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  "body": "\n## Example\n\nThe *Whitney umbrella* is the surface in $\\R^3$ with equation\n$$x^2 = y^2 z.$$\nIt contains the image of the smooth map $f \\colon \\R^2 \\to \\R^3$, $f(u, v) = (uv, u, v^2)$, since $(uv)^2 = u^2 \\cdot v^2$.\n\nCheck of the singularity: the [[differential|6P9PXQDSEN]] of $f$ has matrix\n$$\\begin{pmatrix} v \u0026 u \\\\ 1 \u0026 0 \\\\ 0 \u0026 2v \\end{pmatrix}.$$\nIts first two rows show that it has rank $2$ unless $u = 0$.\nFor $u = 0$, its first and third rows show that it has rank $2$ unless $v = 0$.\nSo the origin is the only [[singular point|1JG3HFWX3T]] of $f$, and there the rank is $1$.\n\nThe points $(0, 0, z)$ with $z \u003e 0$ have the two preimages $(0, \\pm \\sqrt{z})$, so the image crosses itself along the positive $z$-axis.\nThe origin has the single preimage $(0, 0)$ and is called the pinch point.\nThe points $(0, 0, z)$ with $z \u003c 0$ satisfy the equation but are not in the image of $f$; they form the handle of the umbrella.\n\n## Remarks\n\nThe map germ of $f$ at the origin is the cross-cap.\nWhitney showed that, besides immersion germs, the cross-cap is the only singularity of a generic smooth map from a surface into $3$-space, and that it is stable.\n\n## Explanations\n\n### L1 human\n\nTake a sheet, pinch one point and let the sheet pass through itself along a line starting at that point.\nThe result looks like an umbrella that has been folded and pushed through itself, with a stick, the handle, sticking out the other side.\nThe pinch point is where the map from the plane fails to be an immersion.\nA generic smooth map of a surface into space has only isolated pinch points of this kind as singular points.\n",
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