Question

I Verify Green, Theorem for the field F(x, y) = (x - y) + + x 7 and the region R bounded by the unit circle x² + y² = 1. b) Use the Green's theorem to evoluate the line integral 3 y dx + 2xdy where C is the boundary of the region Os x a , Os y < siax orciented counter clockwise. (Answer : -2)

Answer 1

- Ans. @ Green's Theorem states that If Mcniy) & Naiy) are two functions of afy with continous partial derivatives & Y R is a region bounded by simple closed wwe c', then $ (Mdet Ndy) = SS (2N - 2M) dxdy me mom R Clearly is way) = ł ta? Here, f(a,y) = MitNp = (x- y May) = a-y & c = Unit Cirde first., 145= foc m de + Ndy) = $pa-y) dx + xdy Since, C = unit circle with radius, r= 1 > x= x coso = cos o 4 dx = -sino do & y = rsino = sino & dy = cos o do cou ou o $(Mdx + Nely) = $ (cos o – sino) C-Sino)d0+ card.co40-40 $E cos @sino + dino + cop?o) do = $[**2din0.colo + 1] do - Esin 20 +1] do [compete] [pica Ciste] = x + 2x - 4 = 2x Now, Rus season - Dura) doby = SS (1-+ 1) de dy R

R JJ dedy = a(i) ------ 235.(12 = 28 ( area of circles toe?] Hence, LHS = RHS =) Green's Theorem is. Veri @ We have $ ( 34 dx + gedy) osx 2 x - Using Green's Theorem, we can write =) of By dx + 2x dy = f( (2-3) dady = - If dx dy = -ff dy de - - *J y] nie die * sina x = co ---- &osy & fint = cos X - cogo - -2 -- 3 $ ( 3y dx + 2xdy) = -2
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