Question

D Question 11 12 pts to Consider the vector field F (x, y, z) =< 2x – yz, 2y – az,2z – xy>. a) (3) Is this vector field conservative? Justify your answer. b) (9) Find the amount of work done by this vector field in moving a particle along the curve (t) =< 3cost, cos’t, cos” (2t) > from t = 0 tot = 1

Answer 1

we VA 22 1 20 = OM 0† (21,9,2) = x2x-42, 2y-x2, 22-49] we have check whethe this vector field is conservative or not. know it conservative then f f 20 i tad 5 + 20 È 2x ay = 24-yz, a = ?9-42, a 2 = 22-hy. әр an-yz on integration i we get $(1,7, 2) =22 2y+f, 15,2) - diff worto g. - x 2t - f, 19, 2) ay 7) 24- x2 = ang at ay. of a af, 27. fly, 2) = y2 ++212) - ① 20 a dy on using " in 0 we get d(x, y, z) = x²-xyz+ y2 + +₂12) diff worotoz. al uy tot 2 $₂12) 21 22-14 =-nyt 2 22 (2) a 1 - 2 f2 (27 = 22 f212) =22 using 4 0 in 0 we get $14,7,2l = n²=142+42 +22

5) 7 $14,9, 2) stof=00 ) a f = {21-42, 24722, 22-14) is conservative vector fied. have to find the MOL, we amount work done particle along 9 JT of by this vector tield in moving the curve riela <3cost; cost, ces zr> from t-otot SF.dr =)< 24-42, 24-12, 22-ny).gr 2 dl x² x y zty2422) 2²+ y² + 2²-242 Brosti?+ (ca 363+rcosézt? -3cost.cosit.costat

0+0 4 X 3.0 -( 9+1+/-3) =-7 An.
ANY DOUBT THEN COMMENT PLEASE......I WILL HELP YOU ASAP....THANK YOU................

Answer 2

evaluate the line integral ∫_c▒f ⃗ •dr ⃗ , given that f ⃗(x,y,z)=xy^2 i-yz j+3x^2 z k along the curve c from the point a(0,0,0)to point b(1,2,3), where c is parametrized by x=t,y=2t^2,and z=3t^3 .