Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 15.8 Teorema de Stokes 1137
15.8 Ejercicios
En los ejercicios 1 a 6, hallar el rotacional del campo vectorial F.
En los ejercicios 7 a 10, verificar el teorema de Stokes evaluando
como integral de línea e integral doble.
En los ejercicios 11 a 20, utilizar el teorema de Stokes para evaluar
Utilizar un sistema algebraico por computadora y
verificar los resultados. En cada uno de los casos, C está orientada
en sentido contrario a las manecillas del reloj como se vio
anteriormente.
16.
17.
sobre de un pétalo de
en el
primer octante
18.
la porción en el primer octante de
sobre
19.
es el vector unitario normal a la superficie, dirigido hacia abajo.
20.
la porción en el primer octante de sobre x 2 y 2 a 2 Movimiento de un líquido En los ejercicios 21 y 22, el movimiento
de un líquido en un recipiente cilíndrico de radio 1 se
describe mediante el campo de velocidad
Hallar
donde S es la superficie superior del recipiente
cilíndrico.
21. 22.
25. Sean f y g funciones escalares con derivadas parciales continuas,
y supóngase que C y S satisfacen las condiciones del teorema
de Stokes. Verificar cada una de las identidades siguientes.
26. Demostrar los resultados del ejercicio 25 para las funciones
y Sea S el hemisferio
27. Sea C un vector constante. Sea S una superficie orientada con
vector unitario normal N, limitada o acotada por una curva suave
C. Demostrar que
S C N dS 1 2C
C r dr.
z 4 x 2 y 2 .
gx, y, z z.
f x, y, z xyz
Fx, y, z zi yk
Fx, y, z i j 2k
6, find the curl of the vector field F.
7–10, verify Stokes’s Theorem by evaluating
as a line integral and as a double integral.
–20, use Stokes’s Theorem to evaluate
r algebra system to verify your results. In each
ted counterclockwise as viewed from above.
o cuyos vértices son
o cuyos vértices son
over
in the first octant
-octant portion of
over
wnward unit normal to the surface.
-octant portion of
over
Motion of a Liquid
In Exercises 21 and 22, the motion of a
liquid in a cylindrical container of radius 1 is described by the
velocity field Find where is the
upper surface of the cylindrical container.
21. 22.
25. Let and be scalar functions with continuous partial derivatives,
and let and satisfy the conditions of Stokes’s
Theorem. Verify each identity.
a)
b) c)
26. Demonstrate the results of Exercise 25 for the functions
and Let be the hemisphere
27. Let be a constant vector. Let be an oriented surface with a
unit normal vector bounded by a smooth curve Prove that
S
C
N dS
1
2 C C r dr.
C.
N,
S
C
z 4 x 2 y 2 .
S
g x, y, z z.
f x, y, z
xyz
C f g g f dr 0
C f f dr 0
C f g dr S f g N dS
S
C
g
f
F x, y, z zi yk
F x, y, z i j 2k
S
S rot F
N dS,
F x, y, z .
x 2 y 2 a 2
z x 2
x 2 y 2 a 2
xyzi yj zk,
0 y a
0 x a,
xyzi yj zk
x 2 y 2 16
x 2 z 2 16
x 2 y 2 16
yzi 2 3y j x 2 y 2 k,
r 2 sen 2
2x
3y
ln x 2 y 2 i arctan x y j k
x 2 y 2
x 2 i z 2 j xyzk
x 2 y 2
z 2 i yj zk
z ≥ 0
x 2 y 2 ,
4xzi yj 4xyk
z ≥ 0
x 2 y 2 ,
z 2 i 2xj y 2 k
0, 0, 2
1, 1, 1 ,
0, 0, 0 ,
arctan x y i ln x2 y 2 j k
0, 0, 2
0, 2, 0 ,
2, 0, 0 ,
2yi 3zj xk
C F
dr.
0 y a
0 x a,
z 2 i x 2 j y 2 k
z
12, primer octante
xyzi yj zk
x 2 y 2
y z i x z j x y k
z 0
x 2 y 2 ,
y z i x z j x y k
F
dr
arcsen yi 1 x 2 j y 2 k
e x2 y 2 i e y2 z 2 j xyzk
x sen yi y cos xj yz 2 k
2zi 4x 2 j arctan xk
x 2 i y 2 j x 2 k
2y z i e z j xyzk
15.8 Stokes’s Theorem 1137
ercises
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
23. State Stokes’s Theorem.
24. Give a physical interpretation of curl.
WRITING ABOUT CONCEPTS
28. Verify Stokes’s Theorem for each given vector field and
upward oriented surface. Is the line integral or the double
integral easier to set up? to evaluate? Explain.
