z
6xyz
2 ye 3xy y 2 22.
28. 24. 26. fx, gx, y y,
y, z x sen 1
fx, y x 2 2y 2
3x x x2 4 2 y
y cos z
4y 4y xz
2 z 2
29. 27. 23. gx, gh y, y z 5xzxy ln x y 30. h x arcsen yz
25. fx, y, 6xyz
ye 3xy x2 y 2 28. 24.
gx, y, y z sen
26. fx, y, y x 2 z 4yxz
27.
29. gx, y, z z ye 28.
30.
gx, y, z
3x cos x 4y 2y
z 2
ln hx, arcsen In 29. 25.
Exercises hfx, y, y, z
6xyz
31–34, lnverify x ythat the 30. 26. fx,
vector hx, y, y, z
field is x z
yconservative.
arcsen x 2 x
z
4yz
xz
y z 2
27. 29. En los gx, h x, ejercicios y, y, zxy 31 lnye a x34, x2 verificar y 28. 30. que gx, hx, el y, y, campo vectorial es con-
In Exercises 31–34, verify that the vector field y
is zx arcsen conservative. xz
yz xz y
In 27.
servativo. Exercises gx,
31. y, y z 31–34, z ye verify that the 28. vector gx, y, field z 1is 29. h x, y, z xy2 xy i ln x 2 yjy
32. 30. Fhx, y, y z 2 31. x
x
z conservative.
yi arcsen
x xjyz
y
In Exercises 31–34, verify that the vector field is conservative.
29. h xy2 2 yj
32.
31. 2
yi xj
33.
In Exercises x, x, y, y z xy ln x y 30. hx, y, z 1 x arcsen yz
31–34, sen i yi verify x 2 x yj
cos that yj the 32.
34.
vector F x, y
field is conservative.
31. 1
33. xy i x 2 2
yi xj
x
In Exercises 31–34, verify yj that the 32. vector F x, yfield 34.
x, xis 2 xy yi
conservative.
yi xj
33. 31. xj
In Exercises i
35–38, x 2 determine yj 1
34. 32. F x, y
2 33. whether the vector field is
In conservative. Exercises Justify 35–38, your determine answer.
xyx yi xj
31. 1
i x 2 yj 32. 34. F x, y
2
yi xj
whether x
the vector field In
is
33. Exercises conservative. En los ejercicios Justify 35–38,
35 determine
a your 38, determinar answer.
34. whether Fsi x, el y the
1
vector
campo vectorial es conservativo.
3xj
35.
5y
xy yi xj field is
conservative. 33. In Exercises Justify 35–38, your determine answer. 1
34. whether F x, y the vector field is
xy yi xj
35. conservative. x, 5y Justify 2
2 yi your 3xj answer.
36. 35. In Exercises F x, y 5y 35–38, y e 2x yi y yi determine 3xj xj whether the vector field is
In conservative. 2 35. Exercises
36.
F x,
x, y Justify 5y 35–38, 2 2x yi your determine y 3xjanswer.
whether the vector field is
2
36. conservative. F x, y Justify yi xj
37.
1
35.
x, 5y y e 2x y your yi
x 2 y i answer. xj
36.
x, 2 yi 3xj j 2 37.
x, y e 2x y yi xj
35. F x, y 5y
2
yi
36.
2 38.
y e 1 3xj
37. F x, y
2
2x y yi
2 xj
37.
x 2 1 y yi 36. F x, y
2 xj
1 xy 38.
x, y e 2x y yi
x 2 y i xj j
1
2 yi 38.
xj
37. F x, y
xy yi
In Exercises 39– x 2 48, y determine xj
38.
1 1 xy
37. F x, y
2 yi xj whether the vector field is
In conservative. Exercises If 39– it x1 2 is, 1
48, find xy y i determine a potential whether function the for vector the vector field field.
In
is
38. Exercises F x, y 39– 48, determine yi xj whether the vector field is
conservative. If it is, find potential function for the vector 39. yi 1 1
38. conservative. In Exercises F x, y If 39– it is, 48, xjfind xydetermine yi a potential xj 40. whether function F x, y the for 3x 2 the y 2 ivector 2xfield 3 field.
yj field. is
39. 41. conservative. En los x, ejercicios If 2xyi it 1
39 is, xja find xy
48, x 2 j determinar a potential 40. 42.
function si x, el campo for 3x xe 2x2 vectorial the 2 y 2yi 2x 3 es yj xj field. conservativo.
