Whitney umbrella

Example

The Whitney umbrella is the surface in $\R^3$ with equation $$x^2 = y^2 z.$$ It contains the image of the smooth map $f \colon \R^2 \to \R^3$, $f(u, v) = (uv, u, v^2)$, since $(uv)^2 = u^2 \cdot v^2$.

Check of the singularity: the differential of $f$ has matrix $$\begin{pmatrix} v & u \\ 1 & 0 \\ 0 & 2v \end{pmatrix}.$$ Its first two rows show that it has rank $2$ unless $u = 0$. For $u = 0$, its first and third rows show that it has rank $2$ unless $v = 0$. So the origin is the only singular point of $f$, and there the rank is $1$.

The points $(0, 0, z)$ with $z > 0$ have the two preimages $(0, \pm \sqrt{z})$, so the image crosses itself along the positive $z$-axis. The origin has the single preimage $(0, 0)$ and is called the pinch point. The points $(0, 0, z)$ with $z < 0$ satisfy the equation but are not in the image of $f$; they form the handle of the umbrella.