Question

Problem 4. (15 points each) Let F(x, y, z) = (0, x, y) G(x, y, z) = (2x, z, y) + (x, y, z) = (3y, 2x, z). (a) For each field, either find a scalar potential function or prove that none exists. (b) For each field, either find a vector potential function or prove that none exists. (c) Let F(t) = (2, 2t, t2). For which of these vector fields is ñ a flow line? Justify your answer.

Answer 1

Soli Given? Zet F(x, y, z) = (0,4%) į (-0,9,+) = (2x, 2,4) & À (x, y2) = 134,2,2) ® to find a scalar potential function is. VAB (Fide {where ř - field vector de line element} for cortesion Co-ondinate system de = dxi tdy i td ze >> Vp = S(xf typ) Cok @ + dyi +dzť) - VF = Siedy tydz) vp = syt yz teret a Now Vo = S(221 + zityk) (drit dyptdzk) 1 = x² tyz + 4z + CZ VG = x² tayz +(2. And, Now NA = Sh.de Slay? + ax9 +2²) (drli tady? t dak) = Slay de tasedy tzdz) A = sny + 2 / +63 6 If à is vector potential vector field F = F x Ā & vector field must be a solenoidal Vector field = x. (W x H) = F =o - -

V Nou Now, FF - Congo in the A) (x + 38) Wł otoco . And E. Ĝ = ( 2 h + 2 + 2 F) Carl + 2} tyk) N =atoto =d to Nowe ZH - (afte ita £) (3yi taxjt?k) ol = 0+0+1 =1 to i only te vector ř (x, y, z) = 10, x, y) field only this field satisfies the solerwidal exists which admits polential vector, Let 5 (t) = (a, a, 13 . - where x=2, yzat , z=t² Let I flow line of F is path an ñ such that d5710) = f (CH) It en dit sal ) - ( det hele tehty) = 10, 2, at) = 10,49) .: DALD = F (3,492) F (oby, 2) sector field n is a flow line. For