Math2015_09_T5_Solns - (60 Math 2015 Applied Multivariate...

Info icon This preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

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

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

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

View Full Document Right Arrow Icon
Image of page 6
Image of page 7

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

View Full Document Right Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full 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