Calculo 2 De dos variables_9na Edición - Ron Larson
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1068 CAPÍTULO 15 Análisis vectorial
En los ejercicios 53 a 56, usar un sistema algebraico por computadora
y representar el rotacional del campo vectorial.
En los ejercicios 57 a 62, determinar si el campo vectorial F es
conservativo. Si lo es, calcular una función potencial para él.
En los ejercicios 63 a 66, calcular la divergencia del campo vectorial
F.
En los ejercicios 67 a 70, calcular la divergencia del campo vectorial
F en el punto dado.
En los ejercicios 75 y 76, calcular
En los ejercicios 77 y 78, hallar
77.
78.
En los ejercicios 79 y 80, hallar
En los ejercicios 81 y 82, hallar div (rot F) · ( F).
81.
82.
En los ejercicios 83 a 90, demostrar la propiedad para los campos
vectoriales F y G y la función escalar ƒ. (Suponer que las
derivadas parciales requeridas son continuas.)
90. div(rot F) 0 (Teorema 15.3)
En los ejercicios 91 a 93, sea
y
91. Probar que 92. Probar que
93. Probar que
¿Verdadero o falso?
En los ejercicios 95 a 98, determinar si la
declaración es verdadera o falsa. Si es falsa, explicar por qué o
dar un ejemplo que demuestre su falsedad.
95. Si F(x, y) 4xi y 2 j, entonces cuando (x, y) Æ
(0, 0).
96. Si y está en el eje y positivo, entonces
el vector apunta en la dirección y negativa.
97. Si es un campo escalar, entonces el rotacional tiene sentido.
98. Si es un campo vectorial y rot F = 0, entonces F es irrotacional
pero no conservativo.
F
f
f
x, y
Fx, y 4xi y 2 j
Fx, y → 0
f n nf n2 F.
1
f F f 3.
ln f F f 2.
Fx, y, z .
fx, y, z
Fx, y, z xi yj zk,
Fx, y, z x 2 zi 2xzj yzk
Fx, y, z xyzi yj zk
Fx, y, z x 2 zi 2xzj yzk
Fx, y, z xyzi yj zk
Desarrollo de conceptos
71. Definir un campo vectorial en el plano y en el espacio. Dar
algunos ejemplos físicos de campos vectoriales.
72. ¿Qué es un campo vectorial conservativo y cuál es su criterio
en el plano y en el espacio?
73. Definir el rotacional de un campo vectorial.
74. Definir la divergencia de un campo vectorial en el plano y en
el espacio.
9. 10.
11. 12.
13. 14.
15. 16.
In Exercises 17–20, use a computer algebra system to graph
several representative vectors in the vector field.
17.
18.
19.
20.
47.
48.
In Exercises 49–52, find curl F for the vector field at the given
point.
49.
50.
51.
52. 3, 2, 0
F x, y, z e xyz i j k
0, 0, 1
F x, y, z e x sen yi e x cos yj
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 3
F x, y, z xyzi xyzj xyzk
Punto
Campo vectorial
F x, y
2xi
2yj
x 2 y 2 2
F x, y e x cos yi sen yj
F x, y, z xi yj zk
F x, y, z
xi yj zk
x 2 y 2 z 2
F x, y 2y 3x i 2y 3x j
F x, y
1
8 2xyi y 2 j
F x, y, z xi yj zk
F x, y, z i j k
F x, y x 2 y 2 i j
F x, y 4xi yj
F x, y
xi
F x, y, z
3yj
F x, y yi 2xj
F x, y yi xj
CAS
In Exercises 53–56, use a computer algebra system to find the
rot F for the vector field.
53.
54.
55.
56.
In Exercises 57–62, determine whether the vector field F is
conservative. If it is, find a potential function for the vector field.
57.
58.
59.
60.
61.
62.
In Exercises 63–66, find the divergence of the vector field F.
63.
64.
65.
66.
In Exercises 67–70, find the divergence of the vector field F at
the given point.
67.
68.
69.
70.
In Exercises 75 and 76, find rot
75. 76.
In Exercises 77 and 78, find rot
77.
78.
In Exercises 79 and 80, find
79. 80.
In Exercises 81 and 82, find
81.
82.
In Exercises 83–90, prove the property for vector fields F and G
and scalar function
(Assume that the required partial
derivatives are continuous.)
83.
84.
85.
86.
87.
88.
89.
90. (Theorem 15.3)
In Exercises 91–93, let
and let
91. Show that 92. Show that
93. Show that
True or False?
In Exercises 95–98, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
95. If then as
96. If and is on the positive -axis, then
the vector points in the negative -direction.
97. If is a scalar field, then rot is a meaningful expression.
98. If is a vector field and then is irrotational but
not conservative.
F
rot F 0,
F
f
f
y
y
x, y
F x, y 4xi y 2 j
x, y → 0, 0 .
F x, y → 0
F x, y 4xi y 2 j,
f n nf n 2 F.
1
f
F
f 3.
ln f
F
f 2.
fx, y, z F x, y, z .
F x, y, z
xi yj zk,
div rot F 0
div f F f div F f F
f F f F f F
f F F
div F G rot F G F rot G
div F G div F div G
rot f f 0
rot F G rot F rot G
f.
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
div rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
div F G F G .
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
F G F G .
3, 2, 1
F x, y, z ln xyz i j k
3, 0, 0
F x, y, z e x sen yi e x cos yj z 2 k
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 1
F x, y, z xyzi xyj zk
Punto
Campo vectorial
F x, y, z ln x 2 y 2 i xyj ln y 2 z 2 k
F x, y, z senxi cos yj z 2 k
F x, y xe x i ye y j
F x, y x 2 i 2y 2 j
F x, y, z
x
x 2 y 2 i y
x 2 y 2 j k
F x, y, z
z
y i
xz
y 2 j
x
y k
F x, y, z ye z i ze x j xe y k
F x, y, z sen zi sen xj sen yk
F x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 k
F x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zk
F x, y, z x 2 y 2 z 2 i j k
F x, y, z sen x y i sen y z j sen z x k
F x, y, z
yz
y z i xz
x z j xy
x
y k
F x, y, z arctan x y i ln x2 y 2 j k
1068 Chapter 15 Vector Analysis
CAS
71. Define a vector field in the plane and in space. Give some
physical examples of vector fields.