(a)
square with vertices
(b)
the portion of the paraboloid
that lies
below the plane z 4
z x 2 y 2
S:
F x, y, z z 2 i x 2 j y 2 k
0, 1, 0
1, 1, 0 ,
1, 0, 0 ,
0, 0, 0 ,
C:
F x, y, z e y z i
CAPSTONE
29. Let
Prove or disprove that there is a vector-valued function
with the
following properties.
(i)
have continuous partial derivatives for all
(ii) Curl
for all
(iii)
This problem was composed by the Committee on the Putnam Prize Competition.
© The Mathematical Association of America. All rights reserved.
F x, y, 0 G x, y .
x, y, z 0, 0, 0 ;
F 0
x, y, z 0, 0, 0 ;
P
N,
M,
Px, y, z
N x, y, z ,
F x, y, z M x, y, z ,
G x, y
y
x 2 4y 2, x
x 2 4y 2, 0 .
PUTNAM EXAM CHALLENGE
Fx, y, z.
z x 2
S:
x 2 y 2 ≤ a 2
Fx, y, z xyzi yj zk,
N
0 ≤ y ≤ a
0 ≤ x ≤ a,
S: z x 2 ,
Fx, y, z xyzi yj zk
x 2 y 2 16
x 2 z 2 16
S:
x 2 y 2 ≤ 16
Fx, y, z yzi 2 3yj x 2 y 2 k,
r 2 sin 2
z 9 2x 3y
S:
Fx, y, z lnx 2 y 2 i arctan x y j k
S: z 4 x 2 y 2
Fx, y, z x 2 i z 2 j xyzk
C F dr.
C
F T ds C F dr
Desarrollo de conceptos
23. Enunciar el teorema de Stokes.
24. Dar una interpretación física del rotacional.
sen
In Exercises 1–6, find the curl of the vector field F.
1.
2.
3.
4.
5.
6.
In Exercises 7–10, verify Stokes’s Theorem by evaluating
as a line integral and as a double integral.
7.
8.
9.
10.
In Exercises 11–20, use Stokes’s Theorem to evaluate
Use a computer algebra system to verify your results. In each
case,
is oriented counterclockwise as viewed from above.
11.
triángulo cuyos vértices son
12.
triángulo cuyos vértices son
13.
14.
15.
16.
17.
over
in the first octant
18.
the first-octant portion of
over
19.
is the downward unit normal to the surface.
20.
the first-octant portion of
over
Motion of a Liquid
In Exercises 21 and 22, the motion of a
liquid in a cylindrical container of radius 1 is described by the
velocity field Find where is the
upper surface of the cylindrical container.
21. 22.
25. Let and be scalar functions with continuous partial derivatives,
and let and satisfy the conditions of Stokes’s
Theorem. Verify each identity.
a)
b) c)
26. Demonstrate the results of Exercise 25 for the functions
and Let be the hemisphere
27. Let be a constant vector. Let be an oriented surface with a
unit normal vector bounded by a smooth curve Prove that
S
C
N dS
1
2 C C r dr.
C.
N,
S
C
z 4 x 2 y 2 .
S
g x, y, z z.
f x, y, z
xyz
C f g g f dr 0
C f f dr 0
C f g dr S f g N dS
S
C
g
f
F x, y, z zi yk
F x, y, z i j 2k
S
S rot F
N dS,
F x, y, z .
x 2 y 2 a 2
z x 2
S:
x 2 y 2 a 2
F x, y, z xyzi yj zk,
N
0 y a
0 x a,
S: z x 2 ,
F x, y, z xyzi yj zk
x 2 y 2 16
x 2 z 2 16
S:
x 2 y 2 16
F x, y, z yzi 2 3y j x 2 y 2 k,
r 2 sen 2
z 9 2x 3y
S:
F x, y, z ln x 2 y 2 i arctan x y j k
S: z 4 x 2 y 2
F x, y, z x 2 i z 2 j xyzk
S: z 4 x 2 y 2
F x, y, z z 2 i yj zk
z ≥ 0
S: z 9 x 2 y 2 ,
F x, y, z 4xzi yj 4xyk
z ≥ 0
S: z 1 x 2 y 2 ,
F x, y, z z 2 i 2xj y 2 k
0, 0, 2
1, 1, 1 ,
0, 0, 0 ,
C:
F x, y, z arctan x y i ln x2 y 2 j k
0, 0, 2
0, 2, 0 ,
2, 0, 0 ,
C:
F x, y, z 2yi 3zj xk
C
C F
dr.
