Question

Let F = (yºzcosx -ax®z)T + 2z'ysinx + (byz sinx - x) 6p a) Find the values of a and b for which is conservative force field. [10p b) For the values of a and bobtained above, find a potential function for F 9ple) Find the work done in moving an object in this field over the curve C:x=sin(t), y t, z= tº from t -0 to t = 1/2

Hello could you please help me out with this applied engineering mathematic problem?

Answer 1

Solution 3) Let us Consider giver vector feld. O (a) + = y*23c8 * - 2x3z, 27°y SinX , byʻz-sinzxy)=(4,,5) The above Equation can be completa > B F = (M,N,P) H < y'zBcos X -ax3z N → 2 z3y sinx p> by zesina-84 We know that , Ñ is conservative Vector field if my = Nx , Ne=py, Mz = Px) het dy * Z*cx = Q 23y cos at (My=Na) • Nz = Py =) 67°y fon x + 2by2 "sou 6=ab 16=3] Mz = Px 3yrzrcasa -ax² =by z "cosx-423 16=3 Tasul (b) there we have to find potential function F. We know that f' is potential function Then consider that are fy=fz Fq=fz

Fx = y'=3 ces X 4222 0 Fy= 22²y sinx F2 = 3y* 2% Sinx -qu -> @ Nexo consider Egno Fx=y^2 3 cos x - 4x3.7 integrating with respect to x. =) pay ?ZSinx xºz +(4,2)+ differentiate with respect to g. Fy- Dyz'senx + ¢y(4,2) =22yana {sy &? @ } = ¢y (4,2) ze het o is function of Z only ie--, $,(z) put in question Now F= y2 3 Sin x – 847 +, (2) differentiating with respect to z F2 = 3y+z7 Sine *+ CD.(21) '=32%80X-24 - [4(2))0 > os Constant function let d =C where as CER Now Substitute this for Equations we get 1 Fay 2592 - X°2+1 is it is potential function

☺ het Sf.dr = [ECU ? 8(t)=(sint, t, t2) F(t) = +962)3 8n (sint) (Sin t]€? F(t) = 48 sin (sin t) - Ersina (7) SF- dra work done - [8 Sin (en t.) - 4* sin« ct) ] - 03859 () - Ja SMO) - = 9488:53 (0.0174) - 958969 256 = Ź = 37.064 (0.0174) -2.4672 0.6444 -2.4672 -1.8222