Can you give me an example of a transformation of a plane which is invertible but still has a jacobian vanishing somewhere

Question

Note that I'm not asking for a linear transformation. A linear transformation which is invertible would have a non vanishing jacobian for sure but not true for transformations in general. Any examples?

Answer 1

Consider for example $f(x,y) = (x^3,y)$. Then $f$ is smooth, one-to-one and onto but

$$ Df|_{(x,y)} = \begin{pmatrix} 3x^2 & 0 \\ 0 & 1 \end{pmatrix} $$

is not invertible at $(x,y) = (0,0)$.