0 y a
0 x a,
S: z y 2 ,
F x, y, z z 2 i x 2 j y 2 k
S: 6x 6y z 12, primer octante
F x, y, z xyzi yj zk
S: z 1 x 2 y 2
F x, y, z y z i x z j x y k
z 0
S: z 9 x 2 y 2 ,
F x, y, z y z i x z j x y k
C
F
T ds
C
F
dr
F x, y, z arcsen yi 1 x 2 j y 2 k
F x, y, z e x2 y 2 i e y2 z 2 j xyzk
F x, y, z x sen yi y cos xj yz 2 k
F x, y, z 2zi 4x 2 j arctan xk
F x, y, z x 2 i y 2 j x 2 k
F x, y, z 2y z i e z j xyzk
15.8 Stokes’s Theorem 1137
15.8 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
23. State Stokes’s Theorem.
24. Give a physical interpretation of curl.
WRITING ABOUT CONCEPTS
28. Verify Stokes’s Theorem for each given vector field and
upward oriented surface. Is the line integral or the double
integral easier to set up? to evaluate? Explain.
(a)
square with vertices
(b)
the portion of the paraboloid
that lies
below the plane z 4
z x 2 y 2
S:
F x, y, z z 2 i x 2 j y 2 k
0, 1, 0
1, 1, 0 ,
1, 0, 0 ,
0, 0, 0 ,
C:
F x, y, z e y z i
CAPSTONE
29. Let
Prove or disprove that there is a vector-valued function
with the
following properties.
(i)
have continuous partial derivatives for all
(ii) Curl
for all
(iii)
This problem was composed by the Committee on the Putnam Prize Competition.
© The Mathematical Association of America. All rights reserved.
F x, y, 0 G x, y .
x, y, z 0, 0, 0 ;
F 0
x, y, z 0, 0, 0 ;
P
N,
M,
Px, y, z
N x, y, z ,
F x, y, z M x, y, z ,
G x, y
y
x 2 4y 2, x
x 2 4y 2, 0 .
PUTNAM EXAM CHALLENGE
In Exercises 1–6, find the curl of the vector field F.
1.
2.
3.
4.
5.
6.
In Exercises 7–10, verify Stokes’s Theorem by evaluating
as a line integral and as a double integral.
7.
8.
9.
10.
In Exercises 11–20, use Stokes’s Theorem to evaluate
Use a computer algebra system to verify your results. In each
case,
is oriented counterclockwise as viewed from above.
11.
triángulo cuyos vértices son
12.
triángulo cuyos vértices son
13.
14.
15.
16.
17.
over
in the first octant
18.
the first-octant portion of
over
19.
is the downward unit normal to the surface.
20.
the first-octant portion of
over
Motion of a Liquid
In Exercises 21 and 22, the motion of a
liquid in a cylindrical container of radius 1 is described by the
velocity field Find where is the
upper surface of the cylindrical container.
21. 22.
25. Let and be scalar functions with continuous partial derivatives,
and let and satisfy the conditions of Stokes’s
Theorem. Verify each identity.
a)
b) c)
26. Demonstrate the results of Exercise 25 for the functions
and Let be the hemisphere
27. Let be a constant vector. Let be an oriented surface with a
unit normal vector bounded by a smooth curve Prove that
S
C
N dS
1
2 C C r dr.
C.
N,
S
C
z 4 x 2 y 2 .
S
g x, y, z z.
f x, y, z
xyz
C f g g f dr 0
C f f dr 0
C f g dr S f g N dS
S
C
g
f
F x, y, z zi yk
F x, y, z i j 2k
S
S rot F
N dS,
F x, y, z .
x 2 y 2 a 2
z x 2
S:
x 2 y 2 a 2
F x, y, z xyzi yj zk,
N
0 y a
0 x a,
S: z x 2 ,
F x, y, z xyzi yj zk
x 2 y 2 16
x 2 z 2 16
S:
x 2 y 2 16
F x, y, z yzi 2 3y j x 2 y 2 k,
r 2 sen 2
z 9 2x 3y
S:
F x, y, z ln x 2 y 2 i arctan x y j k
S: z 4 x 2 y 2
F x, y, z x 2 i z 2 j xyzk
S: z 4 x 2 y 2
F x, y, z z 2 i yj zk
z ≥ 0
S: z 9 x 2 y 2 ,
F x, y, z 4xzi yj 4xyk
z ≥ 0
S: z 1 x 2 y 2 ,
F x, y, z z 2 i 2xj y 2 k
0, 0, 2
1, 1, 1 ,
0, 0, 0 ,
C:
F x, y, z arctan x y i ln x2 y 2 j k
0, 0, 2
0, 2, 0 ,
2, 0, 0 ,
C:
F x, y, z 2yi 3zj xk
C
C F
dr.
