Jacobians

Shows the effect of a change of variables in two dimensions on area units, using the Jacobian to approximate the transformed areas.
How to use   ||   Examples   ||   Other Notes


How to Use
The text input fields marked "x=f (u,v)=", "y=g(u,v)=", and "J(u,v)= fugv-fvgu=" can accept a wide variety of expressions to represent functions, and the buttons under the graph allow various manipulations of the graph coordinates. The text input fields marked "u=" and "v=" can accept any decimal numbers. The "Grid" button provides a pop-up window which allows changes to the bounds and spacing of the coordinate grid shown in the left graph, and the transformed grid in the right graph.
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)
g(u,v)=usinh(v)

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.