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====== Octave: Surface, Contour (Isolines), Gradient ======

===== Functions of Two Variables =====

{{:phy2:ss2013:surf.jpg?direct&300|}}

In Octave analyse the function f(x,y) = x^2*y - 2*y.

  * Understand what a meshgrid is, how it is generated and how it is used.\\ Define x=[-2:0.1:2]; and y=[-1:0.1:1]; and generate the meshgrid yielding X and Y.

  * Define Z=(X.^2).*Y-2*Y; . Z is the matrix containing the function f evaluated at all points of the meshgrid.\\ Generate a surface plot in the first figure window.

  * Generate a contour plot in the second figure window.

  * Use the function [dX,dY] = gradient(Z); to estimate the gradient of f at each point of the meshgrid.

  * Open a third figure window. Add a contour plot, hold the plot, and add a vector arrow plot (octave buit-in function quiver).\\ How are contour lines and gradient vectors related?

==== Solution ====

Download the script by klicking on the file name in the box header. Execute the file in Octave. Compare the results with the gradients determined in the lecture. 

<file matlab surf_cont_grad.m>

% surf_cont_grad
% Plot surface, contour lines and gradient of functions of two variables
% 2015-04-22, Rolf Becker

x = [-2:2]
y = [-1:1]

[XX,YY] = meshgrid(x,y) % explain

%break

x = [-2:0.1:2]; % x values
y = [-1:0.1:1]; % y values

[X,Y] = meshgrid(x,y);

Z=(X.^2).*Y-2*Y; % the function f(x,y)

[dX,dY] = gradient(Z);

figure(1)
surf(X,Y,Z)
xlabel("x"); ylabel("y");zlabel("f(x,y)")

figure(2)
contour(X,Y,Z)
xlabel("x"); ylabel("y");title("Contour lines of f(x,y)")
grid on 

figure(2)
contour(X,Y,Z)
xlabel("x"); ylabel("y");title("Contour lines of f(x,y)")
grid on 

figure(3)
contour(X,Y,Z)
hold on
quiver(X,Y,dX,dY)
hold off
xlabel("x"); ylabel("y");title("Gradient f(x,y)")
grid on 

</file>