Question

3. Consider the vector field F(x,y) = (27x D = {(1,y): 0 < rº + y2 <2}. +ya) defined on the region D where a) Directly compute SF. Tds using the definition of the line integral, where C is the unit circle oriented counterclockwise. b). Use Theorem 3.3 (Test for Conservative Vector Fields) from the text to determine if F is conservative. Is your answer consistent with part a)? If not, what is the source of the discrepancy?

Answer 1

Given Vector field is F(ay) =< -2 The vector field is defined on the aegion Du{(x,y): 0<x+ykey. Now we have to find (F.Tds ... ) las where c is a unit circle oriented in the Counter facture directon. Take the bare meten xir cost yarsint 1 ! x = cost, y=sint xzty = sosht t sint 1. ostean - x = cost ss = sint dx = a sintat » dy = costat श = 1 - sint (-sintet + est castelt ī sinnt at a costat = (sin²t + cast at lidt = [t] au - =[PT-0]. ( 2T ../fords =21

We know that, Let F = pitai be a rector field as an open and simply connected regioa D. Then it p and a halle continuous partia ntinuous partial derivky and al-20 the retor fixed .. the ind a t and is of conservative a narative. - > <P, QS Here Ē = < 442 xt yea PESQ = (2+ 7) (1) - (-4) (25) = (2+3) a) ~(22) (1+y) & foto x² + y) a E = (-22 ) is 24 y Hence the vector field conservative. th > aty 1.274 = F =< ***

→ of a = 1 = y a od si a put a = 4 x= ugua da = y du y du -dna tan' (2) + c =ofit us to = - taniu) fris) = - tan ( 2 ) + $14). y X

ay Samas Sada finis) = -tan (²) + $in)

Choose & ing = 0 and $19) , weget finis) = -tan (2) Hence the potential function is fre,y) =-tan' (%). Ventication fta' })] vontca)-4. y txt - E=(- 2 27 } y vesitied trye