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  "body": "\n## Definition\n\nLet $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]] of dimensions $m$ and $n$, and let $f \\colon M \\to N$ be a smooth map.\nFor $p \\in M$, let $df_p \\colon T_p M \\to T_{f(p)} N$ be the [[differential|6P9PXQDSEN]] of $f$ at $p$.\nThe point $p$ is a *singular point*, or *singularity*, of $f$ if the [[rank|0ZZXSBTR4J]] of $df_p$ is less than $\\min(m, n)$.\nOtherwise $p$ is a *regular point* of $f$.\nThe set of singular points of $f$ is denoted $\\Sigma(f)$.\n\n## Remarks\n\nExample: $f \\colon \\R^2 \\to \\R^2$, $f(x, y) = (x, y^2)$, has $df_{(x,y)} = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 2y \\end{pmatrix}$, of rank $1 \u003c 2$ exactly when $y = 0$.\nSo $\\Sigma(f)$ is the line $y = 0$, along which $f$ folds the plane.\nNon-example: for $f \\colon \\R \\to \\R^2$, $f(t) = (t, t^2)$, the differential $df_t$ has matrix $(1, 2t)^T$ of rank $1 = \\min(1, 2)$ at every $t$, so $f$ has no singular points.\nA point is regular if and only if $f$ is an [[immersion|6RW68JGBHC]] near it (when $m \\le n$) or a [[submersion|BMY55EW3RF]] near it (when $m \\ge n$).\nFor $N = \\R$ the singular points are the [[critical points|0YTCMFB4HC]] of $f$, where $df_p = 0$.\nSome authors call every point with $df_p$ not surjective a critical point; this agrees with the definition above when $m \\ge n$.\n",
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