Calculo 2 De dos variables_9na Edición - Ron Larson

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1068 CAPÍTULO 15 Análisis vectorialEn los ejercicios 53 a 56, usar un sistema algebraico por computadoray representar el rotacional del campo vectorial.En los ejercicios 57 a 62, determinar si el campo vectorial F esconservativo. Si lo es, calcular una función potencial para él.En los ejercicios 63 a 66, calcular la divergencia del campo vectorialF.En los ejercicios 67 a 70, calcular la divergencia del campo vectorialF en el punto dado.En los ejercicios 75 y 76, calcularEn los ejercicios 77 y 78, hallar77.78.En los ejercicios 79 y 80, hallarEn los ejercicios 81 y 82, hallar div (rot F) · ( F).81.82.En los ejercicios 83 a 90, demostrar la propiedad para los camposvectoriales F y G y la función escalar ƒ. (Suponer que lasderivadas parciales requeridas son continuas.)90. div(rot F) 0 (Teorema 15.3)En los ejercicios 91 a 93, seay91. Probar que 92. Probar que93. Probar que¿Verdadero o falso?En los ejercicios 95 a 98, determinar si ladeclaración es verdadera o falsa. Si es falsa, explicar por qué odar 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, entoncesel 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 irrotacionalpero no conservativo.Fffx, yFx, y 4xi y 2 jFx, y → 0f n nf n2 F. 1f 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 yzkFx, y, z xyzi yj zkFx, y, z x 2 zi 2xzj yzkFx, y, z xyzi yj zkDesarrollo de conceptos71. Definir un campo vectorial en el plano y en el espacio. Daralgunos ejemplos físicos de campos vectoriales.72. ¿Qué es un campo vectorial conservativo y cuál es su criterioen 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 enel espacio.9. 10.11. 12.13. 14.15. 16.In Exercises 17–20, use a computer algebra system to graphseveral 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 givenpoint.49.50.51.52. 3, 2, 0F x, y, z e xyz i j k0, 0, 1F x, y, z e x sen yi e x cos yj2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 3F x, y, z xyzi xyzj xyzkPuntoCampo vectorialF x, y2xi2yjx 2 y 2 2F x, y e x cos yi sen yjF x, y, z xi yj zkF x, y, zxi yj zkx 2 y 2 z 2F x, y 2y 3x i 2y 3x jF x, y18 2xyi y 2 jF x, y, z xi yj zkF x, y, z i j kF x, y x 2 y 2 i jF x, y 4xi yjF x, yxiF x, y, z3yjF x, y yi 2xjF x, y yi xjCASIn Exercises 53–56, use a computer algebra system to find therot F for the vector field.53.54.55.56.In Exercises 57–62, determine whether the vector field F isconservative. 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 atthe given point.67.68.69.70.In Exercises 75 and 76, find rot75. 76.In Exercises 77 and 78, find rot77.78.In Exercises 79 and 80, find79. 80.In Exercises 81 and 82, find81.82.In Exercises 83–90, prove the property for vector fields F and Gand scalar function(Assume that the required partialderivatives are continuous.)83.84.85.86.87.88.89.90. (Theorem 15.3)In Exercises 91–93, letand let91. Show that 92. Show that93. Show thatTrue or False?In Exercises 95–98, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.95. If then as96. If and is on the positive -axis, thenthe 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 butnot conservative.Frot F 0,Fffyyx, yF x, y 4xi y 2 jx, y → 0, 0 .F x, y → 0F x, y 4xi y 2 j,f n nf n 2 F.1fFf 3.ln fFf 2.fx, y, z F x, y, z .F x, y, zxi yj zk,div rot F 0div f F f div F f Ff F f F f Ff F Fdiv F G rot F G F rot Gdiv F G div F div Grot f f 0rot F G rot F rot Gf.F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkdiv rot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykdiv F G F G .F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkrot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykF G F G .3, 2, 1F x, y, z ln xyz i j k3, 0, 0F x, y, z e x sen yi e x cos yj z 2 k2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 1F x, y, z xyzi xyj zkPuntoCampo vectorialF x, y, z ln x 2 y 2 i xyj ln y 2 z 2 kF x, y, z senxi cos yj z 2 kF x, y xe x i ye y jF x, y x 2 i 2y 2 jF x, y, zxx 2 y 2 i yx 2 y 2 j kF x, y, zzy ixzy 2 jxy kF x, y, z ye z i ze x j xe y kF x, y, z sen zi sen xj sen ykF x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 kF x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zkF x, y, z x 2 y 2 z 2 i j kF x, y, z sen x y i sen y z j sen z x kF x, y, zyzy z i xzx z j xyxy kF x, y, z arctan x y i ln x2 y 2 j k1068 Chapter 15 Vector AnalysisCAS71. Define a vector field in the plane and in space. Give somephysical examples of vector fields.72. What is a conservative vector field, and how do you test forit 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 inspace.WRITING ABOUT CONCEPTS94. (a) Sketch several representative vectors in the vector fieldgiven by(b) Sketch several representative vectors in the vector fieldgiven by(c) Explain any similarities or differences in the vectorfields y G x, y .F x, yG x, yxiyjx 2 y 2.F x, yxiyjx 2 y 2.CAPSTONEIn Exercises 53–56, use a computer algebra system to find therot F for the vector field.53.54.55.56.In Exercises 57–62, determine whether the vector field F isconservative. 