MATH
Math2015_09_T5_Solns

# Math2015_09_T5_Solns - (60 Math 2015 Applied Multivariate...

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Unformatted text preview: (60 Math 2015 Applied Multivariate and Vector Calculus: Test5 Friday November 27, 2009 10:30am to 11:20am Name: SOLO”! to N g Student Number: Instructions: Complete all 4 of the following problems in the space provided. Notes and calculators are not permitted. All cell phones and pagers are to be turned off. 1. (a) Show that the vector ﬁeld g WM“ lit; ~ F(x, y) = (3x2 — 3y2 — 2x)i + (—Gwy + 2y)j is conservative and then ﬁnd a potential function f (x, y) such that F = V f . (b) Use your result in part (a) to compute the line integral /CF(\$,y) -dr Where C is any curve connecting the points (0, 0) and (1,3). Le} m: ”\$23 Jag-”Ill. N‘ = 4013 + 2}) 9M —‘ 3 " (9 DE : “(a I ,\ 3‘3 “A ’DBL ‘3 CD 2. Using spherical coordinates, evaluate the triple integral /// x2+y2+22dV D Where D is the region that lies above the xy-plane, below the cone 4b = 7r/ 4, and between the spherespzaandp=bfor0<a<b. S me‘k) 4' A Lei! l: EMSii/I47 ta; 89499.4(} (D a : §w5<l> D a “/4 o b 3 “71/ :- Z‘IT ‘ Sq l) 31w \$m<l>cl~l>cl§ -— I : _. b . “It 211‘ Sq 8'5 [w\$s¥1")+ d? - er b 2. __ A z: 2 5 T2 at g E 2184] -llb‘l~ﬂ4§ -'® 5!: IF Ab 0k 0: SEN M \$1, oofr { Q; l H d» se @ 3. Evaluate the line integral 5 Win k6 . /F-dr C Where F(m,y, z) = (y +1)i— (a: + 4)j + (a: + y + 7r)k. Do this when (a) C is the helical curve a: = cost, y = sint7 z = t/Tl', for 0 g t 3 7r. Repeat the calculation of the integral when G is the straight line between the points (1,0,0) and (—1,0, 1). Comparing your results from parts (a) and (b), What can you conclude about F? A [00 Ht) : ng‘bﬁmt ,‘t/TL‘] Dét éTr ‘ lull” ‘5 ['Sln‘t, cost, UTE} if. E“ AV 3 g . FlH/7 Al?“ I Evil/H, —- wst’ilr, (/0‘5bl3in‘l7 +TVF}\:SM+, (495%, l?‘\ Al? . : - Sint Lewd“) —- cosh \Qos’c M>+Tlc [met Pant +1T> :: -—sm7’{;—we}t + l "=%M’V --‘}oosl7 +i[w\$h +sinjc) l"[ 2 l%~I)Sml7 Jr (i-Ll-) wSlT Ti TT 8.4%,&: 3 SD LlT—qswg +[lE—4vwaJc (PC "" © : —— (ﬂit—170095 \: ~l- l‘%_\—~'+>S\Cn’c \r = we) {no ° (by) Who“ C \b a ‘QTMQMk Qm, 1"}me C‘s0,o3 mot Lv-l,o,1) W7: "50+ JCS U‘nM :1: k,\‘0,|3> (1,0,0) :6- 2,0,1) 5) PW): (Lo,o)+ {:LJAO”) : (4‘—LbIOI+/7 bété‘ =3 ‘6‘ Hi): (4'2, 0‘ I > A "‘ A Fm = FF'lth; = \IJb—s, -\-Lt+ﬂ~[—2,o.u]c\t : \:Z +£1~£h+w>3dﬁ~ = (Ir-'1 *Z-mzk \.¢ 3 E'd‘? = 3‘ hpqdvcjcﬁ» —-® 0 9mm. (mull/‘5 do (:0 W [b) are, dgg‘QUWhQ CD 4. Consider a region D of the xy plane whose boundary C is a piecewise smooth simple closed curve. Further, suppose the functions qS and d) have continuous ﬁrst and second order partial derivatives on D. Use Green’s theorem to prove the following important identity g WWIH‘l% . £¢V¢-dr=—j€¢v¢‘dr- Fxrsl“ WscMr LHs- (4} M z 521,» A} = 4, D1? Tm 53 Ctrwn‘s “ll/124mm éc Zl’vxl'dﬂé‘ : :M'Bcﬁ'aj’1L4DR—1iz’9j23’ 2:1 D lM/ 2‘3 92,3") b3 92:, +9397LECM ‘ ll (33; 223 - at w W Smut gear was D 3 3‘3 3‘0 CLI‘L W‘lu/LOOJ‘) 1——-(i) 3mmc‘ﬂMLD, WeCdﬂrn/lé) (ll/UL (2H5, Lb‘l’ M: [Ya—5:! "l= ‘fal‘l’ Aﬁam, Smd. Sam Wkaﬁg, GAL MAIL/10003 22.1 § 3% . hwb 93311/ 1 CD ...
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