%  illustrates movement on a surface

   xgrid = -2:0.2:2;
 
    [X,Y] = meshgrid(xgrid, xgrid);
    Z = X.*Y;
     surf(X,Y,Z);
        colormap copper;
        hold on
        
 t = 0:(0.1*pi):(2*pi);    % sequence of t-values
 x = cos(t);
 y = sin(t);               % x,y move in a circle in xy-plane
 z = x.*y;   
      plot3( x, y, z, 'LineWidth',2)     % parametric equation of the intersection of cylinder with saddle
  
  % let's say the ring is the road, z is the elevation and dz/dt is how fast the elevation is
  % changing. Find dz/dt for  t = pi/6
    x0 = cos(pi/6);
    y0 = sin(pi/6);
    z0 = x0*y0;
    plot3(x0,y0,z0,'*')
  
  hold off