72. What is a conservative vector field, and how do you test for
it in the plane and in space?
73. Define the rot of a vector field.
74. Define the divergence of a vector field in the plane and in
space.
WRITING ABOUT CONCEPTS
94. (a) Sketch several representative vectors in the vector field
given by
(b) Sketch several representative vectors in the vector field
given by
(c) Explain any similarities or differences in the vector
fields y G x, y .
F x, y
G x, y
xi
yj
x 2 y 2.
F x, y
xi
yj
x 2 y 2.
CAPSTONE
In Exercises 53–56, use a computer algebra system to find the
rot F for the vector field.
53.
54.
55.
56.
In Exercises 57–62, determine whether the vector field F is
conservative. If it is, find a potential function for the vector field.
57.
58.
59.
60.
61.
62.
In Exercises 63–66, find the divergence of the vector field F.
63.
64.
65.
66.
In Exercises 67–70, find the divergence of the vector field F at
the given point.
67.
68.
69.
70.
In Exercises 75 and 76, find rot
75. 76.
In Exercises 77 and 78, find rot
77.
78.
In Exercises 79 and 80, find
79. 80.
In Exercises 81 and 82, find
81.
82.
In Exercises 83–90, prove the property for vector fields F and G
and scalar function
(Assume that the required partial
derivatives are continuous.)
83.
84.
85.
86.
87.
88.
89.
90. (Theorem 15.3)
In Exercises 91–93, let
and let
91. Show that 92. Show that
93. Show that
True or False?
In Exercises 95–98, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
95. If then as
96. If and is on the positive -axis, then
the vector points in the negative -direction.
97. If is a scalar field, then rot is a meaningful expression.
98. If is a vector field and then is irrotational but
not conservative.
F
rot F 0,
F
f
f
y
y
x, y
F x, y 4xi y 2 j
x, y → 0, 0 .
F x, y → 0
F x, y 4xi y 2 j,
f n nf n 2 F.
1
f
F
f 3.
ln f
F
f 2.
fx, y, z F x, y, z .
F x, y, z
xi yj zk,
div rot F 0
div f F f div F f F
f F f F f F
f F F
div F G rot F G F rot G
div F G div F div G
rot f f 0
rot F G rot F rot G
f.
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
div rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
div F G F G .
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
F G F G .
3, 2, 1
F x, y, z ln xyz i j k
3, 0, 0
F x, y, z e x sen yi e x cos yj z 2 k
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 1
F x, y, z xyzi xyj zk
Punto
Campo vectorial
F x, y, z ln x 2 y 2 i xyj ln y 2 z 2 k
F x, y, z senxi cos yj z 2 k
F x, y xe x i ye y j
F x, y x 2 i 2y 2 j
F x, y, z
x
x 2 y 2 i y
x 2 y 2 j k
F x, y, z
z
y i
xz
y 2 j
x
y k
F x, y, z ye z i ze x j xe y k
F x, y, z sen zi sen xj sen yk
F x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 k
F x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zk
F x, y, z x 2 y 2 z 2 i j k
F x, y, z sen x y i sen y z j sen z x k
F x, y, z
yz
y z i xz
x z j xy
x
y k
F x, y, z arctan x y i ln x2 y 2 j k
1068 Chapter 15 Vector Analysis
CAS
71. Define a vector field in the plane and in space. Give some
physical examples of vector fields.
72. What is a conservative vector field, and how do you test for
it in the plane and in space?
73. Define the rot of a vector field.
74. Define the divergence of a vector field in the plane and in
space.
WRITING ABOUT CONCEPTS
94. (a) Sketch several representative vectors in the vector field
given by
(b) Sketch several representative vectors in the vector field
given by
(c) Explain any similarities or differences in the vector
fields y G x, y .
F x, y
G x, y
xi
yj
x 2 y 2.
F x, y
xi
yj
x 2 y 2.
CAPSTONE
In Exercises 53–56, use a computer algebra system to find the
rot F for the vector field.
53.
54.
55.
56.
In Exercises 57–62, determine whether the vector field F is
conservative. If it is, find a potential function for the vector field.
57.
58.
59.
60.
61.
62.
In Exercises 63–66, find the divergence of the vector field F.
63.
64.
65.
66.
In Exercises 67–70, find the divergence of the vector field F at
the given point.
67.
68.
69.
70.
In Exercises 75 and 76, find rot
75. 76.
In Exercises 77 and 78, find rot
77.
78.
In Exercises 79 and 80, find
79. 80.
In Exercises 81 and 82, find
81.
82.
In Exercises 83–90, prove the property for vector fields F and G
and scalar function
(Assume that the required partial
derivatives are continuous.)
83.
84.
85.
86.
87.
88.
89.
90. (Theorem 15.3)
In Exercises 91–93, let
and let
91. Show that 92. Show that
93. Show that
True or False?
In Exercises 95–98, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
95. If then as
96. If and is on the positive -axis, then
the vector points in the negative -direction.
97. If is a scalar field, then rot is a meaningful expression.
98. If is a vector field and then is irrotational but
not conservative.
F
rot F 0,
F
f
f
y
y
x, y
F x, y 4xi y 2 j
x, y → 0, 0 .
F x, y → 0
F x, y 4xi y 2 j,
f n nf n 2 F.
1
f
F
f 3.
ln f
F
f 2.
fx, y, z F x, y, z .
F x, y, z
xi yj zk,
div rot F 0
div f F f div F f F
f F f F f F
f F F
div F G rot F G F rot G
div F G div F div G
rot f f 0
rot F G rot F rot G
f.
