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  "body": "\n## Definition\n\nLet $\\pi \\colon E \\to M$ be a smooth [[fiber bundle|CX4M8K61AB]], let $k \\ge 0$, and let $\\pi_k \\colon J^k\\pi \\to M$ be its $k$-th [[jet bundle|HC3VBBCJQP]].\nLet $s \\colon U \\to E$ be a smooth local section of $\\pi$ over an open set $U \\subseteq M$.\nThe *$k$-th prolongation* of $s$ is the map\n$$j^k s \\colon U \\to J^k\\pi, \\qquad p \\mapsto j^k_p s,$$\nwhere $j^k_p s$ is the [[$k$-jet|1QE0Z0756D]] of $s$ at $p$.\nIt is a smooth local section of $\\pi_k$.\n\nA local section $\\sigma$ of $\\pi_k$ is *holonomic* if $\\sigma = j^k s$ for some local section $s$ of $\\pi$.\n\n## Remarks\n\nIn adapted coordinates $(x^i, u^\\alpha, u^\\alpha_I)$, if $s$ has components $s^\\alpha(x)$, then\n$$j^k s(x) = \\Big(x^i,\\ s^\\alpha(x),\\ \\frac{\\partial^{\\abs{I}} s^\\alpha}{\\partial x^I}(x)\\Big)_{1 \\le \\abs{I} \\le k}.$$\nIf $\\sigma = j^k s$, then $s = \\pi_{k,0} \\circ \\sigma$, so $s$ is determined by $\\sigma$.\n\nExample: for $\\pi \\colon \\R \\times \\R \\to \\R$ and $s(x) = x^2$, the first prolongation is $j^1 s(x) = (x, x^2, 2x)$.\n\nNon-example: the section $\\sigma(x) = (x, 0, 1)$ of $J^1\\pi \\to \\R$ for the same bundle is not holonomic.\nIf $\\sigma = j^1 s$, then $s = \\pi_{1,0} \\circ \\sigma = 0$, so $j^1 s(x) = (x, 0, 0) \\ne \\sigma(x)$.\n\nA [[partial differential equation|GC5BDZ6QBC]] of order $k$ for sections of $\\pi$ can be given as a subset $\\mathcal{R} \\subseteq J^k\\pi$.\nA local section $s$ is a solution exactly when $j^k s$ takes its values in $\\mathcal{R}$.\n",
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