Question

6. (4) (a) Is F(x, y, z) = <e'siny, e cosx, esiny > a conservative vector field? Justify. (4) (b) Is F incompressible? Explain. Is it irrotational? Explain. (8) (c) The vector field F(x,y,z)= < 6xy+ e?, 6yx²+zcos(y), sin(y)+xe?> is conservative. Find the potential function f. That is, the function f such that Vf= F. Use a process. Don't guess and check.

Answer 1

curl F = 1 Sem? ☺ 9) Given, f(1.902)= Tezsiny , e casx , e "sing) Guslf -l î Ï Âl - 0 (excoy - ecos x) = [ 2 = le coy - e cosas lesing ecosa ensing I Cersing-e?sing) t ê (e?shna - ecoy) here, cullf to fence, F is not conservative vector field b) for Ini compressible i divergence 8. F must be o . them o potez lesinge te le cos x) + (e"siny) - 0 +0+0 -0 . " blence, f is compressible. for notational curl of F must be o- then curlf to: ( from part. a) . » F is not an imotational. c) Given, F(214,2)= ( oxyhte², 64 x 4.2 cosy, sing they for conservative, Jl. . porest ble Cuel F .. .

cul fel î î (laya 12yx] fonyhtet Gaya or coy doing on - î ( cosy - Cosy) - ĵ (eZ-e? sio -ġ.otico zo " fus conservative potential function, considers Öf=F. - bržte? . f = 6yx2+cony fer * = winy +ze? from any one of 63y2 +e? Intepate with respect to a flavy; 2) = 32%y2 +2e? t fy (972) iume e at lie where f(312) is arbitary in hartial desuitive of ear 6 . I constants in derms of with respect to y then y and af on the Gay + 0 + wifi a lyst tz cosy = kyk + sf ( from a o frezcoy f, (4,2)= 2 going + f2 (2) ( where fa(2) is consta in deras ng 2) then f(x19,2)= 3x²y? t u e²d? siny et fa (2) - 6 again partial derivative of egne ③ with respect to the then we get L me?wing + oste sing tae?= xe²f sing + t2 ( from (3) I

a tą (2)2€ (where cas arbitrany constant) I flang,2)= 8a²y + xe2 at Zsing of - potential function -------