Consider the parabolic transformation [math]x=u^2-v^2[/math] and [math]y=2uv[/math].[br][br]The Jacobian establishes a relationship between the area of the [color=#0000ff]blue[/color] square (left) and the area of the [color=#ff0000]red[/color] region (right). [br][br][b]Instructions:[/b][br][list][*]Drag the [color=#38761d]green[/color] point (on the left) to change the position of the [color=#0000ff]blue[/color] square. Observe what happens to the [color=#ff0000]red[/color] region.[/*][*]Drag the slider [math]\Delta[/math], which determines the side of the [color=#0000ff]blue[/color] square. See what happens to [math]\Delta A=\text{Jacobian}\times\text{Area of square}[/math][i],[/i] for small values of [math]\Delta[/math].[br][/*][*]Zoom in or out, if necessary.[br][/*][/list]