Show that the differential udr+ady x + y (x, y)(0,0), is closed. Show that it is not independent of path on any annulus centered at 0. Solution P = -y x+y2 (x+y)+y(2y) (x+y2) == Because / = 2+y2 (x+y)-x(2x) (x2+y2)2 = y2-2 (x+y2) == = (x+y2)2 === Q/ax,the differential is closed and we can use Green's Theorem f Pdx+Qdy= == $ -y x + y I dx+ dy == + y ++= fydx+xdy= |2|=r 11|3| == = (1+1) (1+1) dx dy |z|