ISM_T11_C16_A - CHAPTER 16 INTEGRATION IN VECTOR FIELDS...

Info icon This preview shows pages 1–3. Sign up to view the full content.

CHAPTER 16 INTEGRATION IN VECTOR FIELDS 16.1 LINE INTEGRALS 1. t ( t) x t and y 1 t y 1 x (c) r i j œ " Ê œ œ Ê œ Ê 2. t x 1, y 1, and z t (e) r i j k œ Ê œ œ œ Ê 3. (2 cos t) (2 sin t) x 2 cos t and y 2 sin t x y 4 (g) r i j œ Ê œ œ Ê œ Ê # # 4. t x t, y 0, and z 0 (a) r i œ Ê œ œ œ Ê 5. t t t x t, y t, and z t (d) r i j k œ Ê œ œ œ Ê 6. t (2 2t) y t and z 2 2t z 2 2y (b) r j k œ Ê œ œ Ê œ Ê 7. t 1 2t y t 1 and z 2t y 1 (f) r j k œ Ê œ œ Ê œ Ê a b # # z 4 8. (2 cos t) (2 sin t) x 2 cos t and z 2 sin t x z 4 (h) r i k œ Ê œ œ Ê œ Ê # # 9. (t) t (1 t) , 0 t 1 2 ; x t and y 1 t x y t ( t) 1 r i j i j j œ Ÿ Ÿ Ê œ Ê œ œ œ Ê œ " œ d d dt dt r r ¸ ¸ È f(x y z) ds f(t 1 t 0) dt (1) 2 dt 2 t 2 Ê ß ß œ ß ß œ œ œ ' ' ' C 0 0 1 1 ¸ ¸ Š È È È d dt r " ! 10. (t) t (1 t) , 0 t 1 2; x t, y 1 t, and z 1 x y z 2 r i j k i j œ Ÿ Ÿ Ê œ Ê œ œ œ œ Ê d d dt dt r r ¸ ¸ È t (1 t) 1 2 2t 2 f(x y z) ds (2t 2) 2 dt 2 t 2t 2 œ œ Ê ß ß œ œ œ ' ' C 0 1 È È È c d # " ! 11. (t) 2t t (2 2t) , 0 t 1 2 2 4 1 4 3; xy y z r i j k i j k œ Ÿ Ÿ Ê œ Ê œ œ d d dt dt r r ¸ ¸ È (2t)t t (2 2t) f(x y z) ds 2t t 2 3 dt 3 t t 2t 3 2 œ Ê ß ß œ œ œ œ ' ' C 0 1 a b ˆ # $ # " " # # # " ! 2 2 13 3 3 12. (t) (4 cos t) (4 sin t) 3t , 2 t 2 ( 4 sin t) (4 cos t) 3 r i j k i j k œ Ÿ Ÿ Ê œ 1 1 d dt r 16 sin t 16 cos t 9 5; x y 16 cos t 16 sin t 4 f(x y z) ds (4)(5) dt Ê œ œ œ œ Ê ß ß œ ¸ ¸ È È È d dt r # # # # # # ' ' C 2 2 20t 80 œ œ c d # # 1 1 1 13. (t) ( 2 3 ) t( 3 2 ) (1 t) (2 3t) (3 2t) , 0 t 1 3 2 r i j k i j k i j k i j k œ œ Ÿ Ÿ Ê œ d dt r 1 9 4 14 ; x y z (1 t) (2 3t) (3 2t) 6 6t f(x y z) ds Ê œ œ œ œ Ê ß ß ¸ ¸ È È d dt r ' C (6 6t) 14 dt 6 14 t 6 14 3 14 œ œ œ œ ' 0 1 È È È È Š ˆ ‰ t 2 " ! " # 14. (t) t t t , 1 t 3 ; r i j k i j k œ Ÿ Ÿ _ Ê œ Ê œ œ œ d d dt dt x y z t t t 3t 3 3 3 r r ¸ ¸ È È È È f(x y z) ds 3 dt lim 1 1 Ê ß ß œ œ œ œ ' ' C 1 Š È ˆ È 3 3t t b 1 _ " " b Ä _
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

998 Chapter 16 Integration in Vector Fields 15. C : (t) t t , 0 t 1 2t 1 4t ; x y z t t 0 t t 2t " # # # # r i j i j œ Ÿ Ÿ Ê œ Ê œ œ œ œ d d dt dt r r ¸ ¸ È È È k k since t 0 f(x y z) ds 2t 1 4t dt 4t (5) 5 5 1 ;   Ê ß ß œ œ " œ œ ' ' C 0 1 È Š a b È # " " " " # $Î# $Î# " ! 6 6 6 6 C : (t) t , 0 t 1 1; x y z 1 1 t 2 t # # # # r i j k k œ Ÿ Ÿ Ê œ Ê œ œ œ d d dt dt r r ¸ ¸ È È f(x y z) ds 2 t (1) dt 2t t 2 ; therefore f(x y z) ds Ê ß ß œ œ œ œ ß ß ' ' ' C 0 C 1 a b # $ " " " ! 3 3 3 5 f(x y z) ds f(x y z) ds 5 œ ß ß ß ß œ ' ' C C 5 3 6 È # 16. C : (t) t , 0 t 1 1; x y z 0 0 t t " # # # r k k œ Ÿ Ÿ Ê œ Ê œ œ œ d d dt dt r r ¸ ¸ È È f(x y z) ds t (1) dt ; Ê ß ß œ œ œ ' ' C 0 1 a b # " ! " t 3 3 C : (t) t , 0 t 1 1; x y z 0 t 1 t 1 # # r j k j œ Ÿ Ÿ Ê œ Ê œ œ œ d d dt dt r r ¸ ¸ È È È f(x y z) ds t 1 (1) dt t t 1 ; Ê ß ß œ œ œ œ ' ' C 0 1 ˆ È 2 2 3 3 3 $Î# " ! " C : (t) t , 0 t 1 1; x y z t 1 1 t $ # r i j k i œ Ÿ Ÿ Ê œ Ê œ œ œ d d dt dt r r ¸ ¸ È È f(x y z) ds (t)(1) dt f(x y z) ds f ds f ds f ds Ê ß ß œ œ œ Ê ß ß œ œ ' ' ' ' ' ' C 0 C C C C 1 ˆ t 2 3 3 " !
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern