Question

l.a. FIND F SUCH THAT F= vf IF F = (2xY-2, x+22, 24-2X=) b. FIND THE WORK DONE UNDER IN MOVING A BODY PROM (-3,-2-D TO (1,2,3)

Answer 1

Given F- ८२१५-२, २+२८ , २५-२५z> ) -> DXF O k । ने 08 २४५-३२ १२ २+२८ २Y- =fgfg-22) -]-3[apan +2 [Padam fayd] (२५-२72) २५५-३ 02 - [२-२] -5-2--22]] + * [२श-(a)] - 0 7x150 Cure of vector field F is zero -> it can be expressed as gradient of Potential function of --> f If $= potential function

1 @ F = (224-7?)* + (x?az) jt (29_22Z)Ê it F3 + F3 ^ F = => f. 214-2 F2= x+22 F3 = 24-202 f=f let f f«,4,7) = afit af af Or a jt ay f 5x af dx = F = axy-z? >> (= constant f- ffidx = Jaxy-z pas ff1,3,72 x -xz+9622+ > gyz= function of Gand Z af ay = F2 = x 422 $= 5 (vitaz)dy = Xyt 29ż+ 9(8,72 sto 2 g (1,2)= function of land Z

de F3 = 23_222 $= Ję dz = 5(24-202)d2 t( -> g(x,y)= function of randy = 2yz -xz2+91949) from 0 and ① and ② f= x/y _ x2 + 2y ztc (=cauffant. २ b Fodo we have work done = SF- JXF zo cure of f is zero É is is conservative field from fundamental theorem of line intez yal's for conservative field's the line integral of ren servative vector Sield's is in dependent of path and is given as B work done $(6) - f(A) t- Potential fanetich (moving from A to B = FodÝ by field

A= (-3,-1,-1), B= (1,2,3) work done = f(1,2,3) - f(-3,-27) by field 1 $(1,2,3)= (092) - (1) 1)6332 +2(Q)(3)+c -2-9+12+C $11,2,3) = 576 $(-3,-3,-1)=(-7-2)-(376-14 2621104C =-18+3 turc २ --11tc Werk done by field - (5+ )- (-ilfc) - 16