How to Use
Enter a change of variables from (u,v) coordinates to (x,y) coordinates (in the form x=f (u,v) and y=g(u,v)) into the text input fields marked "x=f(u,v)=" and "y=g(u,v)=".

    Example: x=f (u,v)=u-v, y=g(u,v)=u+v

Enter the Jacobian J(u,v)=fu gv - fv gu into the text input field marked "J(u,v)=fugv-fvgu=".

    Example: J(u,v)=2

Click the "Graph" button (this button also refreshes the graph).

Move the blue rectangle in the left graph.

    Example: (u,v)=(2,3)

This can be done in any of 3 ways:

(The mouse has no effect in the right graph.)

To erase the graph and all input fields, click the "Clear" button.


Examples
 Rectilinear:
x=f (u,v)=u-v
y=g(u,v)=u+v

 Polar:
f (u,v)=ucos(v)
g(u,v)=usin(v)

 Hyperbolic:
f (u,v)=ucosh(v) =u(ev+e-v)/2
g(u,v)=usinh(v) =u(ev-e-v)/2


Other Notes
The area of the blue rectangle in the left graph is shown in the field marked "Left Area=".

With the point (u,v), shown in the left graph by the position of the red point, the value of the Jacobian J(u,v) provides a way to approximate the area of the blue region in the right graph, as the product of the left area and J(u,v), shown in the field marked "Area*J(u,v)=". The approximation is more accurate for smaller values of the area in the left graph.