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
div rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
div F G F G .
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
F G F G .
3, 2, 1
F x, y, z ln xyz i j k
3, 0, 0
F x, y, z e x sen yi e x cos yj z 2 k
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 1
F x, y, z xyzi xyj zk
Punto
Campo vectorial
F x, y, z ln x 2 y 2 i xyj ln y 2 z 2 k
F x, y, z senxi cos yj z 2 k
F x, y xe x i ye y j
F x, y x 2 i 2y 2 j
F x, y, z
x
x 2 y 2 i y
x 2 y 2 j k
F x, y, z
z
y i
xz
y 2 j
x
y k
F x, y, z ye z i ze x j xe y k
F x, y, z sen zi sen xj sen yk
F x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 k
F x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zk
F x, y, z x 2 y 2 z 2 i j k
F x, y, z sen x y i sen y z j sen z x k
F x, y, z
yz
y z i xz
x z j xy
x
y k
F x, y, z arctan x y i ln x2 y 2 j k
1068 Chapter 15 Vector Analysis
CAS
71. Define a vector field in the plane and in space. Give some
physical examples of vector fields.
72. What is a conservative vector field, and how do you test for
it in the plane and in space?
73. Define the rot of a vector field.
74. Define the divergence of a vector field in the plane and in
space.
WRITING ABOUT CONCEPTS
94. (a) Sketch several representative vectors in the vector field
given by
(b) Sketch several representative vectors in the vector field
given by
(c) Explain any similarities or differences in the vector
fields y G x, y .
F x, y
G x, y
xi
yj
x 2 y 2.
F x, y
xi
yj
x 2 y 2.
CAPSTONE
In Exercises 53–56, use a computer algebra system to find the
rot F for the vector field.
53.
54.
55.
56.
In Exercises 57–62, determine whether the vector field F is
conservative. If it is, find a potential function for the vector field.
57.
58.
59.
60.
61.
62.
In Exercises 63–66, find the divergence of the vector field F.
63.
64.
65.
66.
In Exercises 67–70, find the divergence of the vector field F at
the given point.
67.
68.
69.
70.
In Exercises 75 and 76, find rot
75. 76.
In Exercises 77 and 78, find rot
77.
78.
In Exercises 79 and 80, find
79. 80.
In Exercises 81 and 82, find
81.
82.
In Exercises 83–90, prove the property for vector fields F and G
and scalar function
(Assume that the required partial
derivatives are continuous.)
83.
84.
85.
86.
87.
88.
89.
90. (Theorem 15.3)
In Exercises 91–93, let
and let
91. Show that 92. Show that
93. Show that
True or False?
In Exercises 95–98, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
95. If then as
96. If and is on the positive -axis, then
the vector points in the negative -direction.
97. If is a scalar field, then rot is a meaningful expression.
98. If is a vector field and then is irrotational but
not conservative.
F
rot F 0,
F
f
f
y
y
x, y
F x, y 4xi y 2 j
x, y → 0, 0 .
F x, y → 0
F x, y 4xi y 2 j,
f n nf n 2 F.
1
f
F
f 3.
ln f
F
f 2.
fx, y, z F x, y, z .
F x, y, z
xi yj zk,
div rot F 0
div f F f div F f F
f F f F f F
f F F
div F G rot F G F rot G
div F G div F div G
rot f f 0
rot F G rot F rot G
f.
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
div rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
div F G F G .
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
F G F G .
3, 2, 1
F x, y, z ln xyz i j k
3, 0, 0
F x, y, z e x sen yi e x cos yj z 2 k
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 1
F x, y, z xyzi xyj zk
Punto
Campo vectorial
F x, y, z ln x 2 y 2 i xyj ln y 2 z 2 k
F x, y, z senxi cos yj z 2 k
F x, y xe x i ye y j
F x, y x 2 i 2y 2 j
F x, y, z
x
x 2 y 2 i y
x 2 y 2 j k
F x, y, z
z
y i
xz
y 2 j
x
y k
F x, y, z ye z i ze x j xe y k
F x, y, z sen zi sen xj sen yk
F x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 k
F x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zk
F x, y, z x 2 y 2 z 2 i j k
F x, y, z sen x y i sen y z j sen z x k
F x, y, z
yz
y z i xz
x z j xy
x
y k
F x, y, z arctan x y i ln x2 y 2 j k
1068 Chapter 15 Vector Analysis
CAS
71. Define a vector field in the plane and in space. Give some
physical examples of vector fields.
72. What is a conservative vector field, and how do you test for
it in the plane and in space?
73. Define the rot of a vector field.
74. Define the divergence of a vector field in the plane and in
space.
WRITING ABOUT CONCEPTS
94. (a) Sketch several representative vectors in the vector field
given by
(b) Sketch several representative vectors in the vector field
given by
(c) Explain any similarities or differences in the vector
fields y G x, y .
F x, y
G x, y
xi
yj
x 2 y 2.
F x, y
xi
yj
x 2 y 2.
CAPSTONE
In Exercises 53–56, use a computer algebra system to find the
rot F for the vector field.
53.
54.
55.
56.
In Exercises 57–62, determine whether the vector field F is
conservative. If it is, find a potential function for the vector field.
57.
58.
59.
60.
61.
62.
In Exercises 63–66, find the divergence of the vector field F.
63.
64.
65.
66.
In Exercises 67–70, find the divergence of the vector field F at
the given point.
67.
68.
69.
70.
In Exercises 75 and 76, find rot
75. 76.
In Exercises 77 and 78, find rot
77.
78.
In Exercises 79 and 80, find
79. 80.
In Exercises 81 and 82, find
81.
82.
In Exercises 83–90, prove the property for vector fields F and G
and scalar function
(Assume that the required partial
derivatives are continuous.)
83.
84.
85.
86.
87.
88.
89.
90. (Theorem 15.3)
In Exercises 91–93, let
and let
91. Show that 92. Show that
93. Show that
True or False?