0 y a
0 x a,
S: z y 2 ,
F x, y, z z 2 i x 2 j y 2 k
S: 6x 6y z 12, primer octante
F x, y, z xyzi yj zk
S: z 1 x 2 y 2
F x, y, z y z i x z j x y k
z 0
S: z 9 x 2 y 2 ,
F x, y, z y z i x z j x y k
C
F
T ds
C
F
dr
F x, y, z arcsen yi 1 x 2 j y 2 k
F x, y, z e x2 y 2 i e y2 z 2 j xyzk
F x, y, z x sen yi y cos xj yz 2 k
F x, y, z 2zi 4x 2 j arctan xk
F x, y, z x 2 i y 2 j x 2 k
F x, y, z 2y z i e z j xyzk
15.8 Stokes’s Theorem 1137
15.8 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
23. State Stokes’s Theorem.
24. Give a physical interpretation of curl.
WRITING ABOUT CONCEPTS
28. Verify Stokes’s Theorem for each given vector field and
upward oriented surface. Is the line integral or the double
integral easier to set up? to evaluate? Explain.
(a)
square with vertices
(b)
the portion of the paraboloid
that lies
below the plane z 4
z x 2 y 2
S:
F x, y, z z 2 i x 2 j y 2 k
0, 1, 0
1, 1, 0 ,
1, 0, 0 ,
0, 0, 0 ,
C:
F x, y, z e y z i
CAPSTONE
29. Let
Prove or disprove that there is a vector-valued function
with the
following properties.
(i)
have continuous partial derivatives for all
(ii) Curl
for all
(iii)
This problem was composed by the Committee on the Putnam Prize Competition.
© The Mathematical Association of America. All rights reserved.
F x, y, 0 G x, y .
x, y, z 0, 0, 0 ;
F 0
x, y, z 0, 0, 0 ;
P
N,
M,
Px, y, z
N x, y, z ,
F x, y, z M x, y, z ,
G x, y
y
x 2 4y 2, x
x 2 4y 2, 0 .
PUTNAM EXAM CHALLENGE
In Exercises 1–6, find the curl of the vector field F.
1.
2.
3.
4.
5.
6.
In Exercises 7–10, verify Stokes’s Theorem by evaluating
as a line integral and as a double integral.
7.
8.
9.
10.
In Exercises 11–20, use Stokes’s Theorem to evaluate
Use a computer algebra system to verify your results. In each
case,
is oriented counterclockwise as viewed from above.
11.
triángulo cuyos vértices son
12.
triángulo cuyos vértices son
13.
14.
15.
16.
17.
over
in the first octant
18.
the first-octant portion of
over
19.
is the downward unit normal to the surface.
20.
the first-octant portion of
over
Motion of a Liquid
In Exercises 21 and 22, the motion of a
liquid in a cylindrical container of radius 1 is described by the
velocity field Find where is the
upper surface of the cylindrical container.
21. 22.
25. Let and be scalar functions with continuous partial derivatives,
and let and satisfy the conditions of Stokes’s
Theorem. Verify each identity.
a)
b) c)
26. Demonstrate the results of Exercise 25 for the functions
and Let be the hemisphere
27. Let be a constant vector. Let be an oriented surface with a
unit normal vector bounded by a smooth curve Prove that
S
C
N dS
1
2 C C r dr.
C.
N,
S
C
z 4 x 2 y 2 .
S
g x, y, z z.
f x, y, z
xyz
C f g g f dr 0
C f f dr 0
C f g dr S f g N dS
S
C
g
f
F x, y, z zi yk
F x, y, z i j 2k
S
S rot F
N dS,
F x, y, z .
x 2 y 2 a 2
z x 2
S:
x 2 y 2 a 2
F x, y, z xyzi yj zk,
N
0 y a
0 x a,
S: z x 2 ,
F x, y, z xyzi yj zk
x 2 y 2 16
x 2 z 2 16
S:
x 2 y 2 16
F x, y, z yzi 2 3y j x 2 y 2 k,
r 2 sen 2
z 9 2x 3y
S:
F x, y, z ln x 2 y 2 i arctan x y j k
S: z 4 x 2 y 2
F x, y, z x 2 i z 2 j xyzk
S: z 4 x 2 y 2
F x, y, z z 2 i yj zk
z ≥ 0
S: z 9 x 2 y 2 ,
F x, y, z 4xzi yj 4xyk
z ≥ 0
S: z 1 x 2 y 2 ,
F x, y, z z 2 i 2xj y 2 k
0, 0, 2
1, 1, 1 ,
0, 0, 0 ,
C:
F x, y, z arctan x y i ln x2 y 2 j k
0, 0, 2
0, 2, 0 ,
2, 0, 0 ,
C:
F x, y, z 2yi 3zj xk
C
C F
dr.
