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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.\nA *Pfaffian system* on $M$ is an [[exterior differential system|901CCKQHVY]] $\\mathcal{I}$ generated, as a differential ideal, by [[one-forms|02Z5B2HA3F]] $\\theta^1, \\dots, \\theta^s$.\nSo $\\mathcal{I}$ is the algebraic ideal generated by $\\theta^1, \\dots, \\theta^s$ and $d\\theta^1, \\dots, d\\theta^s$.\nOne usually asks that $\\theta^1, \\dots, \\theta^s$ are linearly independent at each point.\nThen $\\mathcal{D} = \\set{ v \\in TM : \\theta^a(v) = 0 \\text{ for } a = 1, \\dots, s }$ is a distribution of rank $n - s$, and an immersed submanifold is an integral manifold of $\\mathcal{I}$ exactly when it is tangent to $\\mathcal{D}$.\n\n## Remarks\n\nThe system is *completely integrable* if $d\\theta^a \\wedge \\theta^1 \\wedge \\dots \\wedge \\theta^s = 0$ for $a = 1, \\dots, s$, that is, if each $d\\theta^a$ lies in the algebraic ideal generated by the $\\theta^b$.\nBy the [[Frobenius theorem|5Q4BHBFXJY]], this holds exactly when every point lies on an integral manifold of dimension $n - s$.\n\nExample: on $\\R^3$, the system generated by $\\theta = dz$ has $d\\theta = 0$, so it is completely integrable.\nIts integral surfaces are the planes $z = c$.\n\nExample: on $\\R^3$, let $\\theta = dz - y\\, dx$.\nThen $d\\theta = dx \\wedge dy$ and $\\theta \\wedge d\\theta = dz \\wedge dx \\wedge dy \\ne 0$, so the system is not completely integrable.\nIt has integral curves, such as $t \\mapsto (t, c, ct)$, but no integral surfaces.\n\nNon-example: the exterior differential system on $\\R^3$ generated by the [[two-form|0E3G0GRPZ6]] $dx \\wedge dy$ is not a Pfaffian system.\nIt contains no nonzero one-form, so it is not generated by one-forms.\n",
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