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by: Donny Graham

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# Multivariable Calculus MTH 234

Donny Graham
MSU
GPA 3.57

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## Popular in Mathematics (M)

This 5 page Class Notes was uploaded by Donny Graham on Saturday September 19, 2015. The Class Notes belongs to MTH 234 at Michigan State University taught by Irina Kadyrova in Fall. Since its upload, it has received 20 views. For similar materials see /class/207303/mth-234-michigan-state-university in Mathematics (M) at Michigan State University.

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Date Created: 09/19/15
Math 234 FS 2011 Review for Exam 4 Problems with Stokes and Divergence Theorems Problem 30 Find the curl for each Effector eld F n g I xquotMMW39MNK WWNW a F 3x2yye 2i6y6cosxzzji zzk A duff VK TWQi V A gm cwle l n he gm WM L f f W 5 dry MM t b F 4xi4yj2k V X 7 4 V L I M a W mm1 k lt1 Math 234 FS 2011 Review for Exam 4 Problems with Stokes and Divergence Theorems Problem 31 a Use the surface integral in Stokes Theorem to calculate the circulation ofthe eld F y2 zzic2 y2jx2 y2k around the curve C the square bounded by the lines x 18 y 18 in xy plane counterclockwise when Viewing I from above 2562 4 aimemzl C yr7 k 2T 0 39 l I wequot a 3 A2 33 W2 4 R01 p and M mrMMZ wee222 V4 WWW W a MM at n w x Elf4 2 Ma ay 2 1 xw iw A exf Z fls t 2 f t X quot651 4 f1 3 Mepr w 2257 t W 7 l I K yexwfe fe fwtfi 1 way mi tuber51 a2 5 L 1 W ie a 5 I A x a W 1 1 53 r 9 C x 06X g s 2 15 t WW W y 53 w g L 539 o xw O bF x2i4xj22k Cis the ellipse 25x2 16y2 39 atrv It 3 A 2 wmx X Z N J effK 4 my L M M W 1 a kg t3 2 Ea w W WM 4 1 w 7 e a 39 3 6 r 7 547 Mutt1 W I 4 EV 39 317 aft e 4 e z 1 a hwWHkIRH AL I P WA 4279 22 52g y j f e g u an M J z Math 234 FS 2011 Review for Exam 4 Problems with Stokes and DiVergence Theorems Problem 32 Use Stokes Theorem to calculate the counterclockwise circulation of the vector eld F around the curve C F 2xy2iy zzj 3z x2k Cis the boundary of the portion of the surface 2x y 2 1 that lies in the rst octant and is oriented by the eld of normal unit vectors pointed away A fj zji from the origin x r a a 2 A x X 511 r m 39L39 39 quot v f 4 E quot7 VM 1 J quot quot I L me A 1 t w W v g 7 j 3 sz39gVilla f S W m 7 f z 392 I f P quotMum f k 1 t 3 N 9 le We r 77 M t fastest I 1y L24Ll L 27 g3quot 42 quot y t N K I g Mg l 3 a Jigw ggt155sz til z u 9 3M e e 7 myW Z XMQO 1 f2 1 A 773 39 M 4 t 2322th quot57 32 0 O Math 234 FS 2011 Review for Exam 4 Problems with Stokes and Divergence Theorems Problem 33 Find the divergence for the following vector elds a F2xy2iy zzj 3z x2k b F 3x2yyey22i5y6cosxzzjxyex2y 22k c F ysinzixsinzjxycoszk x2y2 lxzy2 07 m 24 3zo V 5 quot 301 52 9 VF 63ltj3 0 VE O0 x3024 gm 39 5 J 2 5x 00 v 25 9 HOW M We 2 2 1 z 21 4 gr 2 54 X W 1 z 55 39i quot2lt7 39Zquot 5X3 quotxZwWFJ 2 JW w 2quot 5quotquot ill 12va M W M m 344 33 O Math 234 FS 2011 Review for Exam 4 Problems with Stokes and Divergence Theorems Problem 34 Use the Divergence Theorem to nd the outward ux of F across the boundary S of the region D a F3xy2i3x2yjz3k D bounded by the surface 2 44 x2 y2 and the xyplane Express the triple integral in part a as an iterated spherical integral and evaluate x F a 2 2 lFfwx E fffcwvfolx VF 3lt723x23 25x2 22 7 76239223 2amp0 ffl3CX27 32quot Evy ng x2yz 222 g 5 eaZ wra bnazt 7F 1 2 2222f2 2 a 9 GiijPZMMjOPdgaQ PL 02 o 277 ff 2 y r 3 3fd0l5439w 0 J df3392 39 rl 094M 2 2 o 0 0 g 32 7F 36 5 7quotquot b Fxziyzjk D is the upper cap cut from the solid sphere x2 y2 22 S 25 by the plane 2 3 PM mimice fffvi ow v 5 C75 ObiF VF Z 0ZZHMQQQ MIMImtbe l Faxx fIJZEQU zigzagrzl dv rdrolaole 6 l m idzs rl o r 21 jz foiedrole O O 5 I Y 2V2er 739 zamr Jrelr Li 2 m r2 ri dr z 0 a a o 3 51 227916 93 l V r muf er r3de 2ZTI6 g f Va 0 rr v627 9quot672377 5

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