0 y a
0 x a,
S: z y 2 ,
F x, y, z z 2 i x 2 j y 2 k
S: 6x 6y z 12, primer octante
F x, y, z xyzi yj zk
S: z 1 x 2 y 2
F x, y, z y z i x z j x y k
z 0
S: z 9 x 2 y 2 ,
F x, y, z y z i x z j x y k
C
F
T ds
C
F
dr
F x, y, z arcsen yi 1 x 2 j y 2 k
F x, y, z e x2 y 2 i e y2 z 2 j xyzk
F x, y, z x sen yi y cos xj yz 2 k
F x, y, z 2zi 4x 2 j arctan xk
F x, y, z x 2 i y 2 j x 2 k
F x, y, z 2y z i e z j xyzk
15.8 Stokes’s Theorem 1137
15.8 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
23. State Stokes’s Theorem.
24. Give a physical interpretation of curl.
WRITING ABOUT CONCEPTS
28. Verify Stokes’s Theorem for each given vector field and
upward oriented surface. Is the line integral or the double
integral easier to set up? to evaluate? Explain.
(a)
square with vertices
(b)
the portion of the paraboloid
that lies
below the plane z 4
z x 2 y 2
S:
F x, y, z z 2 i x 2 j y 2 k
0, 1, 0
1, 1, 0 ,
1, 0, 0 ,
0, 0, 0 ,
C:
F x, y, z e y z i
CAPSTONE
29. Let
Prove or disprove that there is a vector-valued function
with the
following properties.
(i)
have continuous partial derivatives for all
(ii) Curl
for all
(iii)
This problem was composed by the Committee on the Putnam Prize Competition.
© The Mathematical Association of America. All rights reserved.
F x, y, 0 G x, y .
x, y, z 0, 0, 0 ;
F 0
x, y, z 0, 0, 0 ;
P
N,
M,
Px, y, z
N x, y, z ,
F x, y, z M x, y, z ,
G x, y
y
x 2 4y 2, x
x 2 4y 2, 0 .
PUTNAM EXAM CHALLENGE
Preparación del examen Putman
29. Sea
Demostrar o refutar que hay una función vectorial Fx, y, z
Mx, y, z,
con las propiedades siguientes.
i) tienen derivadas parciales continuas en todo
ii) Rot
para todo
iii)
Este problema fue preparado por el Committee on the Putnam Prize Competition.
© The Mathematical Association of America. Todos los derechos reservados.
Fx, y, 0 Gx, y.
x, y, z 0, 0, 0;
F 0
x, y, z 0, 0, 0;
P
N,
M,
Px, y, z
Nx, y, z,
Gx, y y
x 2 4y 2,
x
x 2 4y 2, 0 .
In Exercises 1–6, find the curl of the vector field F.
1.
2.
3.
4.
5.
6.
In Exercises 7–10, verify Stokes’s Theorem by evaluating
as a line integral and as a double integral.
7.
8.
9.
10.
In Exercises 11–20, use Stokes’s Theorem to evaluate
Use a computer algebra system to verify your results. In each
case,
is oriented counterclockwise as viewed from above.
11.
triángulo cuyos vértices son
12.
triángulo cuyos vértices son
13.
14.
15.
16.
17.
over
in the first octant
18.
the first-octant portion of
over
19.
is the downward unit normal to the surface.
20.
the first-octant portion of
over
Motion of a Liquid
In Exercises 21 and 22, the motion of a
liquid in a cylindrical container of radius 1 is described by the
velocity field Find where is the
upper surface of the cylindrical container.
21. 22.
25. Let and be scalar functions with continuous partial derivatives,
and let and satisfy the conditions of Stokes’s
Theorem. Verify each identity.
a)
b) c)
26. Demonstrate the results of Exercise 25 for the functions
and Let be the hemisphere
27. Let be a constant vector. Let be an oriented surface with a
unit normal vector bounded by a smooth curve Prove that
S
C
N dS
1
2 C C r dr.
C.
