Calculo 2 De dos variables_9na Edición - Ron Larson

1053714_1502.qxp 10/27/08 1:44 PM Page 1080
CAPÍTULO 15 Análisis vectorial
1080 Chapter 15 15 Vector Analysis
1080
1080 Chapter 15 Vector Analysis
Chapter 15 Vector Analysis
En In In los Exercises ejercicios 27–32, a evaluate 32, evaluar
1080 Chapter 15 Vector Analysis
In Exercises 27–32, evaluate
In Exercises 27–32, evaluate
F dr dr C C
C
In FExercises F dr
dr 27–32, evaluate
donde where C
Cestá is is represented representa by by por
rr t rt. t ..
C
27. where 27. F
F
where
x, Cx, dr
yis y
C is
represented xi xi
represented yj yj
by r t .
by r t .
C
C: C: 27.
rrF tt x, y ti ti xi tj, tj,
yj 0 tt 1
27. F x, y xi yj
where
28. 28.
F x, x,
C
yC: y
is represented
r xyi xyi t ti yj yj tj,
by r
0
t .
t 1
C: r t ti tj, 0 t 1
27.
C: FC: 28. x, rryF tt x, y xi 4 cos cos xyi yj
ti ti yj 4 sen sen tj, tj,
0 tt 2
28.
x, xyi yj
29. 29.
C: F x, x, r yC: yt r 3xi 3xi ti 4 tj,
4yj 4yj cos 0ti t 4 sen 1 tj, 0 t 2
C: t 4 cos ti 4 sen tj, 0 t 2
28.
C: FC: 29. x, rry F tt x, y cos xyi cos ti ti 3xi yj
sen sen 4yj
tj, tj,
0 tt 2
29.
x, 3xi 4yj
30. 30.
C: F x, x, r yC: yt r 3xi 3xi t4 cos cos 4yj ti 4yj ti 4 sen tj, tj, 00 t t 22
C: t cos ti sen tj, 0 t 2
29.
C: FC: 30. x, rry F tt x, y ti 3xi ti 3xi 4yj
4 4yj
t 2 t
30.
x, 3xi 4yj
2 j, j,
2 tt 2
31. 31.
C: F x, x, r y, C: y, t zz r tcos xyi xyi ti ti xzj xzj sen4 C: t ti 4 t 2 tj,
yzk yzk t
j,
2 0j,
t 2 t2
2
2 t 2
30.
C: FC: 31. x, rry F tt x, y, ti 3xi tiz t 2 t
31.
x, y, z xyi 2 j4yj
xyi j 2tk, 2tk, xzj 0 yzk
tt 1
xzj yzk
32. 32.
C: F x, x, r y, C: y, t zzr ti xx C: t ti 2 2 iiti t 2 y4 y
j 2 2 jtj 2tk,
2 jt 2 zj,
z 2 2 2tk, k 2 0 t t 2 1
0 t 1
1
1
31.
C: FC: 32. x, rry, F tt zx, y, 2 z sen xyi sen
32.
x, x 2 ti ti x
i y 2 xzj i 2 2 ycos j z 2 j 2 yzk
tj tj z
k
2 k2 2t 2t t22 2 k, k,
0 tt
C: rC: t r ti 2 t 1
CAS En In In
los Exercises C: t
ejercicios 33
233 sen
33 and and ti 2 sen j ti 2tk, 2 cos 0 tjt CAS
y 34, 34, 34, 2 cos
utilizar use use tj
un
a sistema
computer 2t 2 k, 2t 1
2 k, 0 t
0
algebraico algebra t
por system
compu-
to to
tadora 32.
evaluate F x,
y y,
the calcular
the z integral x
In Exercises 2 i y
In Exercises 33 and
la 33 integral and 2 j z
34, 2 k
CAS
use a computer algebra system to
CAS
34, use a computer 1
C: algebra system to
evaluate r t the 2 sen integral ti 2 cos tj 2t 2 k, 0 t
evaluate the integral
F dr dr
CAS In C C
C Exercises 33 and 34, use a computer algebra system to
evaluate
F
F dr the
dr
integral
donde where C
C
está is is represented
representa by by
C
por
rr t rt.
t ..
33. where 33. FFx, F
where
Cx, dr
y, is y, z zrepresented z
C is represented
xx 2 2 zi zi
6yj 6yj
by
by r t yz yz .
