(915,#33) Let

Show that P_y = Q_x, but the line integral of F is not independent of the path.>

Solution: Direct computation shows that P_y = Q_x :>

Let C be the circle of radius 1, centered at the origin, with the parametric equations x = cos(t), y = sin(t), t = 0..2 Pi. Then C is a simple closed loop. But the integral of F on this closed curve is not 0:>

This does not contradict Theorem 6, for the hypothesis of that theorem requires continuous partials for P and Q on the domain enclosed by the circle.