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 atthe given point.67.68.69.70.In Exercises 75 and 76, find rot75. 76.In Exercises 77 and 78, find rot77.78.In Exercises 79 and 80, find79. 80.In Exercises 81 and 82, find81.82.In Exercises 83–90, prove the property for vector fields F and Gand scalar function(Assume that the required partialderivatives are continuous.)83.84.85.86.87.88.89.90. (Theorem 15.3)In Exercises 91–93, letand let91. Show that 92. Show that93. Show thatTrue or False?In Exercises 95–98, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.95. If then as96. If and is on the positive -axis, thenthe 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 butnot conservative.Frot F 0,Fffyyx, yF x, y 4xi y 2 jx, y → 0, 0 .F x, y → 0F x, y 4xi y 2 j,f n nf n 2 F.1fFf 3.ln fFf 2.fx, y, z F x, y, z .F x, y, zxi yj zk,div rot F 0div f F f div F f Ff F f F f Ff F Fdiv F G rot F G F rot Gdiv F G div F div Grot f f 0rot F G rot F rot Gf.F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkdiv rot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykdiv F G F G .F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkrot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykF G F G .3, 2, 1F x, y, z ln xyz i j k3, 0, 0F x, y, z e x sen yi e x cos yj z 2 k2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 1F x, y, z xyzi xyj zkPuntoCampo vectorialF x, y, z ln x 2 y 2 i xyj ln y 2 z 2 kF x, y, z senxi cos yj z 2 kF x, y xe x i ye y jF x, y x 2 i 2y 2 jF x, y, zxx 2 y 2 i yx 2 y 2 j kF x, y, zzy ixzy 2 jxy kF x, y, z ye z i ze x j xe y kF x, y, z sen zi sen xj sen ykF x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 kF x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zkF x, y, z x 2 y 2 z 2 i j kF x, y, z sen x y i sen y z j sen z x kF x, y, zyzy z i xzx z j xyxy kF x, y, z arctan x y i ln x2 y 2 j k1068 Chapter 15 Vector AnalysisCAS71. Define a vector field in the plane and in space. Give somephysical examples of vector fields.72. What is a conservative vector field, and how do you test forit 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 inspace.WRITING ABOUT CONCEPTS94. (a) Sketch several representative vectors in the vector fieldgiven by(b) Sketch several representative vectors in the vector fieldgiven by(c) Explain any similarities or differences in the vectorfields y G x, y .F x, yG x, yxiyjx 2 y 2.F x, yxiyjx 2 y 2.CAPSTONEIn Exercises 53–56, use a computer algebra system to find therot F for the vector field.53.54.55.56.In Exercises 57–62, determine whether the vector field F isconservative. 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 atthe given point.67.68.69.70.In Exercises 75 and 76, find rot75. 76.In Exercises 77 and 78, find rot77.78.In Exercises 79 and 80, find79. 80.In Exercises 81 and 82, find81.82.In Exercises 83–90, prove the property for vector fields F and Gand scalar function(Assume that the required partialderivatives are continuous.)83.84.85.86.87.88.89.90. (Theorem 15.3)In Exercises 91–93, letand let91. Show that 92. Show that93. Show thatTrue or False?In Exercises 95–98, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.95. If then as96. If and is on the positive -axis, thenthe 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 butnot conservative.Frot F 0,Fffyyx, yF x, y 4xi y 2 jx, y → 0, 0 .F x, y → 0F x, y 4xi y 2 j,f n nf n 2 F.1fFf 3.ln fFf 2.fx, y, z F x, y, z .F x, y, zxi yj zk,div rot F 0div f F f div F f Ff F f F f Ff F Fdiv F G rot F G F rot Gdiv F G div F div Grot f f 0rot F G rot F rot Gf.F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkdiv rot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykdiv F G F G .F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkrot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykF G F G .3, 2, 1F x, y, z ln xyz i j k3, 0, 0F x, y, z e x sen yi e x cos yj z 2 k2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 1F x, y, z xyzi xyj zkPuntoCampo vectorialF x, y, z ln x 2 y 2 i xyj ln y 2 z 2 kF x, y, z senxi cos yj z 2 kF x, y xe x i ye y jF x, y x 2 i 2y 2 jF x, y, zxx 2 y 2 i yx 2 y 2 j kF x, y, zzy ixzy 2 jxy kF x, y, z ye z i ze x j xe y kF x, y, z sen zi sen xj sen ykF x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 kF x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zkF x, y, z x 2 y 2 z 2 i j kF x, y, z sen x y i sen y z j sen z x kF x, y, zyzy z i xzx z j xyxy kF x, y, z arctan x y i ln x2 y 2 j k1068 Chapter 15 Vector AnalysisCAS71. Define a vector field in the plane and in space. Give somephysical examples of vector fields.72. What is a conservative vector field, and how do you test forit 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 inspace.WRITING ABOUT CONCEPTS94. (a) Sketch several representative vectors in the vector fieldgiven by(b) Sketch several representative vectors in the vector fieldgiven by(c) Explain any similarities or differences in the vectorfields y G x, y .F x, yG x, yxiyjx 2 y 2.F x, yxiyjx 2 y 2.CAPSTONEIn Exercises 53–56, use a computer algebra system to find therot F for the vector field.53.54.55.56.In Exercises 57–62, determine whether the vector field F isconservative. 