S 17-2 - Math 200, Spring 2010 Worksheet #29 Name Kt;...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

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

View Full DocumentRight Arrow Icon
Background image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 200, Spring 2010 Worksheet #29 Name Kt; Section 17-2 Stokes’ Theorem 1. Find (a) the curl and (b) the divergence of the veg—tor field F = (x3 w y)i + y5j+ ezk 't 'J CurlL-is) =- VX-t'ft 1 33'- 3 i! .3. 5 a : _ .3 a .3. 3 2 7 9 $3.7 :1; it (‘5‘? W» m "#(X'r)iax7 '%("‘7’7 A :(o~o)o—b,6+1>=<olo)1) . e. _ -: _. '3 WW) - V‘T‘ ‘ azlwl+$rstieea=3$+5yl+e£ 2. Compute the curl of (a) F(x, y,z) = xi+ yj and (b) F(x,y,z) = —yi+xj and interpret each graphically. '5 A a. L ' K qw'wxm"): 43:3; g? 3K)=(°lo,0>=-6‘—4 o 7 WW Pb‘t‘wvtom |;\ \Jeci'br ‘Reiol ix g); g =<O,O,2> _,__ ‘4 mun-u ‘7 0 ‘ Graph of (x,y) HEW is ro‘iw’rion m fie veg-{or field 3. Compute the divergence of (a) F(x, y,z) = xi+ yj and Q) CU-fl (-7,) KID) :- (b) F(x,y,z) = -yi+xj and interpret each graphically. “ax :' }+ 1+0 ~' 1 ‘H‘e‘e {5 “9+ Ho)’ 00"” ‘Pl‘em c». fOiNQ' 3*) “‘1 <~v,x,o> = SEA-y) ritxlt‘eéeo mm 0 z 0 +h€~<Q no ne'f‘ ‘HUX coffin» c~ few? 4. Determine whether or not the vector field is conservative. (a) F(x,y,z)=wzzi+x2yzzj+x3yzzk 3; R i g1 g, 3’1. 1.71. z<1xz2- 1?, - L 1.2" Mi Mi: X31} ) 237 J'nyz Qxyifl’xz x > CL) &.\\3 (Ely/O) - Curl : v.9“? : 51-5) 50 F13 he‘l‘ Conger-wallet (b) F(xay’z) = ezi+j+xezk ‘3‘ 'A n 3 r i L ‘ k ~* - ' e R Cur-1U“) “ A i a 2 I; 6 aggmcl 6mg,” 0 h; M at : (Duo/e -6) 6—D? ‘ Or a“) F k C ‘hnms Partial e ‘ K62 “SAL LOmfomrd fine-hops Hm (we. 0“ derivaitai- Heme F is conSEruxldse . 5. Verify that Stokes’ Theorem is true for the vector field F(x, 32,2) = x2i+ y2 j+ 22k , where S is the part of the paraboloid 2 =1 — x2 — y2 that lies above the xy-pane and S has upward orientation. Skw M £15 a? = Sam—x a?) as S :‘(t)= ( cast, Slat)o)) 09.3211 (tl -‘ (ma-t, Cosf,0> F(ra~)).E‘ea+ = OJ _L D1: [(oslt(~5fnt) {- SI'vxut' cos 1;] d1 3, a- : ?_l_3 311" g; S]: g} 3 start gsan‘t} —0 I o X r\. _2L ‘hb . I so 5; Cur-1L?) A—S‘ — Tum-core j F‘ds :jfurHB-ds l5 {Jan'th “Y ‘CM' F¢4x§ytf> 6. Use Stokes’ Theorem to calculate the flux of the curl of F(x, y,z) = (22, y2,x2> through the hemisphere x2 + y2 + 22 =1, x Z 0, oriented with normal pointing away from the yz —plane. The bovmdma curve C [.5 ‘HM’ c'tmle y'all-31:51} 'er ye-Pleme chew Cum'lerclockmsg, 28:9 133 Sfflk€5 Till?0f€- ‘Flux 2' jjfiupwEJ‘JE: gF‘d-E fqrumerr—Ca 3'3 J 3.5 :5“): (0) Cost, stint) “‘1' C (H: (O, rsfn’t/ Cost) 17500) 3' {St-hit) cot’tjo) glux or A "’ 2“ Cot-I :- . d " h' .J 44' an an. (F) figF GM '65 F(c(+))'c(+)clt = —s.