The streamlines of a planar fluid flow are the smooth curves traced by the fluid’s individual particles. The vectors F = M(x, y)i + N(x, y)j of the flow’s velocity field are the tangent vectors of the streamlines. Show that if the flow takes place over a simply connected region R (no holes or missing points) and that if M_x + N_y ≠ 0 throughout R, then none of the streamlines in R is closed. In other words, no particle of fluid ever has a closed trajectory in R. The criterion M_x + N_y ≠ 0 is called Bendixson’s criterion for the nonexistence of closed trajectories.