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
How to Use
The text input fields
marked "x=f (u,v)=", "y=g(u,v)=",
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.
- 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)="
- Enter the Jacobian
J(u,v)=fu gv - fv gu
into the text input field marked
- Click the "Graph" button
(this button also refreshes the graph)
- Move the blue rectangle in the left graph
This can be done in any of 3 ways:
(The mouse has no effect in the right graph)
- 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
- To erase the graph and all input fields, click the
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.