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 atthe given point.67.68.69.70.In Exercises 75 and 76, find rot75. 76.In Exercises 77 and 78, find rot77.78.In Exercises 79 and 80, find79. 80.In Exercises 81 and 82, find81.82.In Exercises 83–90, prove the property for vector fields F and Gand scalar function(Assume that the required partialderivatives are continuous.)83.84.85.86.87.88.89.90. (Theorem 15.3)In Exercises 91–93, letand let91. Show that 92. Show that93. Show thatTrue or False?In Exercises 95–98, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.95. If then as96. If and is on the positive -axis, thenthe 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 butnot conservative.Frot F 0,Fffyyx, yF x, y 4xi y 2 jx, y → 0, 0 .F x, y → 0F x, y 4xi y 2 j,f n nf n 2 F.1fFf 3.ln fFf 2.fx, y, z F x, y, z .F x, y, zxi yj zk,div rot F 0div f F f div F f Ff F f F f Ff F Fdiv F G rot F G F rot Gdiv F G div F div Grot f f 0rot F G rot F rot Gf.F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkdiv rot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykdiv F G F G .F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkrot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykF G F G .3, 2, 1F x, y, z ln xyz i j k3, 0, 0F x, y, z e x sen yi e x cos yj z 2 k2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 1F x, y, z xyzi xyj zkPuntoCampo vectorialF x, y, z ln x 2 y 2 i xyj ln y 2 z 2 kF x, y, z senxi cos yj z 2 kF x, y xe x i ye y jF x, y x 2 i 2y 2 jF x, y, zxx 2 y 2 i yx 2 y 2 j kF x, y, zzy ixzy 2 jxy kF x, y, z ye z i ze x j xe y kF x, y, z sen zi sen xj sen ykF x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 kF x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zkF x, y, z x 2 y 2 z 2 i j kF x, y, z sen x y i sen y z j sen z x kF x, y, zyzy z i xzx z j xyxy kF x, y, z arctan x y i ln x2 y 2 j k1068 Chapter 15 Vector AnalysisCAS71. Define a vector field in the plane and in space. Give somephysical examples of vector fields.72. What is a conservative vector field, and how do you test forit 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 inspace.WRITING ABOUT CONCEPTS94. (a) Sketch several representative vectors in the vector fieldgiven by(b) Sketch several representative vectors in the vector fieldgiven by(c) Explain any similarities or differences in the vectorfields y G x, y .F x, yG x, yxiyjx 2 y 2.F x, yxiyjx 2 y 2.CAPSTONEIn Exercises 53–56, use a computer algebra system to find therot F for the vector field.53.54.55.56.In Exercises 57–62, determine whether the vector field F isconservative. 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 atthe given point.67.68.69.70.In Exercises 75 and 76, find rot75. 76.In Exercises 77 and 78, find rot77.78.In Exercises 79 and 80, find79. 80.In Exercises 81 and 82, find81.82.In Exercises 83–90, prove the property for vector fields F and Gand scalar function(Assume that the required partialderivatives are continuous.)83.84.85.86.87.88.89.90. (Theorem 15.3)In Exercises 91–93, letand let91. Show that 92. Show that93. Show thatTrue or False?In Exercises 95–98, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.95. If then as96. If and is on the positive -axis, thenthe 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 butnot conservative.Frot F 0,Fffyyx, yF x, y 4xi y 2 jx, y → 0, 0 .F x, y → 0F x, y 4xi y 2 j,f n nf n 2 F.1fFf 3.ln fFf 2.fx, y, z F x, y, z .F x, y, zxi yj zk,div rot F 0div f F f div F f Ff F f F f Ff F Fdiv F G rot F G F rot Gdiv F G div F div Grot f f 0rot F G rot F rot Gf.F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkdiv rot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykdiv F G F G .F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkrot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykF G F G .3, 2, 1F x, y, z ln xyz i j k3, 0, 0F x, y, z e x sen yi e x cos yj z 2 k2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 1F x, y, z xyzi xyj zkPuntoCampo vectorialF x, y, z ln x 2 y 2 i xyj ln y 2 z 2 kF x, y, z senxi cos yj z 2 kF x, y xe x i ye y jF x, y x 2 i 2y 2 jF x, y, zxx 2 y 2 i yx 2 y 2 j kF x, y, zzy ixzy 2 jxy kF x, y, z ye z i ze x j xe y kF x, y, z sen zi sen xj sen ykF x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 kF x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zkF x, y, z