Question

Is the following vector field conservative? A(T (y 4x3 z, 2z3y s, 3y2z2 sin x - a) COS x _ If so, what is the potential ø(F) of that vector field A? (see worksheet 17)

Answer 1

Dek Salution : ) conservative vector field. A rector field is called conservative F(x, y, z) = "fit fait F₂ k els cul f = 0 Cul F = VXF - F Fa Fa] a - A rector field Ě is called a conservative I rector field is it the gradient of some scalar function. That is it there exist function f such that Ē - Df = < fær fy, fz > then f is called potential function for F. How we have check the given rector field is conservative or not .. À (7²) = (y2y3 cosa- 4x²2, 23y sinx, 34 22 sinx-x4) Tie Å (7) = (y² 23 cosx - 4x3z)i + 273y sinxj + (3y²z² sinx-x4)

we find, curl à (8) = V Ã(8) cel ) Ý j to do zai2-4x2 27 มิก: 2 งมีก2 - -Boteīns - 24) - 2 (2z8ysinz)]: [*%8v?sinx-x4)- (092?cos2-42°3) ]j + [22sina) -2, (yºz%cosa= 42% 7) Z д Toyz² sinx - 6y z² sin x] i- [3y²z² cosx - x3 – 3y2² cosx + 4 x²] [224 Cost - 2473 cosx ]k - Coli - (0)j + Cok 20 1. Cuel À ) = 0 - Hence Ř (8) is consevative field. How to find potential senction for a (8) using second definition, if ħ (8 ) es conserwatire rector Held then there excist sunction or such that, À (7) - V¢ ) = < Pas dy, oz? then 6 (7) is called potential function of Ã?

15. RCF) = < 057"), Qy (7"), Þz (7) (y²² cost = 4x32 , 27² y sinx , 3y²z² sinx - 24)=<daily, de de = y²z3 cosx - 4x32 - © dy = 223y sinx - ☺ z = 3y?z? sin x - xc4 0 from ean , 8x = y²z3 carx - 4x32 $(79)Bxcon) ?z3 cose -40c3z)dx is o con la fy2-3 sinx - 24 z)+ ecy,z) – Diff. 4 coiret y partially, dy (%) = 24 z3 sinxt Cy (Y,Z) But from *, 0417°) = 273y sina from ② and ③, 2y23 sinxt Cy (Y,Z) = 2z3y sina => Cy (417)=0

as CyMiz)= 0 Cly,z) : S Cycy, z)dy Sody =) (912) = otco (2) : Dom ean ④ $(8)= y2 z3 sinx-x4 2¢ ¢ (2) again diff. above eqn wort ? idz (%)= 3y2z2 sinxa x4 + 1,(2) But from @ O2 = 3y2z? sinc-ac4 3y²z2 sine- x4+4, (2) = 3y²z² sinxa x4 => Clę (2) = 0 C, (Z) = 0 from ear is our potential $(5)= tunction y² z3 sina- x4z for à (8).