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Done with Matlab.

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Source code:

Lx1=-1.7; Lx2=2.3; Ly1=-2; Ly2=2;  % box dimensions
N=60; % split the box into N x N grid
[X, Y]=meshgrid(Lx1:1/N:Lx2, Ly1:1/N:Ly2); % the grid

f=inline('-((y+1).^4/25+(x-1).^4/10+x.^2+y.^2-1)');   % draw the level sets of f
fx=inline('-2/5*(x-1).^3-2*x'); fy=inline('-4/25*(y+1).^3-2*y'); % partial deriv  
Z=f(X, Y); % the function value

figure(1); clf; hold on; axis equal; axis off; % pop up a figure
h=0.5; % spacing between heights
v=[-20:h:0.8 0.85]; % the heights
[c,h] = contour(X, Y, Z, v, 'b'); % the level sets at those heights

x0=0.1333; y0=-0.0666; % coordinates of the top of the hill
delta=0.01; % descend from the top of the hill with this step size
Angles=linspace(0, 2*pi, 20); % will draw 19 descent curves with Angles(i)

for i=1:length(Angles)
   x=x0+0.1*cos(Angles(i)); y=y0+0.1*sin(Angles(i)); % starting point
   Curve_x=[x]; Curve_y=[y]; % will hold a descent curve following the gradient

   % descend from the hill 
   for j=1:500
      x=x-delta*fx(x);
      y=y-delta*fy(y);
      Curve_x=[Curve_x x]; Curve_y=[Curve_y y]; % append the updated values

      if max(abs(x), abs(y)) > 5 % stop when going beyond the picture frame
	 break;
      end
   end
   plot(Curve_x, Curve_y, 'r') % plot the curve of steepest descent
end

axis([Lx1 Lx2 Ly1 Ly2]); % the picture frame

saveas(gcf, 'level_grad.eps', 'psc2') % save as color postscript. Use gimp to convert
% to png. Does anybody know how to reduce aliasing (gimp helps, but only a bit)?

Original upload log edit

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• (del) (cur) 18:41, 23 December 2004 . . en:User:Oleg_Alexandrov Oleg Alexandrov ( en:User_talk:Oleg_Alexandrov Talk) . . 358x336 (71326 bytes) (I made this. { { PD } })