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  "body": "\n## Definition\n\nLet $M$ be a compact [[smooth manifold|F45VQ47GPV]] without boundary, and let $E$ and $F$ be smooth [[vector bundles|P4DZM975BV]] over $M$.\nLet $P \\colon \\Gamma(E) \\to \\Gamma(F)$ be a [[differential operator|CGBX1N9V8H]] of order $m$ with [[principal symbol|S8GC34D5ZG]] $\\sigma_m(P)$.\nThe operator $P$ is *elliptic* if the linear map\n$$\\sigma_m(P)(x, \\xi) \\colon E_x \\to F_x$$\nis invertible for every $x \\in M$ and every nonzero $\\xi \\in T^*_x M$.\n\n## Remarks\n\nWhen $\\dim M \\ge 1$, the condition forces $E$ and $F$ to have the same rank, since an invertible linear map exists only between spaces of equal dimension.\nOn an open subset of $\\R^n$ with $E$ and $F$ trivial of rank $1$, it is the usual condition for an [[elliptic operator|XY8GEQSWPR]].\n\nExample: let $(M, g)$ be a closed Riemannian manifold and let $\\Delta = -\\operatorname{div} \\operatorname{grad}$ be the [[Laplace–Beltrami operator|F4NWZNTRMD]] on functions.\nIts principal symbol is $\\sigma_2(\\Delta)(x, \\xi) = -\\abs{\\xi}_g^2$, which is a nonzero number for $\\xi \\ne 0$, so $\\Delta$ is elliptic.\n\nNon-example: on a closed manifold $M$ of dimension $n \\ge 2$, the [[exterior derivative|0ZHSTAG8ZY]] $d$ from functions to $1$-forms is not elliptic.\nIts symbol at $\\xi$ is the map $\\R \\to T^*_x M$, $c \\mapsto c \\, \\xi$, which is injective but not surjective, since $T^*_x M$ has dimension $n \u003e 1$.\n\n## Explanations\n\n### L1 human\n\nAn operator on bundles over a closed manifold is elliptic when its top-order part is an invertible matrix in every nonzero frequency direction.\n",
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