£\tco.s‘tat 5 §CM3JEJO 3 0 d 7. Use Stokes’ Theorem to evaluate Heurl(F)-dS where F(x, 31,2) = 2 y coszi+ex sin zj + xeyk and S is S the hemisphere x2 + y2 +z2 = 9, z 2 0 , oriented upward. , Time haunt-10m? turue C .3 Hm circle 91-71-13) 2 =9 owenl-N '(ow'h’v' CIOC’WM‘ n g 3+okest mean— Pumn2+ri3935r Eu] 3 (SCOJ'tlggjhtIOL oftfarr S; 4 u 5——* ._, 21: a”: (“RM/“Mt!” . 1 . : H —a --I‘ 2:." —a \, CUrHF) :15 as l: 914 jF(C(tD.de.t l-(CHJ) 0 Rust 34;“! E” n = (It’ssitcLhwfl g.,.c:,/(3:o>t)e , - ’ . d 'a‘ " — ‘ 351m: ;[‘l331nt +D+Ojclt " "(SJ Ji-éC-aslikrt - {SSH-Ct) O) 30:53 > 0 2 = "'filit ‘i salt] “= ~13“- 0 8. Use Stokes’ Theorem to evaluate jF-ds where F(x, y,z) = xzyi+-1§x3j+xyk and C is the curve C of intersection of the hyperbolic paraboloid z = y2 —x2 and the cylinder 3c2 + y2 =1 oriented counterclockwise as viewed from above. ' 1- 1 S is M Pewt- °¥+Le Surfil‘fi 2 ‘-' 7’1 x1 ’Hm'i lie] uh“? iJnP Um‘l‘ disk D: X +75| 3‘7 S'l‘bkef Theorem , famm’rn‘n ‘HM. Sufi-ace 06 a. afafk 3:30”) :)1_x1. a -..n __ __. __l "- : 2L 1 i F'd‘ " flCUrHF)'dS {Ex/‘1) <X2‘J,y x P 5 h - (”%x)‘37)l‘> = (25-27).) ': gen-10:) - h AR (WM?) 2 \ n =— 3» 2 :< -. X41- t“) : DQX+27L)dH x2 .LI'} : X/ y] x D 73K ) = <x - o) / 7) 111 ' i _. .1 '1 = 2; [txrdrelfl - Jhrrlhr L = 11- o 6 9. Use Stokes’ Theorem to evaluate sz dx+ yz dy + xy dz where C is the boundary of triangle formed by the plane x + 2y + 22 = 2 in the octant x, y, z 2 0 with counterclockwise orientation when viewed from above. 2 _ _ A Pam bathe Surgace as a ans-awn 9.- : 213,1) : l-i‘x-y @(XH): (XJyfl'ix-y) ) ‘6': (15) I’D t‘ r‘ R if; hi “ox-7;“ Ki- 7% x7 - Stofihat = SWEWM‘ r'igpx‘72x‘b°>'<5\z‘z‘>da o "- .L _ :".I.X‘H _ 1 1 1. L ;, "%f[x7‘i)1);_ at); " zj—Jix '1')“; *X-x+l)dx 1- ° 1 -5 a "D 1- ° 33/2 (-§X‘+"§.X“J£)¢1x = at an +{xhi-l o : 2 31. 10. Calculate the work done byaforee F(x,y,z)=(yz,—xz,—xy> in movingaparticle around the boundary of the quarter cylinder x2 + y2 = 4, 0 S x, y, 0 S 2 S 4 , with normal pointing away from th — I . —‘ 9 " \ ‘ CZ “‘5 to.” :7 pots ; ECWMFN-g“ 9(ea):<2coie/lsine,2>,059%o:~e—+ - SS 5. fi:<1(059125;h6’0) 15 H _. _ f r‘ e :j <0 y (mot) - f 3‘ E s 8 .. _ ~ 7- 33 ) It} ) (20355) 25m8/O>J.349 7‘1: :32 —X) 0 6 VI " 1y; . "—‘ . “t 3 ~ +>c ~ - > j/‘Bszédide = jsfie.e( 46 4X “‘7’ a i O o o 6 11. Use Stokes’ Theorem to evaluate “ScurlF-ds where F(x,y,z)= exy coszi+xzzj+xyk and S is the hemisphere x = m , oriented in the direction of the positive x—axis. TL: bemxdanna, curse C :3 'Hne mid- circie 71+ ill-=1 sin *8 yi-P‘M“ ' 11: Let '5, be “he dish yl+215 IJ X: D . 3, Shares Mn? San-e boundmu} as S k C h ‘ he . Thereioe .1 SS J Hie aha nan—0 ems? re r ) CU” [F] {n 1% ‘ “2° ’ (wire-at? = gees -= Sfcwrs’J-as’ —- '5 ’ _ 2- A xrf 5 c 5, ‘51:: 47 )2 x a. —. 55cmfiym's‘ 92°” "2 "1 5' '(Jwr‘ 3" J" ; “ ,7*€ Smi x€~xe we) : (kazydn 3f Ddfico $6r5|)\ufci\0559 n=i5° 434i. kufie z 1 A 1‘ Same Grien‘imi" {a + :- 1e — -‘ u i” r “Hi surfing. ‘f a i y i MCUrl(F)in :X-)c1' oh St Iii-ENE X20- 12. Let S be the cone with base (boundary) the unit circle x2 + y2 = 1, z = 0 oriented counterclockwrse when viewed from above, and with vertex the point (2, 2, 2). If F(x,y,z) = curl(A)(x,y,z) = (y —x,0,y + 2 +1), calculate the flux of F through S. Suggestion: Find a simpler surface with the same boundary. a (212)?) Since F : CurlCfiJ) :5 1‘5 cs. (Ur-l UPC'i‘OY' ‘FCQH anal 44.8 flux of— F a ‘i‘krouak t; Stir‘gate 5 olefin}; 0.1% an M2 Urierd'eoi boom-lap? ES. 6;, 3o ‘er flux ‘H’W‘GUJA S is He same 6L5 +Le Eluxitiixmuak 0,. A )6 7 Elni'oler Sur‘QaL-e Skim as M unfl— oiiSK X +7 :t)?:o wltk “mud Slug: SSE-d3; : ids = $- <y-x,c>/y+e+1>-§afl .. s L C \- 4 D “by”. 21‘ ’5 1“ 2." .. _ an i I L 5:. :‘R‘ 1%. (fizflhn ‘ 5 Xmas +0+\)rdrd6 =jss‘sa *2 :9 3' o ’H‘fsi 0 o 0 ~ 13. If S is a sphere and F satisfies the hypotheses of Stokes’ Theorem, show that “curlFodS = 0. assume 5 is Geld-eer (d: “'9 Grit?“ “5‘24" Win-t 0k and 12’: H" tind- H‘s. be 4*? umber And Lower hemispheres) raped-11331; ) 0;. S . %- TL“ CUP\(?) A? 2 SS CUI-i (Fl 'djS‘ ‘i‘ “Sickle-1f?de 5 Hi I. I : F .d a + 9.44 lo Sinker V" it it 7 . M 7 6"“ C‘ 7’ “‘3 5"“ 941372qu- orient-94 n, +ke counierclock wise dlfiC'i'iovx at Mile C3. is ska Same Circle Orienleot ta +Le dockmtse ducal-ton. n'eu‘e- £1 “FE-'0‘: :— _§i F'dra ) Sb SSCur-l ': 0 ‘ .3 Homework #29 Reread Section 17.2 (Due 05/07) Do Section 17.2 Exercises: 5, 9, 12, 13, 16, 23, 27 Prepare for next class session: Read Section 17.3 ...
View Full Document

Page1 / 4

S 17-2 - Math 200, Spring 2010 Worksheet #29 Name Kt;...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online