In Exercises 95–98, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
95. If then as
96. If and is on the positive -axis, then
the vector points in the negative -direction.
97. If is a scalar field, then rot is a meaningful expression.
98. If is a vector field and then is irrotational but
not conservative.
F
rot F 0,
F
f
f
y
y
x, y
F x, y 4xi y 2 j
x, y → 0, 0 .
F x, y → 0
F x, y 4xi y 2 j,
f n nf n 2 F.
1
f
F
f 3.
ln f
F
f 2.
fx, y, z F x, y, z .
F x, y, z
xi yj zk,
div rot F 0
div f F f div F f F
f F f F f F
f F F
div F G rot F G F rot G
div F G div F div G
rot f f 0
rot F G rot F rot G
f.
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
div rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
div F G F G .
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
F G F G .
3, 2, 1
F x, y, z ln xyz i j k
3, 0, 0
F x, y, z e x sen yi e x cos yj z 2 k
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 1
F x, y, z xyzi xyj zk
Punto
Campo vectorial
F x, y, z ln x 2 y 2 i xyj ln y 2 z 2 k
F x, y, z senxi cos yj z 2 k
F x, y xe x i ye y j
F x, y x 2 i 2y 2 j
F x, y, z
x
x 2 y 2 i y
x 2 y 2 j k
F x, y, z
z
y i
xz
y 2 j
x
y k
F x, y, z ye z i ze x j xe y k
F x, y, z sen zi sen xj sen yk
F x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 k
F x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zk
F x, y, z x 2 y 2 z 2 i j k
F x, y, z sen x y i sen y z j sen z x k
F x, y, z
yz
y z i xz
x z j xy
x
y k
F x, y, z arctan x y i ln x2 y 2 j k
1068 Chapter 15 Vector Analysis
CAS
71. Define a vector field in the plane and in space. Give some
physical examples of vector fields.
72. What is a conservative vector field, and how do you test for
it in the plane and in space?
73. Define the rot of a vector field.
74. Define the divergence of a vector field in the plane and in
space.
WRITING ABOUT CONCEPTS
94. (a) Sketch several representative vectors in the vector field
given by
(b) Sketch several representative vectors in the vector field
given by
(c) Explain any similarities or differences in the vector
fields y G x, y .
F x, y
G x, y
xi
yj
x 2 y 2.
F x, y
xi
yj
x 2 y 2.
CAPSTONE
In Exercises 53–56, use a computer algebra system to find the
rot F for the vector field.
53.
54.
55.
56.
In Exercises 57–62, determine whether the vector field F is
conservative. If it is, find a potential function for the vector field.
57.
58.
59.
60.
61.
62.
In Exercises 63–66, find the divergence of the vector field F.
63.
64.
65.
66.
In Exercises 67–70, find the divergence of the vector field F at
the given point.
67.
68.
69.
70.
In Exercises 75 and 76, find rot
75. 76.
In Exercises 77 and 78, find rot
77.
78.
In Exercises 79 and 80, find
79. 80.
In Exercises 81 and 82, find
81.
82.
In Exercises 83–90, prove the property for vector fields F and G
and scalar function
(Assume that the required partial
derivatives are continuous.)
83.
84.
85.
86.
87.
88.
89.
90. (Theorem 15.3)
In Exercises 91–93, let
and let
91. Show that 92. Show that
93. Show that
True or False?
In Exercises 95–98, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
95. If then as
96. If and is on the positive -axis, then
the vector points in the negative -direction.
97. If is a scalar field, then rot is a meaningful expression.
98. If is a vector field and then is irrotational but
not conservative.
F
rot F 0,
F
f
f
y
y
x, y
F x, y 4xi y 2 j
x, y → 0, 0 .
F x, y → 0
F x, y 4xi y 2 j,
f n nf n 2 F.
1
f
F
f 3.
ln f
F
f 2.
fx, y, z F x, y, z .
F x, y, z
xi yj zk,
div rot F 0
div f F f div F f F
f F f F f F
f F F
div F G rot F G F rot G
div F G div F div G
rot f f 0
rot F G rot F rot G
f.
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
div rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
div F G F G .
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
F G F G .
3, 2, 1
F x, y, z ln xyz i j k
3, 0, 0
F x, y, z e x sen yi e x cos yj z 2 k
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 1
F x, y, z xyzi xyj zk
Punto
Campo vectorial
F x, y, z ln x 2 y 2 i xyj ln y 2 z 2 k
F x, y, z senxi cos yj z 2 k
F x, y xe x i ye y j
F x, y x 2 i 2y 2 j
F x, y, z
x
x 2 y 2 i y
x 2 y 2 j k
F x, y, z
z
y i
xz
y 2 j
x
y k
F x, y, z ye z i ze x j xe y k
F x, y, z sen zi sen xj sen yk
F x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 k
F x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zk
F x, y, z x 2 y 2 z 2 i j k
F x, y, z sen x y i sen y z j sen z x k
F x, y, z
yz
y z i xz
x z j xy
x
y k
F x, y, z arctan x y i ln x2 y 2 j k
1068 Chapter 15 Vector Analysis
CAS
71. Define a vector field in the plane and in space. Give some
physical examples of vector fields.
72. What is a conservative vector field, and how do you test for
it in the plane and in space?
73. Define the rot of a vector field.
74. Define the divergence of a vector field in the plane and in
space.
WRITING ABOUT CONCEPTS
94. (a) Sketch several representative vectors in the vector field
given by
(b) Sketch several representative vectors in the vector field
given by
(c) Explain any similarities or differences in the vector
fields y G x, y .
F x, y
G x, y
xi
yj
x 2 y 2.
F x, y
xi
yj
x 2 y 2.
CAPSTONE
In Exercises 53–56, use a computer algebra system to find the
rot F for the vector field.
53.
54.
55.
56.
In Exercises 57–62, determine whether the vector field F is
conservative. If it is, find a potential function for the vector field.