N,
S
C
z 4 x 2 y 2 .
S
g x, y, z z.
f x, y, z
xyz
C f g g f dr 0
C f f dr 0
C f g dr S f g N dS
S
C
g
f
F x, y, z zi yk
F x, y, z i j 2k
S
S rot F
N dS,
F x, y, z .
x 2 y 2 a 2
z x 2
S:
x 2 y 2 a 2
F x, y, z xyzi yj zk,
N
0 y a
0 x a,
S: z x 2 ,
F x, y, z xyzi yj zk
x 2 y 2 16
x 2 z 2 16
S:
x 2 y 2 16
F x, y, z yzi 2 3y j x 2 y 2 k,
r 2 sen 2
z 9 2x 3y
S:
F x, y, z ln x 2 y 2 i arctan x y j k
S: z 4 x 2 y 2
F x, y, z x 2 i z 2 j xyzk
S: z 4 x 2 y 2
F x, y, z z 2 i yj zk
z ≥ 0
S: z 9 x 2 y 2 ,
F x, y, z 4xzi yj 4xyk
z ≥ 0
S: z 1 x 2 y 2 ,
F x, y, z z 2 i 2xj y 2 k
0, 0, 2
1, 1, 1 ,
0, 0, 0 ,
C:
F x, y, z arctan x y i ln x2 y 2 j k
0, 0, 2
0, 2, 0 ,
2, 0, 0 ,
C:
F x, y, z 2yi 3zj xk
C
C F
dr.
0 y a
0 x a,
S: z y 2 ,
F x, y, z z 2 i x 2 j y 2 k
S: 6x 6y z 12, primer octante
F x, y, z xyzi yj zk
S: z 1 x 2 y 2
F x, y, z y z i x z j x y k
z 0
S: z 9 x 2 y 2 ,
F x, y, z y z i x z j x y k
C
F
T ds
C
F
dr
F x, y, z arcsen yi 1 x 2 j y 2 k
F x, y, z e x2 y 2 i e y2 z 2 j xyzk
F x, y, z x sen yi y cos xj yz 2 k
F x, y, z 2zi 4x 2 j arctan xk
F x, y, z x 2 i y 2 j x 2 k
F x, y, z 2y z i e z j xyzk
15.8 Stokes’s Theorem 1137
15.8 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
23. State Stokes’s Theorem.
24. Give a physical interpretation of curl.
WRITING ABOUT CONCEPTS
28. Verify Stokes’s Theorem for each given vector field and
upward oriented surface. Is the line integral or the double
integral easier to set up? to evaluate? Explain.
(a)
square with vertices
(b)
the portion of the paraboloid
that lies
below the plane z 4
z x 2 y 2
S:
F x, y, z z 2 i x 2 j y 2 k
0, 1, 0
1, 1, 0 ,
1, 0, 0 ,
0, 0, 0 ,
C:
F x, y, z e y z i
CAPSTONE
29. Let
Prove or disprove that there is a vector-valued function
with the
following properties.
(i)
have continuous partial derivatives for all
(ii) Curl
for all
(iii)
This problem was composed by the Committee on the Putnam Prize Competition.
© The Mathematical Association of America. All rights reserved.
F x, y, 0 G x, y .
x, y, z 0, 0, 0 ;
F 0
x, y, z 0, 0, 0 ;
P
N,
M,
Px, y, z
N x, y, z ,
F x, y, z M x, y, z ,
G x, y
y
x 2 4y 2, x
x 2 4y 2, 0 .
PUTNAM EXAM CHALLENGE
Para discusión
28. Verificar el teorema de Stokes para cada campo vectorial
dado y superficie orientada hacia arriba. ¿Es más fácil
establecer la integral de línea o la integral doble?, ¿de evaluar?
Explicar.
a)
C: cuadrado con vértices (0, 0, 0), (1, 0, 0), (1, 1, 0), (0, 1, 0)
b)
S: la porción del paraboloide z = x 2 + y 2 que yace abajo del
plano z = 4.
In Exercises 1–6, find the curl of the vector field F.
1.
2.
3.
4.
5.
6.
In Exercises 7–10, verify Stokes’s Theorem by evaluating
as a line integral and as a double integral.
7.
8.
9.
10.
In Exercises 11–20, use Stokes’s Theorem to evaluate
Use a computer algebra system to verify your results. In each
case,
is oriented counterclockwise as viewed from above.
11.
triángulo cuyos vértices son
12.
triángulo cuyos vértices son
13.
14.
15.
