### 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
• 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:
• Enter the values directly into the text fields marked "u=" and "v=" and click the "Point" button to redraw the blue rectangle in the new left graph position (without recomputing the transformation)
• Use the mouse to click and drag the red spot at the lower left corner of the blue rectangle in the left graph
• Use the mouse to single-click anywhere in the left graph, moving the blue rectangle to the grid rectangle in which the mouse was clicked
(The mouse has no effect in the right graph)
• To erase the graph and all input fields, click the "Clear" button
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.