\fGiven a vector eld f5 = M; + N3, where M : $92 + $2 ,and N : 4a: 1.The curve C is the bound of the triangle shown below (i.e. curve C is the union of curves C1. C2, and C3). (a) Directly evaluate the line integral 580 1:7" . d? (b) Use Green's Theorem to evaluate the line integral f0 13" . d? vector field F= MitNS Y- X= 3 F = ( xy ? + X? ) it (4X-1)5. Curve 6 is the bound of the triangle 6 = hitlet63 0 7 * ( a) Directly evaluate the line inlegal fo F.IF Part 1: (9, 0) Paint 2: (0, 3 ) JOF . Or = SoOF. IF point 3: (-3,0) SuE- OF = Jutsutsu, = Q 2. if he's conservative ( 0, 0)-7 ( 0, 3 )_6: Filth= dutt's , ests3 ( 0 , 3) -7 ( - 3, 0 ) (2: Felt ) = -+7+13-+)J, - 35ts3. ( 3 2 0) -z ( 0, Q ) ( 3 : [g l t ) = (= 3 +t ) itos ,.. 3stso Along _ 4 : P= ( xy +x 3, 4x-1) = (Q, -1) and Elle) = ], then Along (2 : F = (xy +x 3 , 4x-1) = ((t ) 63-+) 7(+) ?, - 4t-1). F = ( - 9+ + 6 + 2 - + 3 + + 2 , - 4+-1 ) = ( at + 7+2 + 3 , - 4t-1 ) W2= J. F. F,' Wide= 1 Gut +7+2 + 3, -4t - V . (-1, -1) dt. .... = fit?- 3t'+ut* +th- = (12-63+24+3)-(42+63 +84-3) = = 126+6 =-120 .. . / long 68 : F = ( xy 2 +x 3, 4X-7) = (+26teq, 4t-13) Fs' lt ) = i W = = SC F . V, '(tide = J_ 3 (+ 3 6+ + 9, 41-13) . ( 1, 0) dt. - .. .. . . ... .. ..am = fy ( t ? - 6tta ) dt = (3t'- 3+ +at)_; wi- - =0= (-9-27-27) =63 .. . . . SCF. dF = Jutsut Jug =- 3+ (720 ) + 63 =-60 This is not conservative field.