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# S 16-3 - Math 200 Spring 2010 Worksheet#25 Section 16-3...

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Unformatted text preview: Math 200, Spring 2010 Worksheet #25 Section 16-3 Conservative Vector Fields 1. Let F be the gradient of ¢(x,y,z) = xy + 22 . Compute the line integral of F along: (a) The line segment from (1,1,1) to (1,2,2). git-.111 = W111) —cp(I, ,1) = (I-arzzw (I-IH‘) = '+ (a) A circle in the xz-plane. 5mg eve-vi gradient vee‘l'm- ﬁeld is ConSeruwl-[ue (5,1: d4 15 indefpndmro-fM) ’er [me in 4’3th over q closed curve 1‘ O (c) The upper half of a circle of rad1us4 with its center (0,1,1) in the yz—plane, oriented clockwise. Ir Thl‘ts‘ml Pbl'rd' n = (0,4,1) ) +11»...th {Jaini- 3 -‘ (0151‘) 11:10,») on B=(o,‘s,1) 5M: = (W031)- <P(0,—s,|) = 1" 2 r C 1 1s . 2. Compute the work done against the inverse-square vector ﬁeld F(x, y, z) = - ( 2 2 2 )312 (x, y, z) in mov1ng x + y + z aparticle ﬁ'om the point (1,0,0) to thepoint (3,0, 2) along the curve c(t)=(2+t,Jl—I:,t+1),—ISt51. _ 1 ‘L ‘—" A Letr~0x+y+21kﬂnen F()§,Ig):~"§<xm%): “L g>zﬂ§£ . Fa. Smne vet): '~ Jr: Y' ) 1P0+edlci £vnc~l~fm ¢ ._—_ , H1, Tim Work elem. against F 2;? 1‘4 fiche-1ft? (32“) 500 410)]: “[62545] : Mil—L? 10 I 3. Determine whether or not the vector ﬁeld F( x ,)y )(= (x— y) )i+( (—x 2)j F “((31%) is 0» BMJxen'l' vec+ov {talc}? UJI'H\ i : \r' is conservative. If so, ﬁnd its potential function. F1: (X-V) ) F, :(x—g) 3E -.. aﬁ— : ‘D—y“ ‘ 7,1 3X l ‘ ‘d s D ﬁQr-ewcwe ) vec’i‘or £EEII'J- F 15 not— Cen§ervost1ue “019. 'H\°:t g}. 'dA > D :3) F15 no‘i’ CODSPI'U'Ov‘l‘U-Fﬂ- ...... h 4. Determine whether or not the vector ﬁeld F(x, y) = (3 + 2xy)i+(xz'— 3 y?) j is conservative. If so, ﬁnd its potential function LE F =F) V‘- -! . : :2), is. so F 15 1 3x ) canhmhbcE, 9? 2 Usﬁﬂ; Fl- “1‘13 '3‘» 3):)(3+287)ckx1 “32-11-317 +20) . ’ ' 3‘; :iy(3><+>‘7+3(11)= X+a(VI->3(11—3, (t5. _ 3 * C H—Q'ﬂ-Cej 7):}? 7)] I 7x“) 73* WXJ‘J‘BX’rxy-ys-PC» 5. Determine whether or not the vector ﬁeld F(x, y, z) = (2392 — z, x2 + 2 y,1 — x) is conservative. If so, ﬁnd its potential function. Ft F; F: g; (x1417) :ZbF 3E7 lbw-‘55) g - . 31‘! ()~V-) : o —' 3),“2 (X1+Qy) =7 F is conSﬁ'waJrée» 6‘“ CW’PMQI‘ MW) :33. (Ht) = -| ~‘- ??(lxr 2) I. U39. agiiﬁ: 3X) *2- '. (W473) ‘jﬂ’xy— %)dx = pray—XE +3 (7,2) 1 2.05 : z: ‘ 3 "A" 1 e y F; X+J7. jg-ay(x27—x%+alz%)):x+%g =)%?:2/ H?) =51 M. = it»: c. Hence £96912)”; “*2 with L 6. Determine whether or not the vortex vector ﬁeld F(x, y) =< —y , x > is conservative. 