57.
58.
59.
60.
61.
62.
In Exercises 63–66, find the divergence of the vector field F.
63.
64.
65.
66.
In Exercises 67–70, find the divergence of the vector field F at
the given point.
67.
68.
69.
70.
In Exercises 75 and 76, find rot
75. 76.
In Exercises 77 and 78, find rot
77.
78.
In Exercises 79 and 80, find
79. 80.
In Exercises 81 and 82, find
81.
82.
In Exercises 83–90, prove the property for vector fields F and G
and scalar function
(Assume that the required partial
derivatives are continuous.)
83.
84.
85.
86.
87.
88.
89.
90. (Theorem 15.3)
In Exercises 91–93, let
and let
91. Show that 92. Show that
93. Show that
True or False?
In Exercises 95–98, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
95. If then as
96. If and is on the positive -axis, then
the vector points in the negative -direction.
97. If is a scalar field, then rot is a meaningful expression.
98. If is a vector field and then is irrotational but
not conservative.
F
rot F 0,
F
f
f
y
y
x, y
F x, y 4xi y 2 j
x, y → 0, 0 .
F x, y → 0
F x, y 4xi y 2 j,
f n nf n 2 F.
1
f
F
f 3.
ln f
F
f 2.
fx, y, z F x, y, z .
F x, y, z
xi yj zk,
div rot F 0
div f F f div F f F
f F f F f F
f F F
div F G rot F G F rot G
div F G div F div G
rot f f 0
rot F G rot F rot G
f.
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
div rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
div F G F G .
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
F G F G .
3, 2, 1
F x, y, z ln xyz i j k
3, 0, 0
F x, y, z e x sen yi e x cos yj z 2 k
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 1
F x, y, z xyzi xyj zk
Punto
Campo vectorial
F x, y, z ln x 2 y 2 i xyj ln y 2 z 2 k
F x, y, z senxi cos yj z 2 k
F x, y xe x i ye y j
F x, y x 2 i 2y 2 j
F x, y, z
x
x 2 y 2 i y
x 2 y 2 j k
F x, y, z
z
y i
xz
y 2 j
x
y k
F x, y, z ye z i ze x j xe y k
F x, y, z sen zi sen xj sen yk
F x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 k
F x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zk
F x, y, z x 2 y 2 z 2 i j k
F x, y, z sen x y i sen y z j sen z x k
F x, y, z
yz
y z i xz
x z j xy
x
y k
F x, y, z arctan x y i ln x2 y 2 j k
1068 Chapter 15 Vector Analysis
CAS
71. Define a vector field in the plane and in space. Give some
physical examples of vector fields.
72. What is a conservative vector field, and how do you test for
it in the plane and in space?
73. Define the rot of a vector field.
74. Define the divergence of a vector field in the plane and in
space.
WRITING ABOUT CONCEPTS
94. (a) Sketch several representative vectors in the vector field
given by
(b) Sketch several representative vectors in the vector field
given by
(c) Explain any similarities or differences in the vector
fields y G x, y .
F x, y
G x, y
xi
yj
x 2 y 2.
F x, y
xi
yj
x 2 y 2.
CAPSTONE
In Exercises 53–56, use a computer algebra system to find the
rot F for the vector field.
53.
54.
55.
56.
In Exercises 57–62, determine whether the vector field F is
conservative. If it is, find a potential function for the vector field.
57.
58.
59.
60.
61.
62.
In Exercises 63–66, find the divergence of the vector field F.
63.
64.
65.
66.
In Exercises 67–70, find the divergence of the vector field F at
the given point.
67.
68.
69.
70.
In Exercises 75 and 76, find rot
75. 76.
In Exercises 77 and 78, find rot
77.
78.
In Exercises 79 and 80, find
79. 80.
In Exercises 81 and 82, find
81.
82.
In Exercises 83–90, prove the property for vector fields F and G
and scalar function
(Assume that the required partial
derivatives are continuous.)
83.
84.
85.
86.
87.
88.
89.
90. (Theorem 15.3)
In Exercises 91–93, let
and let
91. Show that 92. Show that
93. Show that
True or False?
In Exercises 95–98, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
95. If then as
96. If and is on the positive -axis, then
the vector points in the negative -direction.
97. If is a scalar field, then rot is a meaningful expression.
98. If is a vector field and then is irrotational but
not conservative.
F
rot F 0,
F
f
f
y
y
x, y
F x, y 4xi y 2 j
x, y → 0, 0 .
F x, y → 0
F x, y 4xi y 2 j,
f n nf n 2 F.
1
f
F
f 3.
ln f
F
f 2.
fx, y, z F x, y, z .
F x, y, z
xi yj zk,
div rot F 0
div f F f div F f F
f F f F f F
f F F
div F G rot F G F rot G
div F G div F div G
rot f f 0
rot F G rot F rot G
f.
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
div rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
div F G F G .
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
F G F G .
3, 2, 1
F x, y, z ln xyz i j k
3, 0, 0
F x, y, z e x sen yi e x cos yj z 2 k
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 1
F x, y, z xyzi xyj zk
Punto
Campo vectorial
F x, y, z ln x 2 y 2 i xyj ln y 2 z 2 k
F x, y, z senxi cos yj z 2 k
F x, y xe x i ye y j
F x, y x 2 i 2y 2 j
F x, y, z
x
x 2 y 2 i y
x 2 y 2 j k
F x, y, z
z
y i
xz
y 2 j
x
y k
F x, y, z ye z i ze x j xe y k
F x, y, z sen zi sen xj sen yk
F x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 k
F x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zk
F x, y, z x 2 y 2 z 2 i j k
F x, y, z sen x y i sen y z j sen z x k
F x, y, z
yz
y z i xz
x z j xy
x
y k
F x, y, z arctan x y i ln x2 y 2 j k
1068 Chapter 15 Vector Analysis
CAS
71. Define a vector field in the plane and in space. Give some
physical examples of vector fields.