16.
17.
over
in the first octant
18.
the first-octant portion of
over
19.
is the downward unit normal to the surface.
20.
the first-octant portion of
over
Motion of a Liquid
In Exercises 21 and 22, the motion of a
liquid in a cylindrical container of radius 1 is described by the
velocity field Find where is the
upper surface of the cylindrical container.
21. 22.
25. Let and be scalar functions with continuous partial derivatives,
and let and satisfy the conditions of Stokes’s
Theorem. Verify each identity.
a)
b) c)
26. Demonstrate the results of Exercise 25 for the functions
and Let be the hemisphere
27. Let be a constant vector. Let be an oriented surface with a
unit normal vector bounded by a smooth curve Prove that
S
C
N dS
1
2 C C r dr.
C.
N,
S
C
z 4 x 2 y 2 .
S
g x, y, z z.
f x, y, z
xyz
C f g g f dr 0
C f f dr 0
C f g dr S f g N dS
S
C
g
f
F x, y, z zi yk
F x, y, z i j 2k
S
S rot F
N dS,
F x, y, z .
x 2 y 2 a 2
z x 2
S:
x 2 y 2 a 2
F x, y, z xyzi yj zk,
N
0 y a
0 x a,
S: z x 2 ,
F x, y, z xyzi yj zk
x 2 y 2 16
x 2 z 2 16
S:
x 2 y 2 16
F x, y, z yzi 2 3y j x 2 y 2 k,
r 2 sen 2
z 9 2x 3y
S:
F x, y, z ln x 2 y 2 i arctan x y j k
S: z 4 x 2 y 2
F x, y, z x 2 i z 2 j xyzk
S: z 4 x 2 y 2
F x, y, z z 2 i yj zk
z ≥ 0
S: z 9 x 2 y 2 ,
F x, y, z 4xzi yj 4xyk
z ≥ 0
S: z 1 x 2 y 2 ,
F x, y, z z 2 i 2xj y 2 k
0, 0, 2
1, 1, 1 ,
0, 0, 0 ,
C:
F x, y, z arctan x y i ln x2 y 2 j k
0, 0, 2
0, 2, 0 ,
2, 0, 0 ,
C:
F x, y, z 2yi 3zj xk
C
C F
dr.
0 y a
0 x a,
S: z y 2 ,
F x, y, z z 2 i x 2 j y 2 k
S: 6x 6y z 12, primer octante
F x, y, z xyzi yj zk
S: z 1 x 2 y 2
F x, y, z y z i x z j x y k
z 0
S: z 9 x 2 y 2 ,
F x, y, z y z i x z j x y k
C
F
T ds
C
F
dr
F x, y, z arcsen yi 1 x 2 j y 2 k
F x, y, z e x2 y 2 i e y2 z 2 j xyzk
F x, y, z x sen yi y cos xj yz 2 k
F x, y, z 2zi 4x 2 j arctan xk
F x, y, z x 2 i y 2 j x 2 k
F x, y, z 2y z i e z j xyzk
15.8 Stokes’s Theorem 1137
15.8 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
23. State Stokes’s Theorem.
24. Give a physical interpretation of curl.
WRITING ABOUT CONCEPTS
28. Verify Stokes’s Theorem for each given vector field and
upward oriented surface. Is the line integral or the double
integral easier to set up? to evaluate? Explain.
(a)
square with vertices
(b)
the portion of the paraboloid
that lies
below the plane z 4
z x 2 y 2
S:
F x, y, z z 2 i x 2 j y 2 k
0, 1, 0
1, 1, 0 ,
1, 0, 0 ,
0, 0, 0 ,
C:
F x, y, z e y z i
CAPSTONE
29. Let
Prove or disprove that there is a vector-valued function
with the
following properties.
(i)
have continuous partial derivatives for all
(ii) Curl
for all
(iii)
This problem was composed by the Committee on the Putnam Prize Competition.
© The Mathematical Association of America. All rights reserved.
F x, y, 0 G x, y .
x, y, z 0, 0, 0 ;
F 0
x, y, z 0, 0, 0 ;
P
N,
M,
Px, y, z
N x, y, z ,
F x, y, z M x, y, z ,
G x, y
y
x 2 4y 2, x
x 2 4y 2, 0 .
PUTNAM EXAM CHALLENGE
In Exercises 1–6, find the curl of the vector field F.
1.
2.
3.
4.
5.
6.
In Exercises 7–10, verify Stokes’s Theorem by evaluating
as a line integral and as a double integral.
7.
8.
9.
10.