x 2 y 2 z 2 i j kF x, y, z sen x y i sen y z j sen z x kF x, y, zyzy z i xzx z j xyxy kF x, y, z arctan x y i ln x2 y 2 j k1068 Chapter 15 Vector AnalysisCAS71. Define a vector field in the plane and in space. Give somephysical examples of vector fields.72. What is a conservative vector field, and how do you test forit 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 inspace.WRITING ABOUT CONCEPTS94. (a) Sketch several representative vectors in the vector fieldgiven by(b) Sketch several representative vectors in the vector fieldgiven by(c) Explain any similarities or differences in the vectorfields y G x, y .F x, yG x, yxiyjx 2 y 2.F x, yxiyjx 2 y 2.CAPSTONEIn Exercises 53–56, use a computer algebra system to find therot F for the vector field.53.54.55.56.In Exercises 57–62, determine whether the vector field F isconservative. 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 atthe given point.67.68.69.70.In Exercises 75 and 76, find rot75. 76.In Exercises 77 and 78, find rot77.78.In Exercises 79 and 80, find79. 80.In Exercises 81 and 82, find81.82.In Exercises 83–90, prove the property for vector fields F and Gand scalar function(Assume that the required partialderivatives are continuous.)83.84.85.86.87.88.89.90. (Theorem 15.3)In Exercises 91–93, letand let91. Show that 92. Show that93. Show thatTrue or False?In Exercises 95–98, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.95. If then as96. If and is on the positive -axis, thenthe 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 butnot conservative.Frot F 0,Fffyyx, yF x, y 4xi y 2 jx, y → 0, 0 .F x, y → 0F x, y 4xi y 2 j,f n nf n 2 F.1fFf 3.ln fFf 2.fx, y, z F x, y, z .F x, y, zxi yj zk,div rot F 0div f F f div F f Ff F f F f Ff F Fdiv F G rot F G F rot Gdiv F G div F div Grot f f 0rot F G rot F rot Gf.F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkdiv rot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykdiv F G F G .F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkrot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykF G F G .3, 2, 1F x, y, z ln xyz i j k3, 0, 0F x, y, z e x sen yi e x cos yj z 2 k2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 1F x, y, z xyzi xyj zkPuntoCampo vectorialF x, y, z ln x 2 y 2 i xyj ln y 2 z 2 kF x, y, z senxi cos yj z 2 kF x, y xe x i ye y jF x, y x 2 i 2y 2 jF x, y, zxx 2 y 2 i yx 2 y 2 j kF x, y, zzy ixzy 2 jxy kF x, y, z ye z i ze x j xe y kF x, y, z sen zi sen xj sen ykF x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 kF x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zkF x, y, z x 2 y 2 z 2 i j kF x, y, z sen x y i sen y z j sen z x kF x, y, zyzy z i xzx z j xyxy kF x, y, z arctan x y i ln x2 y 2 j k1068 Chapter 15 Vector AnalysisCAS71. Define a vector field in the plane and in space. Give somephysical examples of vector fields.72. What is a conservative vector field, and how do you test forit 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 inspace.WRITING ABOUT CONCEPTS94. (a) Sketch several representative vectors in the vector fieldgiven by(b) Sketch several representative vectors in the vector fieldgiven by(c) Explain any similarities or differences in the vectorfields y G x, y .F x, yG x, yxiyjx 2 y 2.F x, yxiyjx 2 y 2.CAPSTONEIn Exercises 53–56, use a computer algebra system to find therot F for the vector field.53.54.55.56.In Exercises 57–62, determine whether the vector field F isconservative. 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 atthe given point.67.68.69.70.In Exercises 75 and 76, find rot75. 76.In Exercises 77 and 78, find rot77.78.In Exercises 79 and 80, find79. 80.In Exercises 81 and 82, find81.82.In Exercises 83–90, prove the property for vector fields F and Gand scalar function(Assume that the required partialderivatives are continuous.)83.84.85.86.87.88.89.90. (Theorem 15.3)In Exercises 91–93, letand let91. Show that 92. Show that93. Show thatTrue or False?In Exercises 95–98, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.95. If then as96. If and is on the positive -axis, thenthe 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 butnot conservative.Frot F 0,Fffyyx, yF x, y 4xi y 2 jx, y → 0, 0 .F x, y → 0F x, y 4xi y 2 j,f n nf n 2 F.1fFf 3.ln fFf 2.fx, y, z F x, y, z .F x, y, zxi yj zk,div rot F 0div f F f div F f Ff F f F f Ff F Fdiv F G rot F G F rot Gdiv F G div F div Grot f f 0rot F G rot F rot Gf.F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkdiv rot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykdiv F G F G .F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkrot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykF G F G .3, 2, 1F x, y, z ln xyz i j k3, 0, 0F x, y, z e x sen yi e x cos yj z 2 k2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 1F x, y, z xyzi xyj zkPuntoCampo vectorialF x, y, z ln x 2 y 2 i