Question

Problem 7. Given that each of the following vector fields F is conservative Find a potential function f such that f = F and evaluate fe F dr along the given curve C 1. F(r,y) y C: F(t)(t3- 2t, t3 + 2t), 0 <t<1 2. F(x,y, ) yze"* i + e#* j + xye k C: F(t)(t2 1)i +(2 -1)( -2t)k, 0t 2

Answer 1

# Given each of the following vector fields Ê is conservative, find a potential function of such that & f= f and evaluate - Edñ along the giun curm c.. do Flx,y) = xuy?? + x lyn j C: řid) = (+3–24, +3+28); Ostsi 2. F(X,Y,Z) = 8zen? î tek² ) + xy eR? Å Ć: 814) = (l+1 â+ (42.11 + (-ut) ; ostçe solution : 1. Now let's Herr P = x2y3 & Q = x3yd find the potential function. af is conservat ax from these we can see that f(x,y) = f neys de cor fm, 7) = In syddy Here is the first integrale f(x,y) = x y3 +214) whiwa Reys is constant of integration , Now we had to determine hays so let's diffentiale flincluding the hill, with respect to y ¢ see it equal to a

of - x3ya + R'ly) = x3ya? from this we can see that hllylo. 4191= thilyldy - c grum fore flx,y) = 4,23y3 +C F12,5) = 1x3y$]+) =114,91= $(437!out cao here can check our androver by of a Ř. Nord F. JX = *112-2*3" (3+297) (43.203 * 4*42492).(12:46), #101) = So'ſ(+3–2013 43ta4)' + (+3_273°. (+3+4)3] ft -Ic'! +-873- $#* (+28) In order to compute the line integral, note that to corresponds to the point 19,-) A tal corresponds to the point (-1, 3). so by fundamental thoum ff.dr = $,3) – f(25) = f(-1, 3) -flo, os = { [(1)x27 - 0) = -9 If Fate -9]

3]. Р(х,7, 2) = 73e17; + е** } + ye't k home for a frenz - 5 , e*z ——0 ye11. — o war. to a #1 м, 1, 2) = dze1' + 1% зу }{1, 1, 1) = ge11 + 3 (, Integrating Мод у т, 1, 2) = e12 +3+3,3 — Ш ferm 10, 10, 11, 1) =o в , 19, 2) = ky) Sh., f(x,y,zj = 1е12 + 4+1) Bom HNA - 1, 1, 2) = x +2ly) –® fет 10 4 0 11.17 - 31e77k ty) a k'17) =g. - 471 г +к) I tunce one choice for [fie, 1, 2) = x yen?

dinle THIE 1+ 2+12 i + (+2_1) jt H3_2416. jos tsa to compute Corresponds to the line integral, Note that the point lin o) & tad too Corresponda to (5, 3, 4), to by fundamental thomm [Fodra f 15,3,4) – 11-10) - 3et+ e - Itze po Jefior, sekil