72. What is a conservative vector field, and how do you test for
it in the plane and in space?
73. Define the rot of a vector field.
74. Define the divergence of a vector field in the plane and in
space.
WRITING ABOUT CONCEPTS
94. (a) Sketch several representative vectors in the vector field
given by
(b) Sketch several representative vectors in the vector field
given by
(c) Explain any similarities or differences in the vector
fields y G x, y .
F x, y
G x, y
xi
yj
x 2 y 2.
F x, y
xi
yj
x 2 y 2.
CAPSTONE
Para discusión
94. a) Dibujar varios vectores representativos en el campo vectorial
dado por
b) Dibujar varios vectores representativos en el campo vectorial
dado por
c) Explicar cualquier similitud o diferencia en los campos
vectoriales
In Exercises 53–56, use a computer algebra system to find the
rot F for the vector field.
53.
54.
55.
56.
In Exercises 57–62, determine whether the vector field F is
conservative. If it is, find a potential function for the vector field.
57.
58.
59.
60.
61.
62.
In Exercises 63–66, find the divergence of the vector field F.
63.
64.
65.
66.
In Exercises 67–70, find the divergence of the vector field F at
the given point.
67.
68.
69.
70.
In Exercises 75 and 76, find rot
75. 76.
In Exercises 77 and 78, find rot
77.
78.
In Exercises 79 and 80, find
79. 80.
In Exercises 81 and 82, find
81.
82.
In Exercises 83–90, prove the property for vector fields F and G
and scalar function
(Assume that the required partial
derivatives are continuous.)
83.
84.
85.
86.
87.
88.
89.
90. (Theorem 15.3)
In Exercises 91–93, let
and let
91. Show that 92. Show that
93. Show that
True or False?
In Exercises 95–98, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
95. If then as
96. If and is on the positive -axis, then
the vector points in the negative -direction.
97. If is a scalar field, then rot is a meaningful expression.
98. If is a vector field and then is irrotational but
not conservative.
F
rot F 0,
F
f
f
y
y
x, y
F x, y 4xi y 2 j
x, y → 0, 0 .
F x, y → 0
F x, y 4xi y 2 j,
f n nf n 2 F.
1
f
F
f 3.
ln f
F
f 2.
fx, y, z F x, y, z .
F x, y, z
xi yj zk,
div rot F 0
div f F f div F f F
f F f F f F
f F F
div F G rot F G F rot G
div F G div F div G
rot f f 0
rot F G rot F rot G
f.
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
div rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
div F G F G .
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
F G F G .
3, 2, 1
F x, y, z ln xyz i j k
3, 0, 0
F x, y, z e x sen yi e x cos yj z 2 k
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 1
F x, y, z xyzi xyj zk
Punto
Campo vectorial
F x, y, z ln x 2 y 2 i xyj ln y 2 z 2 k
F x, y, z senxi cos yj z 2 k
F x, y xe x i ye y j
F x, y x 2 i 2y 2 j
F x, y, z
x
x 2 y 2 i y
x 2 y 2 j k
F x, y, z
z
y i
xz
y 2 j
x
y k
F x, y, z ye z i ze x j xe y k
F x, y, z sen zi sen xj sen yk
F x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 k
F x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zk
F x, y, z x 2 y 2 z 2 i j k
F x, y, z sen x y i sen y z j sen z x k
F x, y, z
yz
y z i xz
x z j xy
x
y k
F x, y, z arctan x y i ln x2 y 2 j k
1068 Chapter 15 Vector Analysis
CAS
71. Define a vector field in the plane and in space. Give some
physical examples of vector fields.
72. What is a conservative vector field, and how do you test for
it in the plane and in space?
73. Define the rot of a vector field.
74. Define the divergence of a vector field in the plane and in
space.
WRITING ABOUT CONCEPTS
94. (a) Sketch several representative vectors in the vector field
given by
(b) Sketch several representative vectors in the vector field
given by
(c) Explain any similarities or differences in the vector
fields y G x, y .
F x, y
G x, y
xi
yj
x 2 y 2.
F x, y
xi
yj
x 2 y 2.
CAPSTONE
In Exercises 53–56, use a computer algebra system to find the
rot F for the vector field.
53.
54.
55.
56.
In Exercises 57–62, determine whether the vector field F is
conservative. If it is, find a potential function for the vector field.
57.
58.
59.
60.
61.
62.
In Exercises 63–66, find the divergence of the vector field F.
63.
64.
65.
66.
In Exercises 67–70, find the divergence of the vector field F at
the given point.
67.
68.
69.
70.
In Exercises 75 and 76, find rot
75. 76.
In Exercises 77 and 78, find rot
77.
78.
In Exercises 79 and 80, find
79. 80.
In Exercises 81 and 82, find
81.
82.
In Exercises 83–90, prove the property for vector fields F and G
and scalar function
(Assume that the required partial
derivatives are continuous.)
83.
84.
85.
86.
87.
88.
89.
90. (Theorem 15.3)
In Exercises 91–93, let
and let
91. Show that 92. Show that
93. Show that
True or False?
In Exercises 95–98, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
95. If then as
96. If and is on the positive -axis, then
the vector points in the negative -direction.
97. If is a scalar field, then rot is a meaningful expression.
98. If is a vector field and then is irrotational but
not conservative.
F
rot F 0,
F
f
f
y
y
x, y
F x, y 4xi y 2 j
x, y → 0, 0 .
F x, y → 0
F x, y 4xi y 2 j,
f n nf n 2 F.
1
f
F
f 3.
ln f
F
f 2.
fx, y, z F x, y, z .
F x, y, z
xi yj zk,
div rot F 0
div f F f div F f F
f F f F f F
f F F
div F G rot F G F rot G
div F G div F div G
rot f f 0
rot F G rot F rot G
f.
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
div rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
div F G F G .
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
F G F G .