xyj ln y 2 z 2 kF x, y, z senxi cos yj z 2 kF x, y xe x i ye y jF x, y x 2 i 2y 2 jF x, y, zxx 2 y 2 i yx 2 y 2 j kF x, y, zzy ixzy 2 jxy kF x, y, z ye z i ze x j xe y kF x, y, z sen zi sen xj sen ykF x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 kF x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zkF x, y, z x 2 y 2 z 2 i j kF x, y, z sen x y i sen y z j sen z x kF x, y, zyzy z i xzx z j xyxy kF x, y, z arctan x y i ln x2 y 2 j k1068 Chapter 15 Vector AnalysisCAS71. Define a vector field in the plane and in space. Give somephysical examples of vector fields.72. What is a conservative vector field, and how do you test forit 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 inspace.WRITING ABOUT CONCEPTS94. (a) Sketch several representative vectors in the vector fieldgiven by(b) Sketch several representative vectors in the vector fieldgiven by(c) Explain any similarities or differences in the vectorfields y G x, y .F x, yG x, yxiyjx 2 y 2.F x, yxiyjx 2 y 2.CAPSTONEIn Exercises 53–56, use a computer algebra system to find therot F for the vector field.53.54.55.56.In Exercises 57–62, determine whether the vector field F isconservative. 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 atthe given point.67.68.69.70.In Exercises 75 and 76, find rot75. 76.In Exercises 77 and 78, find rot77.78.In Exercises 79 and 80, find79. 80.In Exercises 81 and 82, find81.82.In Exercises 83–90, prove the property for vector fields F and Gand scalar function(Assume that the required partialderivatives are continuous.)83.84.85.86.87.88.89.90. (Theorem 15.3)In Exercises 91–93, letand let91. Show that 92. Show that93. Show thatTrue or False?In Exercises 95–98, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.95. If then as96. If and is on the positive -axis, thenthe 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 butnot conservative.Frot F 0,Fffyyx, yF x, y 4xi y 2 jx, y → 0, 0 .F x, y → 0F x, y 4xi y 2 j,f n nf n 2 F.1fFf 3.ln fFf 2.fx, y, z F x, y, z .F x, y, zxi yj zk,div rot F 0div f F f div F f Ff F f F f Ff F Fdiv F G rot F G F rot Gdiv F G div F div Grot f f 0rot F G rot F rot Gf.F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkdiv rot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykdiv F G F G .F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkrot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykF G F G .3, 2, 1F x, y, z ln xyz i j k3, 0, 0F x, y, z e x sen yi e x cos yj z 2 k2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 1F x, y, z xyzi xyj zkPuntoCampo vectorialF x, y, z ln x 2 y 2 i xyj ln y 2 z 2 kF x, y, z senxi cos yj z 2 kF x, y xe x i ye y jF x, y x 2 i 2y 2 jF x, y, zxx 2 y 2 i yx 2 y 2 j kF x, y, zzy ixzy 2 jxy kF x, y, z ye z i ze x j xe y kF x, y, z sen zi sen xj sen ykF x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 kF x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zkF x, y, z x 2 y 2 z 2 i j kF x, y, z sen x y i sen y z j sen z x kF x, y, zyzy z i xzx z j xyxy kF x, y, z arctan x y i ln x2 y 2 j k1068 Chapter 15 Vector AnalysisCAS71. Define a vector field in the plane and in space. Give somephysical examples of vector fields.72. What is a conservative vector field, and how do you test forit 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 inspace.WRITING ABOUT CONCEPTS94. (a) Sketch several representative vectors in the vector fieldgiven by(b) Sketch several representative vectors in the vector fieldgiven by(c) Explain any similarities or differences in the vectorfields y G x, y .F x, yG x, yxiyjx 2 y 2.F x, yxiyjx 2 y 2.CAPSTONEPara discusión94. a) Dibujar varios vectores representativos en el campo vectorialdado porb) Dibujar varios vectores representativos en el campo vectorialdado porc) Explicar cualquier similitud o diferencia en los camposvectorialesIn Exercises 53–56, use a computer algebra system to find therot F for the vector field.53.54.55.56.In Exercises 57–62, determine whether the vector field F isconservative. 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 atthe given point.67.68.69.70.In Exercises 75 and 76, find rot75. 76.In Exercises 77 and 78, find rot77.78.In Exercises 79 and 80, find79. 