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

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

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

View Full Document

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

View Full Document
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

{[ snackBarMessage ]}

### Page1 / 4

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

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

View Full Document
Ask a homework question - tutors are online