2 2 r k
t .
C
33.
C: C: 33.
F x, rt rrF y, tt x,
z y, ti tiz x 2 zi t 2 t 2 jxj 2 zi
6yj ln ln tk, tk, 6yj
yz 2 k
1
≤yz 2 tk
t
≤ 3
where C
C:
is represented
r t xi xi
ti yj zk
34. 34.
Fx, C: r y,
t
z z
ti t 1 t 3
2 yjt by
F x, y, z j ln 2 j
r
zk
t
ln
.
tk, 1 t 3
tk,
33. F x, y, z x x xx 2 2 yy 2 2 xi yj zkz z 2 2
34. F x, y, z
2 zi xi6yj yj yz 2 zk k
34.
F
C: x,
rt r y, tt z
ti ti x tj tj 2 t 2 j x
y e 2 e 2 t ln k,
t k, tk, y
z 2 0 ≤1 z t 2 t
≤t 2 3
C: r t Work Work C: In
rIn t Exercises ti xiti tj 35–40, yjtj e t k, zke find find 0 t k, 0 t 2
34. F x, y, z
the the t work work 2
done done by by the the force force field field
Trabajo En los ejercicios 35 a 40, hallar el trabajo realizado
F on on a particle moving x along along the the given given path. path.
por Work el
Work
campo
In
In Exercises de
Exercises 2 y 2 z
fuerzas 35–40, F
35–40, 2 sobre find the una
find
work partícula
the work done
done que by the se
by
mueve
the force
force field a lo
field
C:
largo 35. F 35. on F
a de x, x,
on r t
particle yla y
a
trayectoria
particle tj xi xi moving 2yj 2yj
moving e t k,
dada.
along 0 the t given 2 path.
along the given path.
Work
35.
C: C: 35.
F x, xIn xF Exercises x,
y t, t, y y
xi t 3 xi t
2yj
3
from 35–40, from 2yj
0, 0, find 0
to to the 2, 2, work 8 done by the force field
F on yy
a particle
C: x
moving
t, y C: x t, y
C: x = t, y = t 3 t from 0, 0 to 2, 8y
desde 3 t
along 3 from
the
0, 0
given
to y2, path.
8
35. F x, y
0, 0 hasta 2, 8
y
88
y
(2, (2,
y 8) 8)
xi 2yj
y
y
y
C: x t, y t 11
68
6
8 (2, 8)
3 from 0, 0 to 2, 8
(2, 8)
y 8 (2, 8)
y
1
1
46
4
6
C
1
6
C
8 (2, 8)
C
C
24
2
4
C
4 C
1 C
xx
6
2 C
11
2
xx
x
4 2
x
22 44 66 88
C
C
1
x
1 x
x
1
Figure 2 2 for for 35 35 2 4 6 8
4 6 8 x
Figure for for 36 36
x
2 4 6 8
36. Figure 36.
FFigure x, x, for yy for
35 xx 2 2 35
ii xyj xyj
Figure for 361
x Figure for 36
Figura 2 para 4 35 Figura para 36
36.
C: C: 36.
F x, xxF x,
y cos cos y
x 2 3 i 3 6 t, t,
yxy
xyj
2 8
i sen sen xyj
3 3 tt
from from
1, 1, 0
to to
0, 0, 1
Figure for
36. Fx, C: xy C: 35
x cos x 23 it,
cos y xyj
3 t, y
sen 3 sen from 1, 0 to 0, 1
t from
3 t Figure for 36
1, 0 to 0, 1
36. C: F x, x y cosx 32 t, i y xyj
sen sin 3 t desde 1, 0 hasta 0, 1
C: x cos 3 t, y sen 3 t from 1, 0 to 0, 1
37. 37.
F x, x, yy xi xi yj yj
C:
C: 37.
triángulo counterclockwise F x, y cuyos xi vértices yj
around son the the (0, triangle 0), (1, with with 0) y (0, vertices 1), recorrido
0, 0, 0
,,
37. F x, y xi yj
en 1, 1, C:
sentido 0
, counterclockwise , and and 0, contrario 0, 1
(Hint: a las around See See manecillas Exercise the triangle del 17a.) 17a.) reloj. with (Sugerencia: vertices 0, 0 ,
C: counterclockwise around the triangle with vertices 0, 0 ,
yy
Ver ejercicio 17a.)
yy
37. F x, 1, y 1,
0 , and xi0
, and
0, yj 0, 1 (Hint: See Exercise 17a.)