3, 2, 1
F x, y, z ln xyz i j k
3, 0, 0
F x, y, z e x sen yi e x cos yj z 2 k
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 1
F x, y, z xyzi xyj zk
Punto
Campo vectorial
F x, y, z ln x 2 y 2 i xyj ln y 2 z 2 k
F x, y, z senxi cos yj z 2 k
F x, y xe x i ye y j
F x, y x 2 i 2y 2 j
F x, y, z
x
x 2 y 2 i y
x 2 y 2 j k
F x, y, z
z
y i
xz
y 2 j
x
y k
F x, y, z ye z i ze x j xe y k
F x, y, z sen zi sen xj sen yk
F x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 k
F x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zk
F x, y, z x 2 y 2 z 2 i j k
F x, y, z sen x y i sen y z j sen z x k
F x, y, z
yz
y z i xz
x z j xy
x
y k
F x, y, z arctan x y i ln x2 y 2 j k
1068 Chapter 15 Vector Analysis
CAS
71. Define a vector field in the plane and in space. Give some
physical examples of vector fields.
72. What is a conservative vector field, and how do you test for
it in the plane and in space?
73. Define the rot of a vector field.
74. Define the divergence of a vector field in the plane and in
space.
WRITING ABOUT CONCEPTS
94. (a) Sketch several representative vectors in the vector field
given by
(b) Sketch several representative vectors in the vector field
given by
(c) Explain any similarities or differences in the vector
fields y G x, y .
F x, y
G x, y
xi
yj
x 2 y 2.
F x, y
xi
yj
x 2 y 2.
CAPSTONE
In Exercises 53–56, use a computer algebra system to find the
rot F for the vector field.
53.
54.
55.
56.
In Exercises 57–62, determine whether the vector field F is
conservative. If it is, find a potential function for the vector field.
57.
58.
59.
60.
61.
62.
In Exercises 63–66, find the divergence of the vector field F.
63.
64.
65.
66.
In Exercises 67–70, find the divergence of the vector field F at
the given point.
67.
68.
69.
70.
In Exercises 75 and 76, find rot
75. 76.
In Exercises 77 and 78, find rot
77.
78.
In Exercises 79 and 80, find
79. 80.
In Exercises 81 and 82, find
81.
82.
In Exercises 83–90, prove the property for vector fields F and G
and scalar function
(Assume that the required partial
derivatives are continuous.)
83.
84.
85.
86.
87.
88.
89.
90. (Theorem 15.3)
In Exercises 91–93, let
and let
91. Show that 92. Show that
93. Show that
True or False?
In Exercises 95–98, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
95. If then as
96. If and is on the positive -axis, then
the vector points in the negative -direction.
97. If is a scalar field, then rot is a meaningful expression.
98. If is a vector field and then is irrotational but
not conservative.
F
rot F 0,
F
f
f
y
y
x, y
F x, y 4xi y 2 j
x, y → 0, 0 .
F x, y → 0
F x, y 4xi y 2 j,
f n nf n 2 F.
1
f
F
f 3.
ln f
F
f 2.
fx, y, z F x, y, z .
F x, y, z
xi yj zk,
div rot F 0
div f F f div F f F
f F f F f F
f F F
div F G rot F G F rot G
div F G div F div G
rot f f 0
rot F G rot F rot G
f.
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
div rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
div F G F G .
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
F G F G .
3, 2, 1
F x, y, z ln xyz i j k
3, 0, 0
F x, y, z e x sen yi e x cos yj z 2 k
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 1
F x, y, z xyzi xyj zk
Punto
Campo vectorial
F x, y, z ln x 2 y 2 i xyj ln y 2 z 2 k
F x, y, z senxi cos yj z 2 k
F x, y xe x i ye y j
F x, y x 2 i 2y 2 j
F x, y, z
x
x 2 y 2 i y
x 2 y 2 j k
F x, y, z
z
y i
xz
y 2 j
x
y k
F x, y, z ye z i ze x j xe y k
F x, y, z sen zi sen xj sen yk
F x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 k
F x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zk
F x, y, z x 2 y 2 z 2 i j k
F x, y, z sen x y i sen y z j sen z x k
F x, y, z
yz
y z i xz
x z j xy
x
y k
F x, y, z arctan x y i ln x2 y 2 j k
1068 Chapter 15 Vector Analysis
CAS
71. Define a vector field in the plane and in space. Give some
physical examples of vector fields.
72. What is a conservative vector field, and how do you test for
it in the plane and in space?
73. Define the rot of a vector field.
74. Define the divergence of a vector field in the plane and in
space.
WRITING ABOUT CONCEPTS
94. (a) Sketch several representative vectors in the vector field
given by
(b) Sketch several representative vectors in the vector field
given by
(c) Explain any similarities or differences in the vector
fields y G x, y .
F x, y
G x, y
xi
yj
x 2 y 2.
F x, y
xi
yj
x 2 y 2.
CAPSTONE
In Exercises 53–56, use a computer algebra system to find the
rot F for the vector field.
53.
54.
55.
56.
In Exercises 57–62, determine whether the vector field F is
conservative. If it is, find a potential function for the vector field.
57.
58.
59.
60.
61.
62.
In Exercises 63–66, find the divergence of the vector field F.
63.
64.
65.
66.
In Exercises 67–70, find the divergence of the vector field F at
the given point.
67.
68.
69.
70.
In Exercises 75 and 76, find rot
75. 76.
In Exercises 77 and 78, find rot
77.
78.
In Exercises 79 and 80, find
79. 80.
In Exercises 81 and 82, find
81.
82.
In Exercises 83–90, prove the property for vector fields F and G
and scalar function
(Assume that the required partial
derivatives are continuous.)
83.
84.
85.
86.
87.
88.
89.
90. (Theorem 15.3)
In Exercises 91–93, let
and let
91. Show that 92. Show that
93. Show that
True or False?
In Exercises 95–98, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
95. If then as
96. If and is on the positive -axis, then
the vector points in the negative -direction.
