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