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  "body": "\n## Definition\n\nLet $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]], and let $f \\colon M \\to N$ be a [[smooth|0V33BNKZ9S]] map.\nThe map $f$ is a *submersion* if for every point $p \\in M$ the [[differential|6P9PXQDSEN]] $df_p \\colon T_pM \\to T_{f(p)}N$ is surjective.\n\n## Remarks\n\nEquivalently, the rank of $df_p$ equals $\\dim N$ at every point $p$; when $M$ is not empty this forces $\\dim M \\ge \\dim N$.\nA point $p$ where $df_p$ is surjective is a regular point, so a submersion is a smooth map all of whose points are regular.\nExample: the projection $\\pi \\colon \\R^3 \\to \\R^2$, $\\pi(x, y, z) = (x, y)$, is a submersion.\nIts differential at every point is the linear map $\\pi$ itself, which is onto.\nNon-example: $f \\colon \\R^2 \\to \\R$, $f(x, y) = x^2 + y^2$, is not a submersion.\nAt the origin $df_0 = 0$, which is not surjective onto $\\R$.\nBy the constant rank theorem, near each point a submersion has the local form $(x_1, \\dots, x_m) \\mapsto (x_1, \\dots, x_n)$ in suitable charts.\n",
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