Si lo es, calcular una función potencial y
39. In Exercises F x, y yi 39– 48, xj determine 40. whether F x, y the 3x vector field is
para i 2x
él.
yj
41. In conservative. 39. Exercises 2xyi 2 42.
xe 1
x2 2yi xj
43. x, If 39– it is, 48, xjfind determine a potential function for the vector field.
15y3 i 5xy 2 40. whether x, the 3x
j 44.
vector
41. 42. F x, y xe 2
yi i 2x
2xj
yj
y
field is
F x, y 2xyi x 2 j
2yi xj
conservative. If it is, find 43. y
39. 41. F x, y 15y3 2xyi xj x
5xy a potential function for the 2 44. 40. 42.
F x, x, y 3x xe 2 y
vector field.
j
1
2
yi y 2 i2yi 2xj 2x 3 xj yj
43. 2y
xi yj
45.
39. 46.
2y x i x
41.
F x, 2xyi x 2
j
45. xi yj
x, 46.
x, y 42.
x, xe 1 x2 43. x, x, y yi 15y3 ixj
5xy 2 j 44. 40. F x, y 3x
2yi xj
i 2 5xy 2 2
y yyi 2 i 2xj
yj
j 44.
x, 47.
e xy 2
yiy 2 2xj
41. F x, y 2xyi
1
43. 2y
xi yj
45. F x, y 15y3 2y
cos i yi 5xysen 2 j yj 44.
2 yi
yxi
yj
2 2xj
45.
47.
x, 2xi
cos yi 2yj
sen 46.
x 46. F x, y
x i x 2 j
y j
42. F x, y xe x2 y
2yi xj
y j
x 2 y
43.
yj
48. 47. F x, y 1
2
e i
2y
xi yj
45. x, 2xi
y 2yj
46. F x, y
48.
x 2
5xy 2 j 44. F x, y
x x2 yi
y 2 2xj
cos yi
47.
x, e x cos yi y j
sen yj
y
2y 2xi 2yj
sen yj
xi
2 yj
2
48. 45. F x, y
2 22 46. F x, y
In 47. 48. Exercises x, x, 49–52, e2xi
x i x 2
2 cos yi find 2yj y j
22 sen curl yj
x
F for the vector field y
at the given
In point. 47.
F x, y
Exercises e
49–52, 2xi
cos yiy find 2yj
sen yj
curl for the vector field at the In
given
48. Exercises F x, y 49–52, find curl F for the vector field at the given
point. 2xi
point. In Exercises Campo vectorial 49–52, x 2 y2yj
48. F x, y find 2 2
curl F for the vector Punto field at the given
x
49. point.
y 22
In Exercises Campo F x, y, z vectorial 49–52, xyzi find xyzj curl F xyzk for the vector Campo vectorial
Punto
Punto
2, 1, field 3 at the given
49. 50. In point. En Exercises los Campo x, ejercicios y, vectorial 49–52, xyzix a find 52, xyzj 2xzj curl calcular F xyzk
for el the rotacional vector Punto
2, 1, field del 1, 3campo at the given vectorial
49. F
en
x,
el
y,
punto
z xyzi
dado.
xyzj xyzk 2, 1, 3
point.
50. 51. 49.
Campo x, y, vectorial
exyzi 2x sen yi 2xzj xyzje x cos xyzk yj Punto
0, 2, 0, 1, 1, 13
50. F x, y, z x 2 2xzj yzk 2, 1, 3
51. 52. 49. 50.
Campo x, y, vectorial
xyzix sen yi ixyzj 2xzjj x cos k
xyzk
51. F x, y, z e x sen yi e x cos
yj
yj
Punto
0, 3, 2, 0, 2, 1, 0, 0, 1, 1
03
3
52. 49.
50. 51.
F x, y, 52. x, y, z xyzi
x 2 zi sen yi2xzj e x cos e xyz xyzj
i j kyzk
xyzk yj
3, 2,
2, 0, 2, 1, 0, 3, 2, 1,
3
01
3
50.
51. 52. F x, y, z x e x zi
sen yi i2xzjje x cos
yzk k yj
2,
0, 3, 0, 2, 1,
10
3
51.
52.
F x,
x,
y,
y,
z e x sen xyz yi
i j
e x cos
k
yj 0,
3,
0,
2,
1
0
52. F x, y, z e xyz i j k 3, 2, 0