97. If is a scalar field, then rot is a meaningful expression.
98. If is a vector field and then is irrotational but
not conservative.
F
rot F 0,
F
f
f
y
y
x, y
F x, y 4xi y 2 j
x, y → 0, 0 .
F x, y → 0
F x, y 4xi y 2 j,
f n nf n 2 F.
1
f
F
f 3.
ln f
F
f 2.
fx, y, z F x, y, z .
F x, y, z
xi yj zk,
div rot F 0
div f F f div F f F
f F f F f F
f F F
div F G rot F G F rot G
div F G div F div G
rot f f 0
rot F G rot F rot G
f.
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
div rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
div F G F G .
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
F G F G .
3, 2, 1
F x, y, z ln xyz i j k
3, 0, 0
F x, y, z e x sen yi e x cos yj z 2 k
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 1
F x, y, z xyzi xyj zk
Punto
Campo vectorial
F x, y, z ln x 2 y 2 i xyj ln y 2 z 2 k
F x, y, z senxi cos yj z 2 k
F x, y xe x i ye y j
F x, y x 2 i 2y 2 j
F x, y, z
x
x 2 y 2 i y
x 2 y 2 j k
F x, y, z
z
y i
xz
y 2 j
x
y k
F x, y, z ye z i ze x j xe y k
F x, y, z sen zi sen xj sen yk
F x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 k
F x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zk
F x, y, z x 2 y 2 z 2 i j k
F x, y, z sen x y i sen y z j sen z x k
F x, y, z
yz
y z i xz
x z j xy
x
y k
F x, y, z arctan x y i ln x2 y 2 j k
1068 Chapter 15 Vector Analysis
CAS
71. Define a vector field in the plane and in space. Give some
physical examples of vector fields.
72. What is a conservative vector field, and how do you test for
it in the plane and in space?
73. Define the rot of a vector field.
74. Define the divergence of a vector field in the plane and in
space.
WRITING ABOUT CONCEPTS
94. (a) Sketch several representative vectors in the vector field
given by
(b) Sketch several representative vectors in the vector field
given by
(c) Explain any similarities or differences in the vector
fields y G x, y .
F x, y
G x, y
xi
yj
x 2 y 2.
F x, y
xi
yj
x 2 y 2.
CAPSTONE
In Exercises 53–56, use a computer algebra system to find the
rot F for the vector field.
53.
54.
55.
56.
In Exercises 57–62, determine whether the vector field F is
conservative. If it is, find a potential function for the vector field.
57.
58.
59.
60.
61.
62.
In Exercises 63–66, find the divergence of the vector field F.
63.
64.
65.
66.
In Exercises 67–70, find the divergence of the vector field F at
the given point.
67.
68.
69.
70.
In Exercises 75 and 76, find rot
75. 76.
In Exercises 77 and 78, find rot
77.
78.
In Exercises 79 and 80, find
79. 80.
In Exercises 81 and 82, find
81.
82.
In Exercises 83–90, prove the property for vector fields F and G
and scalar function
(Assume that the required partial
derivatives are continuous.)
83.
84.
85.
86.
87.
88.
89.
90. (Theorem 15.3)
In Exercises 91–93, let
and let
91. Show that 92. Show that
93. Show that
True or False?
In Exercises 95–98, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
95. If then as
96. If and is on the positive -axis, then
the vector points in the negative -direction.
97. If is a scalar field, then rot is a meaningful expression.
98. If is a vector field and then is irrotational but
not conservative.
F
rot F 0,
F
f
f
y
y
x, y
F x, y 4xi y 2 j
x, y → 0, 0 .
F x, y → 0
F x, y 4xi y 2 j,
f n nf n 2 F.
1
f
F
f 3.
ln f
F
f 2.
fx, y, z F x, y, z .
F x, y, z
xi yj zk,
div rot F 0
div f F f div F f F
f F f F f F
f F F
div F G rot F G F rot G
div F G div F div G
rot f f 0
rot F G rot F rot G
f.
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
div rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
div F G F G .
F x, y, z x 2 zi 2xzj yzk
F x, y, z xyzi yj zk
rot F F .
G x, y, z x 2 i yj z 2 k
G x, y, z xi yj zk
F x, y, z xi zk
F x, y, z i 3xj 2yk
F G F G .
3, 2, 1
F x, y, z ln xyz i j k
3, 0, 0
F x, y, z e x sen yi e x cos yj z 2 k
2, 1, 3
F x, y, z x 2 zi 2xzj yzk
2, 1, 1
F x, y, z xyzi xyj zk
Punto
Campo vectorial
F x, y, z ln x 2 y 2 i xyj ln y 2 z 2 k
F x, y, z senxi cos yj z 2 k
F x, y xe x i ye y j
F x, y x 2 i 2y 2 j
F x, y, z
x
x 2 y 2 i y
x 2 y 2 j k
F x, y, z
z
y i
xz
y 2 j
x
y k
F x, y, z ye z i ze x j xe y k
F x, y, z sen zi sen xj sen yk
F x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 k
F x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zk
F x, y, z x 2 y 2 z 2 i j k
F x, y, z sen x y i sen y z j sen z x k
F x, y, z
yz
y z i xz
x z j xy
x
y k
F x, y, z arctan x y i ln x2 y 2 j k
1068 Chapter 15 Vector Analysis
CAS
71. Define a vector field in the plane and in space. Give some
physical examples of vector fields.
72. What is a conservative vector field, and how do you test for
it in the plane and in space?
73. Define the rot of a vector field.
74. Define the divergence of a vector field in the plane and in
space.
WRITING ABOUT CONCEPTS
94. (a) Sketch several representative vectors in the vector field
given by
(b) Sketch several representative vectors in the vector field
given by
(c) Explain any similarities or differences in the vector
fields y G x, y .
F x, y
G x, y
xi
yj
x 2 y 2.
F x, y
xi
yj
x 2 y 2.
CAPSTONE
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