1. A ( L...) : {X1+2L)-X{2x) :- 22;" :3 ”*7" MW)“ H577" Cw” Pm'ﬁd’ ' __ :_‘2_ “(>33- L) f a. " are e W»! ' -_-v 7 + ‘2 ) __ -X 37[ XL‘W‘I) ‘ﬁ " 17—371 3 FIGURE 12 The vortex vector ﬁeld. { X *7} U‘ +7) CM; F 6 not conserroltlte beamse é '{E-ol‘ﬁ' .74 0 Note ~Hnat+ -Hte verify U665" Un‘oL circle. (13 aG)'-‘(Cose) Sn‘oG) 2'32}. ﬁeld '3 not swirl “med-Ed Efﬁe): <-Su§6, “59> 1’ _ 13, _ ztr 2” c HIE- F(g[9)).(_(o)ae :J (—stke, cose>-(-sttlo, Coso> J: ~69 = Qn’ 95 O O O O 7. Evaluate ICFtds,where F(x,y)=1:22i+2yarctanxj and C: x=t2,y=2t, 0951 3k ‘4 ‘ x ‘ . i 4:. _ 21... .. A 1, Mn ﬁat F l3 Con served-we . '37 ( ”12.) - H‘X)’ ‘ 3)! (lyarc'l'anx) - ‘ . 3.89- 1 . 2. 1' F‘M‘ P°"°“"'°~‘ ‘9- We >x ”It—L“ (PM) = Lieu = y‘m‘rw +30) ”5‘1 gtzlftrdanx ‘. (Say-Q = lyurc+anx+3yyl *7) 3‘C7)30 30) =I3lﬂol7 = C . 5° @9093) : ﬁat-Chm + C 3. EValvwl‘B “he {Miami ' lkt'h‘cq faker o‘f‘C??{o} 51010)) ‘l‘écmfmlroa'tn‘l‘I ”cs-(I) : (1'3) {Haﬁz/[Octet's :CD(I,1)-Cp(o,o) = Hatch“ — o = H-‘l/o =2 T? 8. Evaluate [CF-ds,where F(x,y,z)=(2xz+y2)i+2xyj+(x2+322)k and C: x: t:y= -t+=lz 2t— 1, 0<t<1. F‘ F’- F3 I 5kowF \s conSQrUz'Iw-e‘ é; (2x7)=—1),: 37(17‘2'1'71) - A 2 F'Gsol “Obtefifl d)‘ .A( :(Xli 3,2 J: o , ”a (y,2)= Sﬁgllﬁhx tkte) §%¢'X+L(:)= F =5L’:ta) “32,36 Heme ) (ﬁlmy—I = 9‘ 2 *‘X7 ”4-? I: C. I2J=I3ue =2 +(_ 3. Eumcere Hug we grew 1 mm Pamufcra‘to): (qt-1))ﬁrmlmifauir:2m.-(W) £96.43 :CIUCU-as‘: @(I13,l)"50I0/IJ—l) .: é‘("II : 7 9. Find the work done-by the force ﬁeld F(x, y) = e'y i—xe‘y j in moving an object from P(0,1) to Q(2,0) . “-4 _ A- _ .. ._. —-._3 _ - F65”) : (8‘3"Xf‘f >- 5mm 3718?)" "€v=§(("Xe 6‘”) F D (onSPrvoi‘tue . ‘ _ «‘1 _. - ~ Fman. 3:316 1" :CpIYJ‘II: fLK:X€ 7+3“) £1. 37 "‘xe VHF M : wean-a I?) :93”) 5°) 5° 8/7):S<71{7I“7:C- I—Ifnc-e) (135W) :Xe 7+C 1.5 a {JOHA‘I-i‘a.‘ {UM‘g'L‘Oh- 'Qar ? TIM», mi— 3,9561? {was : 491.20% 9%,) = get: 0 2 2 10. Show that the line integral is independent of path and evaluate the integral. IC(1-ye‘x)dx+e‘xdy, C IS any path from (0,1) to (1, 2). "J -x -X__ 3 ( e”) F(X,7)=zl—ye )8 x.) 5mm? 37(1 ")9 x)-—“€ - 5X6 ) .H‘_ and ‘Iihus :‘I‘J Ime UTI‘Qal‘Bvl is Independmfo‘I fa FxQCp'. M‘le— .—7:"‘ 3L}(may!:-ﬂ1-;e“")4x=u+;e"+3(,) 37:13 :6x - ”(H7613 (70:6 +9va =9 3(3)“) ﬂoaty) =6 That-afore 12(1)le” x+7eu 55+ (- I) mono-I-en‘l-rql ‘CUI‘L‘I'C‘MN . (I‘VE W)°I¥+f d7: EVE? do =£ﬁfl.1) ”0(0) :([+29 I) I "i“ h.) F‘s CanseeUK‘J‘IUE’ It: a = .. a . F. F‘. ﬁ‘ 3% :0 ConSQruwl-Nﬂ 11. So what do conservative vector ﬁelds look like? F(x, y) = (2x,2 y) Which of the following 7 vector ﬁelds are conservative? What is not happening in any of the conservative ﬁelds? What is going on in the nonconservative ﬁelds that is not present in the conservative ﬁelds? check crust Pea-HA; 3311:; and JAYF' ”tor Buck veclm- {hit can Sewe‘l'iu-e K\\\‘ ,x\\ \‘ '\\ \\* \\\xa:j//:,k\\ xnfg/jj,x\\\xh \\\\a,////,k\\ xa,(§//,kx\\a- .