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  "body": "\n## Summary\n\nA singularity, or singular point, of a smooth map between [[smooth manifolds|F45VQ47GPV]] is a point where the [[differential|6P9PXQDSEN]] has less than the largest possible rank.\nAt all other points the map is locally a linear projection or a linear inclusion, by the rank theorem.\nSo the singular points are the only places where the local shape of a map can be interesting.\nTheir study and classification is the subject of [[singularity theory|42V5XGZS4F]] of smooth maps.\n\n## Explanations\n\n### L1 human\n\nThink of a smooth map as a way of projecting, folding or pressing one space onto another.\nWhere the map is regular, it looks like a flat projection, with nothing special happening nearby.\nA singular point is a place where the map folds, pinches or crushes directions together.\nFor example, the map $x \\mapsto x^2$ from the line to the line folds the line in half at $0$, and $0$ is its only singular point.\nShadows of curved objects show the same thing: the outline of the shadow is the image of the singular points of the projection.\n",
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