### 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
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Examples
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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*)=*f*_{u} g_{v} - *f*_{v} g_{u}
into the text input field marked
"*J*(*u*,*v*)=
*f*_{u}*g*_{v}-*f*_{v}*g*_{u}="

(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*)=
*f*_{u}*g*_{v}-*f*_{v}*g*_{u}="
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**

**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.