1 (Hint: See Exercise 17a.)
C: y counterclockwise y
around the triangle 33
ywith yvertices 0, 0 ,
(0, (0, 1)
11
1, 1) 0 , and 0, 1 (Hint: See Exercise 17a.)
3
3
y (0, 1)
(0, 1)
y
C
1
1
C
11
C
3 C
(0, 1) C
1
1
xx
1 C
xx
−2 −2 −1 −1 11 22
11
C
x
x
−1 −1
x
x
−2 −1 1 2
C 1
−2 −1 1 1 2
1
−1
Figure for for 37 37 Figure for for −1
38 38
x
x
−2 −1 1 2
38. Figure 38.
Figura FFigure x, x, for yy para for 37
37 yi yi 137 xj xj Figura Figure para for 38 38
Figure for −138
C: C: 38.
Fx, counterclockwise F x, y yi
y yi xj
along along xj the the semicircle yy 4 xx 38. F y xj
2 2
from from
Figure for 2, 2, C:
37 0counterclockwise to to
2, 2, 0 along Figure the semicircle for 38 y 4
C: counterclockwise contorno del semicírculo along the ysemicircle
4 xy 2 desde 4 2, x0
2 x
C:
from
2 from
hasta
39. 38. 39.
F x, x, x, 2, y, yy, zz 2,
0
xi 0
0 to recorrido yi xito
2, 0yj xj yj2, 0
en
5zk 5zk
sentido contrario a las manecillas del
39.
FC:
C: 39.
x, rcounterclockwise reloj
rF tt x, y, 2 z cos cos ti ti xi 2
y, z xi yj along sen yj sen tj tj
5zk the 5zk
tk, semicircle tk,
0 tty 2 4 x 2 from
39. Fx, C: rz
2, y, C: z 0
t zr to
t
2 cos xi 2, 2 cos 0
ti yj ti 2 sen 5zk2 sen tj tk, 0 t 2
tj tk, 0 t 2
39. FC: x, rt y, z z
z 2 xi cos tiyj 2 sen 5zk sin tj tk, 0 ≤ t ≤ 2
zz
2π 2π
C: r t C 2 cos ti 2 sen tj tk, 0 t 2
z
33
z
2π
z
2π z C
z
C
π
22
3
2π
3
C
3
C 11
−3 −3 π
z 2
2π
π
−3 −3
55
2
C
2
1C
1
−3 π −3
5 C 3
−3
−3
5 C 1
33
33
π
33
yy
5
2
−3
−3
x
yy
3
3
3
3
3
Figure 3
for for 39 39 y
C 1
−3
3 −3y
Figure 5
for for 40 40
y
y
y
x 3 3 y
40. Figure 40.
FFigure x, x, for y, y, z39 zfor 39
yzi yzi xzj xzj xyk xyk Figure for 40
Figure for 40
3
3 3 y
y
40.
C: C: 40.
F x, line line F x,
y, z from from y, z
yzi 0, 0, 0, 0, yzi
xzj 0
to to xzj 5, 5,
xyk
3, 3, xyk 2
Figure Figura for C: 39 para
line from
39
0, 0, 0 to 5, Figure Figura
3, 2 for para 40 40
In In Exercises C: line from
41– 41– 44, 44, 0, 0, determine 0 to 5, 3, whether 2
the the work work done done along along the the
40.
path
path FFx, In C
Exercises y, positive, zz yzi 41– negative, xzj
44, determine
xyk
or or zero. zero. whether Explain. the work done along the
In Exercises C: 41– 44, determine whether the work done along the
recta de 0, 0, 0 a 5, 3, 2
41. path 41.
path line C from is positive, 0, 0, 0
C is positive, negative, yy
negative, to 5, 3, or 2 zero. Explain.
or zero. Explain.
In
41.
En Exercises los 41. ejercicios 41– 44, 41 determine a y 44, ydeterminar whether si the el trabajo work done efectuado along the a lo
path largo Cde is la positive, trayectoria negative, C es positivo, or zero. Explain. negativo o cero. Explicar.
41.
C
42. 42.
42.
42.
C
C
42.
C
C
C
y
C
xx
x
x
yy
x
y
y
xx
y
x
x
C
x