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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] and let $\\Omega(M) = \\bigoplus_{k \\ge 0} \\Omega^k(M)$ be the algebra of smooth [[differential forms|0E3G0GRPZ6]] with the [[wedge product|0GRAATMGP1]].\nAn *exterior differential system* on $M$ is a subset $\\mathcal{I} \\subseteq \\Omega(M)$ with these properties.\n\n1. $\\mathcal{I}$ is a two-sided [[ideal|096Z9MP2ME]]: it is closed under addition, and $\\alpha \\wedge \\varphi \\in \\mathcal{I}$ for all $\\alpha \\in \\Omega(M)$ and $\\varphi \\in \\mathcal{I}$.\n2. $\\mathcal{I}$ is graded: if $\\varphi \\in \\mathcal{I}$, then each homogeneous component of $\\varphi$ is in $\\mathcal{I}$.\n3. $\\mathcal{I}$ is closed under the [[exterior derivative|0ZHSTAG8ZY]]: $d\\varphi \\in \\mathcal{I}$ for all $\\varphi \\in \\mathcal{I}$.\n\nAn ideal with these properties is also called a differential ideal.\nAn *integral manifold* of $\\mathcal{I}$ is an [[immersion|6RW68JGBHC]] $\\iota \\colon N \\to M$ with $\\iota^* \\varphi = 0$ for all $\\varphi \\in \\mathcal{I}$.\n\n## Remarks\n\nUsually $\\mathcal{I}$ is given by finitely many generators $\\varphi_1, \\dots, \\varphi_r$.\nThe differential ideal they generate is the algebraic ideal generated by the $\\varphi_a$ and the $d\\varphi_a$.\nAdding the $d\\varphi_a$ changes no integral manifold, because $\\iota^* d\\varphi_a = d(\\iota^* \\varphi_a) = 0$ whenever $\\iota^* \\varphi_a = 0$.\n\nExample: on $M = \\R^3$ with coordinates $(x, u, p)$, let $\\mathcal{I}$ be generated by $\\theta = du - p\\, dx$ and $d\\theta = dx \\wedge dp$.\nFor every smooth function $f$, the curve $x \\mapsto (x, f(x), f'(x))$ is an integral manifold: $\\theta$ pulls back to $(f' - f')\\, dx = 0$, and every $2$-form pulls back to $0$ on a curve.\n\nNon-example: the algebraic ideal generated by $\\theta$ alone fails clause 3.\nEvery element of it has the form $\\alpha \\wedge \\theta$, so every $2$-form in it is zero after wedging with $\\theta$.\nBut $\\theta \\wedge d\\theta = du \\wedge dx \\wedge dp \\ne 0$, so $d\\theta$ is not in that ideal.\n",
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