Question

Exercise 10: a) Compute dy - dx by parameterizing the unit counterclockwise circle centered at the origin and computing the parametric integral. b) Compute and simplify 26 (1) c) Does the Divergence Form of Green's theorem say Lady - dx = PS 26.12) dx dy I when D is the unit radius disk centered at the origin and C is the counterclockwise oriented unit radius circle centered at the origin? Why? or Why not?
please show all parts. thank you.

Answer 1

at the origin. Solution: We parametrize the unit cirele centred out the origin by a= eso, yo sino. NOW dy- x² + y2 = 21 Saro + sino) do do where P and Q are differentiarre findian -sinodo and dy = oso do. da cro.co sind. (-sino) do Rey2 22 42 1. (at) -2ax 2+ y 2 2 yà (at yaj 2 + 1 2 + y2)2 = (2+ y)2 Ag, $S 3 (20)+ ) es + 1 (2+ y ) - 2y. y (2+ y2) = y = x + 2²-g² o sy as the integrand D but С. SO it Sony day y do = 217 does not sutissy Green's the orem. Greren's theorem states that & pda tQdy = Whrenere SS (304) region D ht 2²142 y C and 23142 but here pa differentiale (partial)