80.In Exercises 81 and 82, find81.82.In Exercises 83–90, prove the property for vector fields F and Gand scalar function(Assume that the required partialderivatives are continuous.)83.84.85.86.87.88.89.90. (Theorem 15.3)In Exercises 91–93, letand let91. Show that 92. Show that93. Show thatTrue or False?In Exercises 95–98, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.95. If then as96. If and is on the positive -axis, thenthe 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 butnot conservative.Frot F 0,Fffyyx, yF x, y 4xi y 2 jx, y → 0, 0 .F x, y → 0F x, y 4xi y 2 j,f n nf n 2 F.1fFf 3.ln fFf 2.fx, y, z F x, y, z .F x, y, zxi yj zk,div rot F 0div f F f div F f Ff F f F f Ff F Fdiv F G rot F G F rot Gdiv F G div F div Grot f f 0rot F G rot F rot Gf.F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkdiv rot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykdiv F G F G .F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkrot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykF G F G .3, 2, 1F x, y, z ln xyz i j k3, 0, 0F x, y, z e x sen yi e x cos yj z 2 k2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 1F x, y, z xyzi xyj zkPuntoCampo vectorialF x, y, z ln x 2 y 2 i xyj ln y 2 z 2 kF x, y, z senxi cos yj z 2 kF x, y xe x i ye y jF x, y x 2 i 2y 2 jF x, y, zxx 2 y 2 i yx 2 y 2 j kF x, y, zzy ixzy 2 jxy kF x, y, z ye z i ze x j xe y kF x, y, z sen zi sen xj sen ykF x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 kF x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zkF x, y, z x 2 y 2 z 2 i j kF x, y, z sen x y i sen y z j sen z x kF x, y, zyzy z i xzx z j xyxy kF x, y, z arctan x y i ln x2 y 2 j k1068 Chapter 15 Vector AnalysisCAS71. Define a vector field in the plane and in space. Give somephysical examples of vector fields.72. What is a conservative vector field, and how do you test forit 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 inspace.WRITING ABOUT CONCEPTS94. (a) Sketch several representative vectors in the vector fieldgiven by(b) Sketch several representative vectors in the vector fieldgiven by(c) Explain any similarities or differences in the vectorfields y G x, y .F x, yG x, yxiyjx 2 y 2.F x, yxiyjx 2 y 2.CAPSTONEIn Exercises 53–56, use a computer algebra system to find therot F for the vector field.53.54.55.56.In Exercises 57–62, determine whether the vector field F isconservative. 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 atthe given point.67.68.69.70.In Exercises 75 and 76, find rot75. 76.In Exercises 77 and 78, find rot77.78.In Exercises 79 and 80, find79. 80.In Exercises 81 and 82, find81.82.In Exercises 83–90, prove the property for vector fields F and Gand scalar function(Assume that the required partialderivatives are continuous.)83.84.85.86.87.88.89.90. (Theorem 15.3)In Exercises 91–93, letand let91. Show that 92. Show that93. Show thatTrue or False?In Exercises 95–98, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.95. If then as96. If and is on the positive -axis, thenthe 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 butnot conservative.Frot F 0,Fffyyx, yF x, y 4xi y 2 jx, y → 0, 0 .F x, y → 0F x, y 4xi y 2 j,f n nf n 2 F.1fFf 3.ln fFf 2.fx, y, z F x, y, z .F x, y, zxi yj zk,div rot F 0div f F f div F f Ff F f F f Ff F Fdiv F G rot F G F rot Gdiv F G div F div Grot f f 0rot F G rot F rot Gf.F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkdiv rot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykdiv F G F G .F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkrot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykF G F G .3, 2, 1F x, y, z ln xyz i j k3, 0, 0F x, y, z e x sen yi e x cos yj z 2 k2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 1F x, y, z xyzi xyj zkPuntoCampo vectorialF x, y, z ln x 2 y 2 i xyj ln y 2 z 2 kF x, y, z senxi cos yj z 2 kF x, y xe x i ye y jF x, y x 2 i 2y 2 jF x, y, zxx 2 y 2 i yx 2 y 2 j kF x, y, zzy ixzy 2 jxy kF x, y, z ye z i ze x j xe y kF x, y, z sen zi sen xj sen ykF x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 kF x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zkF x, y, z x 2 y 2 z 2 i j kF x, y, z sen x y i sen y z j sen z x kF x, y, zyzy z i xzx z j xyxy kF x, y, z arctan x y i ln x2 y 2 j k1068 Chapter 15 Vector AnalysisCAS71. Define a vector field in the plane and in space. Give somephysical examples of vector fields.72. What is a conservative vector field, and how do you test forit 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 inspace.WRITING ABOUT CONCEPTS94. (a) Sketch several representative vectors in the vector fieldgiven by(b) Sketch several representative vectors in the vector fieldgiven by(c) Explain any similarities or differences in the vectorfields y G x, y .F x, yG x, yxiyjx 2 y 2.F x, yxiyjx 2 y 2.CAPSTONEIn Exercises 53–56, use a computer algebra system to find therot F for the vector field.53.54.55.56.In Exercises 57–62, determine whether the vector field F isconservative. 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 atthe given point.67.68.69.70.In Exercises 75 and 76, find rot75. 76.In Exercises 77 and 78, find rot77.78.In Exercises 79 and 80, find79. 