In this example we see the transformation of the rectangle \( S = [1,2] \times [0,\pi] \) in the \(r,\theta\)-plane using
\( x= r\cos(\theta),\ y=r\sin(\theta) \)
In this example we see the transformation of the square \( S = [0,1] \times [0,1] \) in the \(u,v\)-plane using
\( x= u + v ,\ y= u-v \)
In this example we see the transformation of the triangle\( S \) with vertices \( (0,0)\ (1,0), (1,1) \) in the \(u,v\)-plane using
\( x= u^2-v^2,\ y=2uv \)