\\\\., ’1’ .x't x wax-t . ’, 4 . i.,’:1’3:_»9¥l|.,j11:,|l‘ll§, '1. ‘1} fp-“‘ \\.1ff1‘1' I“\T\“’/ fl,‘\\ \\,1//‘-’/ J‘\\\\Kl/ /,a‘ \ “ﬂyiffz , \\\\ ;,,\\§\\,,;//, fr, \‘\"\' riff} *\\\\‘ff "tuv ‘\\\‘“ \xli’ll, "«;a«‘; 4 )1 f...- ," f! “‘l“‘. rr ”Suva?” ,“\\“,xx/ ‘\\\“,;// Conservation of Energy Let F be a conservative force ﬁeld such as the gravitational force or electrostatic force; that is, we can write F = Vga. In physics, the potential energy (PE) of an object at location (x,y,z) is deﬁned as P(x, y,z) = -go(x, y, z) , so that F = —VP. The work required to move an object from point A to point B along path C is equal to the change in potential energy: W 2 [CF ods = - I0 VP- ds = P(A) — P(B) . The name “conservative field” stems from the Law of conservation of Energy. Suppose that an object subject to a conservative force ﬁeld F travels along a path c(t) with velocity v z c’(t). By deﬁnition, the object’s kinetic energy (KE) is émv-vand its total energy E at time t is the sum E = KE + PE = émv - v+P(c(t)). The Law of Conservation of Energ states that E is constant in time. 12. Show that the total energy E = KE + PE is constant in time. UH“? New“; Law F : “"3: (where EC = Fit) 3 a E at ”A . 21—i- H CH; émU U + P(€(‘H) ndo‘l'e . by {newest rule. 2 J. a _. A, .4 _, ,p, (Um-Jog; m + _.. ﬂ .4: g 1 tlU U) at Pam) _2v U r2170. 2 .L (1... __, by chain rule ﬁar PL'HA-S 2 m U Ca -‘ *4’ * __, __, __,, _. “WORM C m :3? HM) =\7¥{c(+J)-c (1:) : V'M ‘— F-‘J (sac: F=-U?) ' Conservative versus Nonconservative Forces Generally speaking, gravitational ﬁelds and static electrical ﬁelds are conservative. That means that the work done to move a charged object from one point to another in an electric ﬁeld does not depend on the path taken, but just on the total displacement. Usually, magnetic ﬁelds are not conservative. Also, electric ﬁelds induced by a changing magnetic ﬁeld are nonconservative (see Faraday). Conservative forces: When lifting a book, the work that you do “against gravity” in lifting is stored (somewhere... say “in the gravitational ﬁeld” or “in the Earth/book system”), and is available for kinetic energy of the book once you let go. Gravitational, elastic (Hooke), and electric forces are all conservative. N onconservative forces: When pushing a book, the work that you do “against friction” is apparently lost ~— it is certainly not available to the book as kinetic energy. Forces that do not store energy are called nonconservative or dissi native forces. An ﬁ‘iction-t pe force, such as air resistance, is a nonconservative force. Homework #25 Reread Section 163 (Due 04/28) Do Section 16.3 Exercises: 3, 5, 9, ll, 13, 17, 18, 19, 23 Prepare for next class session: Read Section 16.4 ...
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