ISM_T11_C16_A

ISM_T11_C16_A - CHAPTER 16 INTEGRATION IN VECTOR FIELDS...

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CHAPTER 16 INTEGRATION IN VECTOR FIELDS 16.1 LINE INTEGRALS 1. t ( t) x t and y 1 t y 1 x (c) ri j œ±" ² Ê œ œ²Ê œ²Ê 2. t x 1, y 1, and z t (e) rij 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) j œ±Ê œ œ Ê ± œ Ê ## 4. x t, y 0, and z 0 (a) œÊœ œ œÊ 5. t t x t, y t, and z t (d) rijk œ±± Ê œ œ 6. t (2 2t) y t and z 2 2t z 2 2y (b) rj k œ±² Êœ œ² Êœ² Ê 7. t 1 2t y t 1 and z 2t y 1 (f) k œ² ± Ê œ ² œ ² Ê ab z 4 # 8. (2 cos t) x 2 cos t and z 2 sin t x z 4 (h) 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 j i j j œ± ² ŸŸ Ê œ²Ê œ œ œ²Ê±œ±" ²œ dd dt dt rr ¸¸ È f(x y z) ds f(t 1 t 0) dt (1) 2 dt 2 t 2 Êß ß œ ß ² ß œ œ œ '' ' C0 0 11 Š‹ ÈÈ 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 j k i j œ± ² ± ŸŸ Ê œ²Ê œ œ œ² œ ʲ±² dt dt È t (1 t) 1 2 2t 2 f(x y z) ds (2t 2) 2 dt 2 t 2t 2 œ² ² ±² œ ² Ê ßß œ ² œ ² 1 È cd # " ! 11. (t) 2t t (2 2t) , 0 t 1 2 2 4 1 4 3; xy y z j k i j k ± ² Ÿ Ÿ ± ² Ê œ ± ± œ ± ± dt dt È (2t)t t (2 2t) f(x y z) ds 2t t 2 3 dt 3 t t 2t 3 2 ± ² Ê ß ßœ ² ± œ ² ±œ² ± œ 1 ± ‘ˆ #$ # "" # " ! 22 1 3 33 12. (t) (4 cos t) (4 sin t) 3t , 2 t ( 4 sin t) (4 cos t) 3 j k i j k œ±± ² Ÿ Ÿ Ê œ ² ±± 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 ± ± œ ± œ ± œ Ê ß È d dt r # # C2 2 1 1 20t 80 œœ # ±# 1 1 1 13. (t) ( 2 3 ) t( 3 2 ) (1 t) (2 3t) (3 2t) , 0 t 1 3 2 r ijk i j k œ±± ±² ²² œ²±² ±² ŸŸÊ œ ² ²² d dt r 194 1 4 ; xyz( 1t )( 23 t 32 t )66 t f ( x y z ) d s Ê œ ±±œ ±±œ ²± ² ± ² œ² Ê ß ß È È d dt r ' C (6 6t) 14 dt 6 14 t 6 14 3 14 œ ² œ œ ' 0 1 È È ’“ Š ˆ‰ t 2 # " ! " # 14. (t) t t t , 1 t 3 ; j k i j k ŸŸ_Ê œ±± Ê œ œ œ dt dt x y z t t t 3t 3 È È # # # ### # f(x y z) ds 3 dt lim 1 1 ß œ œ ² œ ² ± œ C1 _ È ± È 3 3t t b 1 # _ " " b Ä_
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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 " # # ## ri j i j œ± ŸŸ Ê œ± Ê œ ± ± ² œ± ²œ± œ dd dt dt rr ¸¸ È È È kk since t 0 f(x y z) ds 2t 1 4t dt 4t (5) 5 5 1 ;  Ê ß ß œ ± œ " ± œ ²œ ² '' C0 1 " È ’“ Š ab È # "" " " # $Î# $Î# " ! 66 6 6 C : (t) t , 0 t 1 1; x y z 1 1 t 2 t # # j k k œ±± ŸŸ Ê œ Ê œ ± ² œ ± ² œ ² dt dt È È f(x y z) ds 2 t (1) dt 2t t 2 ; therefore f(x y z) ds Êß ß œ ² œ ² œ ß ß ' C 1 # ± #$ " ! 33 3 5 f(x y z) ds f(x y z) ds 5 œß ß ±ß ß œ ± CC " # 53 6 È # 16. C : (t) t , 0 t 1 1; x y z 0 0 t t " # rk k œŸ Ÿ Ê œ Êœ ± ² œ ± ² œ ² dt dt È È f(x y z) ds t (1) dt ; ß œ ² œ ² œ ² 1 " # " ! " t $ C : (t) t , 0 t 1 1; x y z 0 t 1 t 1 # # rj k j œ ± ŸŸ Ê œÊ œ ± ² œ± ²œ ² dt dt È ÈÈ f(x y z) ds t 1 (1) dt t t 1 ; ß œ ² œ² œ ² œ ² 1 # ˆ‰ ± È 22 3 $Î# " ! " C : (t) t , 0 t 1 1; x y z t 1 1 t $ # j k i œ±± ŸŸÊ œÊ œ ± ²œ± ²œ dt dt È È f(x y z) ds (t)(1) dt f(x y z) ds f ds f ds f ds Ê ß ßœ œ œ Ê ß ± ± œ ² ± ² ± ' ' ' ''' C 0 C CCC 1 $ "#$ t 2 # " !
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This homework help was uploaded on 09/23/2007 for the course MATH 1910 taught by Professor Berman during the Spring '07 term at Cornell.

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ISM_T11_C16_A - CHAPTER 16 INTEGRATION IN VECTOR FIELDS...

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