80.In Exercises 81 and 82, find81.82.In Exercises 83–90, prove the property for vector fields F and Gand scalar function(Assume that the required partialderivatives are continuous.)83.84.85.86.87.88.89.90. (Theorem 15.3)In Exercises 91–93, letand let91. Show that 92. Show that93. Show thatTrue or False?In Exercises 95–98, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.95. If then as96. If and is on the positive -axis, thenthe 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 butnot conservative.Frot F 0,Fffyyx, yF x, y 4xi y 2 jx, y → 0, 0 .F x, y → 0F x, y 4xi y 2 j,f n nf n 2 F.1fFf 3.ln fFf 2.fx, y, z F x, y, z .F x, y, zxi yj zk,div rot F 0div f F f div F f Ff F f F f Ff F Fdiv F G rot F G F rot Gdiv F G div F div Grot f f 0rot F G rot F rot Gf.F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkdiv rot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykdiv F G F G .F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkrot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykF G F G .3, 2, 1F x, y, z ln xyz i j k3, 0, 0F x, y, z e x sen yi e x cos yj z 2 k2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 1F x, y, z xyzi xyj zkPuntoCampo vectorialF x, y, z ln x 2 y 2 i xyj ln y 2 z 2 kF x, y, z senxi cos yj z 2 kF x, y xe x i ye y jF x, y x 2 i 2y 2 jF x, y, zxx 2 y 2 i yx 2 y 2 j kF x, y, zzy ixzy 2 jxy kF x, y, z ye z i ze x j xe y kF x, y, z sen zi sen xj sen ykF x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 kF x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zkF x, y, z x 2 y 2 z 2 i j kF x, y, z sen x y i sen y z j sen z x kF x, y, zyzy z i xzx z j xyxy kF x, y, z arctan x y i ln x2 y 2 j k1068 Chapter 15 Vector AnalysisCAS71. Define a vector field in the plane and in space. Give somephysical examples of vector fields.72. What is a conservative vector field, and how do you test forit 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 inspace.WRITING ABOUT CONCEPTS94. (a) Sketch several representative vectors in the vector fieldgiven by(b) Sketch several representative vectors in the vector fieldgiven by(c) Explain any similarities or differences in the vectorfields y G x, y .F x, yG x, yxiyjx 2 y 2.F x, yxiyjx 2 y 2.CAPSTONEIn Exercises 53–56, use a computer algebra system to find therot F for the vector field.53.54.55.56.In Exercises 57–62, determine whether the vector field F isconservative. 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 atthe given point.67.68.69.70.In Exercises 75 and 76, find rot75. 76.In Exercises 77 and 78, find rot77.78.In Exercises 79 and 80, find79. 80.In Exercises 81 and 82, find81.82.In Exercises 83–90, prove the property for vector fields F and Gand scalar function(Assume that the required partialderivatives are continuous.)83.84.85.86.87.88.89.90. (Theorem 15.3)In Exercises 91–93, letand let91. Show that 92. Show that93. Show thatTrue or False?In Exercises 95–98, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.95. If then as96. If and is on the positive -axis, thenthe 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 butnot conservative.Frot F 0,Fffyyx, yF x, y 4xi y 2 jx, y → 0, 0 .F x, y → 0F x, y 4xi y 2 j,f n nf n 2 F.1fFf 3.ln fFf 2.fx, y, z F x, y, z .F x, y, zxi yj zk,div rot F 0div f F f div F f Ff F f F f Ff F Fdiv F G rot F G F rot Gdiv F G div F div Grot f f 0rot F G rot F rot Gf.F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkdiv rot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykdiv F G F G .F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkrot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykF G F G .3, 2, 1F x, y, z ln xyz i j k3, 0, 0F x, y, z e x sen yi e x cos yj z 2 k2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 1F x, y, z xyzi xyj zkPuntoCampo vectorialF x, y, z ln x 2 y 2 i xyj ln y 2 z 2 kF x, y, z senxi cos yj z 2 kF x, y xe x i ye y jF x, y x 2 i 2y 2 jF x, y, zxx 2 y 2 i yx 2 y 2 j kF x, y, zzy ixzy 2 jxy kF x, y, z ye z i ze x j xe y kF x, y, z sen zi sen xj sen ykF x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 kF x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zkF x, y, z x 2 y 2 z 2 i j kF x, y, z sen x y i sen y z j sen z x kF x, y, zyzy z i xzx z j xyxy kF x, y, z arctan x y i ln x2 y 2 j k1068 Chapter 15 Vector AnalysisCAS71. Define a vector field in the plane and in space. Give somephysical examples of vector fields.72. What is a conservative vector field, and how do you test forit 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 inspace.WRITING ABOUT CONCEPTS94. (a) Sketch several representative vectors in the vector fieldgiven by(b) Sketch several representative vectors in the vector fieldgiven by(c) Explain any similarities or differences in the vectorfields y G x, y .F x, yG x, yxiyjx 2 y 2.F x, yxiyjx 2 y 2.CAPSTONEIn Exercises 53–56, use a computer algebra system to find therot F for the vector field.53.54.55.56.In Exercises 57–62, determine whether the vector field F isconservative. 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 atthe given point.67.68.69.70.In Exercises 75 and 76, find rot75. 76.In Exercises 77 and 78, find rot77.78.In Exercises 79 and 80, find79. 80.In Exercises 81 and 82, find81.82.In Exercises 83–90, prove the property for vector fields F and Gand scalar function(Assume that the required partialderivatives are continuous.)83.84.85.86.87.88.89.90. (Theorem 15.3)In Exercises 91–93, letand let91. Show that 92. Show that93. Show thatTrue or False?In Exercises 95–98, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.95. If then as96. If and is on the positive -axis, thenthe 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 butnot conservative.Frot F 0,Fffyyx, yF x, y 4xi y 2 jx, y → 0, 0 .F x, y → 0F x, y 4xi y 2 j,f n nf n 2 F.1fFf 3.ln fFf 2.fx, y, z F x, y, z .F x, y, zxi yj zk,div rot F 0div f F f div F f Ff F f F f Ff F Fdiv F G rot F G F rot Gdiv F G div F div Grot f f 0rot F G rot F rot Gf.F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkdiv rot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykdiv F G F G .F x, y, z x 2 zi 2xzj yzkF x, y, z xyzi yj zkrot F F .G x, y, z x 2 i yj z 2 kG x, y, z xi yj zkF x, y, z xi zkF x, y, z i 3xj 2ykF G F G .3, 2, 1F x, y, z ln xyz i j k3, 0, 0F x, y, z e x sen yi e x cos yj z 2 k2, 1, 3F x, y, z x 2 zi 2xzj yzk2, 1, 1F x, y, z xyzi xyj zkPuntoCampo vectorialF x, y, z ln x 2 y 2 i xyj ln y 2 z 2 kF x, y, z senxi cos yj z 2 kF x, y xe x i ye y jF x, y x 2 i 2y 2 jF x, y, zxx 2 y 2 i yx 2 y 2 j kF x, y, zzy ixzy 2 jxy kF x, y, z ye z i ze x j xe y kF x, y, z sen zi sen xj sen ykF x, y, z y 2 z 3 i 2xyz 3 j 3xy 2 z 2 kF x, y, z xy 2 z 2 i x 2 yz 2 j x 2 y 2 zkF x, y, z x 2 y 2 z 2 i j kF x, y, z sen x y i sen y z j sen z x kF x, y, zyzy z i xzx z j xyxy kF x, y, z arctan x y i ln x2 y 2 j k1068 Chapter 15 Vector AnalysisCAS71. Define a vector field in the plane and in space. Give somephysical examples of vector fields.72. What is a conservative vector field, and how do you test forit 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 inspace.WRITING ABOUT CONCEPTS94. (a) Sketch several representative vectors in the vector fieldgiven by(b) Sketch several representative vectors in the vector fieldgiven by(c) Explain any similarities or differences in the vectorfields y G x, y .F x, yG x, yxiyjx 2 y 2.F x, yxiyjx 2 y 2.CAPSTONE
SECCIÓN 15.2 Integrales de línea 106915.2 Integrales de línea■ Comprender y utilizar el concepto de curva suave a trozos.■ Expresar y evaluar una integral de línea.■ Expresar y evaluar una integral de línea de un campo vectorial.■ Expresar y calcular una integral de línea en forma diferencial.xThe Granger CollectionJOSIAH WILLARD GIBBS (1839-1903)Muchos físicos y matemáticos han contribuidoa la teoría y a las aplicaciones descritasen este capítulo, Newton, Gauss, Laplace,Hamilton y Maxwell, entre otros. Sinembargo, el uso del análisis vectorial paradescribir estos resultados se atribuye principalmenteal físico matemático estadounidenseJosiah Willard Gibbs.1Figura 15.71(0, 0, 0)zC 1C 2(1, 2, 0)C = C 1+ C 2+ C 3(1, 2, 1)C 3(0, 2, 0)yCurvas suaves a trozos (o por partes)Una propiedad clásica de los campos gravitatorios (o gravitacionales) es que, sujeto a ciertasrestricciones físicas, el trabajo realizado por la gravedad sobre un objeto que se mueveentre dos puntos en el campo es independiente de la trayectoria que siga el objeto. Una delas restricciones es que la trayectoria debe ser una curva suave a trozos (o por partes).Recuérdese que una curva plana C dada porrt xti ytj,es suave sidxdtyson continuas en [a, b] y no simultáneamente 0 en (a, b). Similarmente, una curva C en elespacio dada porrt xti ytj ztk,es suave sidxdt ,yson continuas en [a, b] y no simultáneamente 0 en (a, b). Una curva C es suave a trozos(o por partes) si el intervalo [a, b] puede dividirse en un número finito de subintervalos, encada uno de los cuales C es suave.EJEMPLO 1dydt ,dydtHallar una parametrización suave a trozosHallar una parametrización suave a trozos de la gráfica C que se muestra en la figura 15.7.Solución Como C consta de tres segmentos de recta C 1, C 2y C 3, se puede construir unaparametrización suave de cada segmento y unirlas haciendo que el último valor de t en C icoincida con el primer valor de t en C i1 , como se muestra a continuación.C 1 : xt 0,C 2 : xt t 1,C 3 : xt 1,dzdtPor tanto, C está dada pora ≤ t ≤ brt 2tj,t 1i 2j,i 2j t 2k,yt 2t,yt 2,yt 2,a ≤ t ≤ b0 ≤ t ≤ 11 ≤ t ≤ 2 .2 ≤ t ≤ 3zt 0,zt 0,zt t 2,Como C 1, C 2y C 3son suaves, se sigue que C es suave a trozos.0 ≤ t ≤ 11 ≤ t ≤ 22 ≤ t ≤ 3Recuérdese que la parametrización de una curva induce una orientación de la curva.Así, en el ejemplo 1, la curva está orientada de manera que la dirección positiva va desde(0, 0, 0), siguiendo la curva, hasta (1, 2, 1). Trátese de obtener una parametrización queinduzca la orientación opuesta.
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Calculo 2 De dos variables_9na Edición - Ron Larson

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