%%%% 
% Example Octave commands

%  Plot the flow field in phase space for a gradient system
%       X'= - grad V(X)
% with potential function V, including the level curves of V

%%% DO THIS FIRST:
% Initialize the grid in phase space where arrows will
% indicate the flow X'=(x',y') at position X=(x,y):
p=-2:0.5:2; [x,y]=meshgrid(p,p);  % 2-D grid 
ral = 0.4;                 % Relative arrow length

% Define the flow field and potential function 
dx = sin(y);
dy = x.*cos(y);
dd = sqrt(dx.*dx + dy.*dy) + 0.01;  % Raw speeds, nonzero 
dy = dy./dd; dx = dx./dd; % Normalized unit arrows

V= -x.*sin(y);    % potential function, so X'= -grad V(X)

clf;                   % clear any previous graphic
quiver(x,y,dx,dy,ral); % Phase portrait = flow field
hold on;               % superimpose the next graphic
contour(x,y,V);        % contour plot = level curves
title("Example 7.2.1, p.223");