Math2015_Worked_Examples_II

Math2015_Worked_Examples_II - Evafium-LL M 091m whys/Q SC...

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Unformatted text preview: Evafium-LL M 091m. whys/Q SC DWViBM JraLch/S *xficfi (Mm (1 sh M 311% gaswuw§ f‘nm “.235 \o (Zmd) Fwfi- $0" (M : >c,+ ‘3 + % N 3 >L P I. ’- \AE AN!) (5: 13M figR/ILQQ d? 5 Am}: 1' 413§+&%/\2 We, was/Mr N Mfii \I : VFW“ (1—233 +L'O»-\)}A4 A /\ = v— 3* K H‘VALL F tad—"b: : (\sl.‘3+ JC(\)-\)"‘) l‘hmu 3L: [+471 ‘33 2/17: 17?: [wt Md” Ham] :\(\+<V)+Lz-m+u~—t>]i‘ + U-H’g * (WOUJCBIQ W‘kwh wwuo Mm ‘h/u. ade’i $Ub5h+vm9 Wx Zfium’kbm kt£§ . “o g 5 EW (159%th db- @- .— HT 1 _> {R [me mmzwr + ‘H‘lld‘k’ 11? 5 Us; + 391?, 5)/ SLBLTF +Z‘h’r law: {iv-M gem) mam Sgla WEerULfiWgTW‘M @1me MEG gaick [Far MCVM m ; ggg 36,9”)ny “W" 3 ‘33 V a \ aux TIL LLC) v 4AM? (waffl {I 6/7 GM: 2 ZTTK c: g1‘11, [‘SSMctJ A$AF ' 4.Prvtht a M a V-(FxG)=(VxF)-G—F~(VXG) (:0 WWW» for the differentiable vector fields F(x, y, z) = F1(x, y, z)i + F2(x, y, z)j + F303, 3/, z)k and G($,y, 2) = G1($,y, Z)i + G206, y, 2H + G3($,y, z)k- - A 2 g {2 ‘- A FXCT: E C; T‘s\= Ill/\(FLGBN 3QL)“B(¥|G‘3’E$&IB CT; A .. Gr. G3 + M ( F|(;.L“t1G*I> ”—0 my LH§E Q'fixt) 2 lLFLCy—5AV3GW)~PZ_(V\CTB-‘\:3cfi\) 3x 33 l ¥6PL"' + ML. at?» “(D 2 " A . M = 2:11;. .. {Es-1R 4; RAE 62L 33 Wt _ m3 "0'12 Eu 3% F, FL F3 A ‘ ‘ 2a , 73f. ft”; 'ba> "“' (D The following is blank for additional workspace if required: -L . .3 ‘WVA‘ C‘AA/wsma 1: Md). [1 % \UIO (Vx§)_: (26%; %§;>-¥L(B:3_ 1:.) .—R~F5 (%1_’57:‘> <> 2 - 01‘an J, ngbfi H: 'th3_ glen ”Dub (Bu, "but, ' 'DDL ‘\ C‘r 'ZFS " (7:3, (3‘: - F 9G3 + P3 2&1 95 ‘35 ‘35 ‘33 ”I C‘q (b53- "\‘ [T7, bits -{- Pa bin —- FLE‘ . m; m: “oi: 7H; : Pit—EC" 3 4’ 9 (sz15) + 2 LEE) -' 3... & F, LNQ [‘33P 2 (Fl 6‘71 23C 3 ) WI J +? > ‘—‘- %( ELECECTQ T )2,“th ‘ CnF§> + 9 ' L“ (D 52-: (R G1 " FlCTc / 3. Evaluate the line integral (a W1M$$ where F(:z:,y, z) = 3:132i + (sz — y)j + zk. Do this when (a) C is the straight line from (0,0,0) to (1,1,1), and (b) when C is the line from (0,0,0) to (1,0,0) to (1,1,0) to (1,1,1). Comparing your results from parts (a) and (b), what can you conclude about F? (m Ft¥31f12+3+123 1: 4L0“) ~__.® M g M: 3' Emmy: a a t 3; (39+ wake» wjdt .2 (9‘1;th a itfi: = 1:. JCD Cb) Bfudl— u? C wk M, 3 PM? C‘,C1,Csm Famowél UL) C. ‘Vfom L0.o,07+v ((1,0,0): mm: H hated) L10 C7, shrew; (hop) 4—,) limo) 1:00: 3+ 19/3 £619") UM) C3 9mm 0,1,0) {>0 (mm) gm 2153ch ttle fl mfilim RML M WWLL W '30 TM Rm) @uld g (M by. WHVWW’O k PMKJJ. MIC/Mafi- A Cum/g (1 mm ?(>—4«3fl\= wamfvt (3:7) 33 0L: avast La: lam/ml? 33$ (-75 1177, "‘ .3 “FM/d ~41 W 2 R F0 A»? :2. g Fth-t))\6 [t7] . P (f) $7 “0 7— La “St +1391“: 3(‘a8m'tj + (Qm$\~~ {>$W\{') {1960563 3 ° Q47, wg‘v‘QWAT *’ 61569847177 ~wgzb> ‘- LZ (,08'59/(7 : ,, (my mam ~qb [svflbAws la ‘ ./ Ferqugitf\';'W\) ,1 L¢1+V>gm at + abwsuc l [\(11 “L Lulk’b > S $MA-Z’E DU: 1 b T L +\ ab 3 ‘ mgzk db _ TIL T] i Lamb >LLOsZ/t10 Jr Llab Lgmltl 1/ A]. o 1. Show that the vector field C? W‘Wfi . 'V F(m, y, z) = (6:1:y3 + 222)i + (9x2y2)j + (4mz + 1)k is conservative and then find a potential function f-(m, y, 2) such that F = V f . UR. MDWdX : [91. “-5 + 1%1’ NLMHATL) ‘; v,q1,zhsl Ru|p5fz): 41%‘4’l l A _. d a '4': Vs mswum‘wx. E VXF :0 ic— ’31“ ~__—, M51 1" EN : “3x 1 V5 as M ‘3 ‘31“ 1: 4-9: '23 : 4?: —————® ‘boc ’0» ‘9.“ = 0 3.? c o 3% ‘33 . C f w WSUVW<\\IUL__ New VOL “twat QAWQ at HN‘GWA -= V§LMM> Mb" MW-tz—Tfiafiw“ 333:5} 5“ —‘ . b3 : aka-3 7" + 9—L "‘ NCNHA 47> M 7’3 =§ ES“ = N—qlzgl = r0 ® ”‘3 M WW“) Wad“ C LA, 0M3 a Emmo’fwh :b % CL‘As%»: qu=> G) NOW 1;: 4x,% 4- %CLW‘%» .4 ADO: ‘\‘ D‘G’) =- WMWJ 5a} + Sm 2 4x2 +1 =5 “D‘L*3= ‘ bu,» -: % +oc ' (Tot vow-WW ‘ Au» ¥DM£3= 35516 +u£+ "1 4— 0<' f® Lawn, UL \~\r~ gfifis a/MQ wag N" ’M. game, ‘DDH/{Eb- Far chAn' Cums 938— M “\o fink) 0m orQQ/Whtm. "Hm WL film W COMM (Lockwm (Quay ohm/t W cm 9367‘ *WL ‘ ...
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