In Exercises 11–20, use Stokes’s Theorem to evaluate
Use a computer algebra system to verify your results. In each
case,
is oriented counterclockwise as viewed from above.
11.
triángulo cuyos vértices son
12.
triángulo cuyos vértices son
13.
14.
15.
16.
17.
over
in the first octant
18.
the first-octant portion of
over
19.
is the downward unit normal to the surface.
20.
the first-octant portion of
over
Motion of a Liquid
In Exercises 21 and 22, the motion of a
liquid in a cylindrical container of radius 1 is described by the
velocity field Find where is the
upper surface of the cylindrical container.
21. 22.
25. Let and be scalar functions with continuous partial derivatives,
and let and satisfy the conditions of Stokes’s
Theorem. Verify each identity.
a)
b) c)
26. Demonstrate the results of Exercise 25 for the functions
and Let be the hemisphere
27. Let be a constant vector. Let be an oriented surface with a
unit normal vector bounded by a smooth curve Prove that
S
C
N dS
1
2 C C r dr.
C.
N,
S
C
z 4 x 2 y 2 .
S
g x, y, z z.
f x, y, z
xyz
C f g g f dr 0
C f f dr 0
C f g dr S f g N dS
S
C
g
f
F x, y, z zi yk
F x, y, z i j 2k
S
S rot F
N dS,
F x, y, z .
x 2 y 2 a 2
z x 2
S:
x 2 y 2 a 2
F x, y, z xyzi yj zk,
N
0 y a
0 x a,
S: z x 2 ,
F x, y, z xyzi yj zk
x 2 y 2 16
x 2 z 2 16
S:
x 2 y 2 16
F x, y, z yzi 2 3y j x 2 y 2 k,
r 2 sen 2
z 9 2x 3y
S:
F x, y, z ln x 2 y 2 i arctan x y j k
S: z 4 x 2 y 2
F x, y, z x 2 i z 2 j xyzk
S: z 4 x 2 y 2
F x, y, z z 2 i yj zk
z ≥ 0
S: z 9 x 2 y 2 ,
F x, y, z 4xzi yj 4xyk
z ≥ 0
S: z 1 x 2 y 2 ,
F x, y, z z 2 i 2xj y 2 k
0, 0, 2
1, 1, 1 ,
0, 0, 0 ,
C:
F x, y, z arctan x y i ln x2 y 2 j k
0, 0, 2
0, 2, 0 ,
2, 0, 0 ,
C:
F x, y, z 2yi 3zj xk
C
C F
dr.
0 y a
0 x a,
S: z y 2 ,
F x, y, z z 2 i x 2 j y 2 k
S: 6x 6y z 12, primer octante
F x, y, z xyzi yj zk
S: z 1 x 2 y 2
F x, y, z y z i x z j x y k
z 0
S: z 9 x 2 y 2 ,
F x, y, z y z i x z j x y k
C
F
T ds
C
F
dr
F x, y, z arcsen yi 1 x 2 j y 2 k
F x, y, z e x2 y 2 i e y2 z 2 j xyzk
F x, y, z x sen yi y cos xj yz 2 k
F x, y, z 2zi 4x 2 j arctan xk
F x, y, z x 2 i y 2 j x 2 k
F x, y, z 2y z i e z j xyzk
15.8 Stokes’s Theorem 1137
15.8 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
23. State Stokes’s Theorem.
24. Give a physical interpretation of curl.
WRITING ABOUT CONCEPTS
28. Verify Stokes’s Theorem for each given vector field and
upward oriented surface. Is the line integral or the double
integral easier to set up? to evaluate? Explain.
(a)
square with vertices
(b)
the portion of the paraboloid
that lies
below the plane z 4
z x 2 y 2
S:
F x, y, z z 2 i x 2 j y 2 k
0, 1, 0
1, 1, 0 ,
1, 0, 0 ,
0, 0, 0 ,
C:
F x, y, z e y z i
CAPSTONE
29. Let
Prove or disprove that there is a vector-valued function
with the
following properties.
(i)
have continuous partial derivatives for all
(ii) Curl
for all
(iii)
This problem was composed by the Committee on the Putnam Prize Competition.
© The Mathematical Association of America. All rights reserved.
F x, y, 0 G x, y .
x, y, z 0, 0, 0 ;
F 0
x, y, z 0, 0, 0 ;
P
N,
M,
Px, y, z
N x, y, z ,
F x, y, z M x, y, z ,
G x, y
y
x 2 4y 2, x
x 2 4y 2, 0 .
PUTNAM EXAM CHALLENGE