Consider the parabolic transformation and x=u2-v2 and y=2uv
The Jacobian establishes a relationship between the area of the blue square (left) and the area of the red region (right).
- Drag the green point (on the left) to change the position of the blue square. Observe what happens to the red region.
- Drag the slider , which determines the side of the blue square. See what happens to △A=Jacobian⨯ Area of square, for small values of △.
- Zoom in or out, if necessary.
Credit: Juan Carlos Ponce Campuzano @GeoGebra