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  "body": "\n## Definition\n\nLet $\\pi \\colon E \\to M$ be a smooth [[fiber bundle|CX4M8K61AB]] over a [[smooth manifold|F45VQ47GPV]] $M$, and let $k \\ge 0$ be an integer.\nA local section of $\\pi$ is a smooth map $s \\colon U \\to E$, with $U \\subseteq M$ open, such that $\\pi \\circ s = \\id_U$.\nFor $p \\in U$, let $j^k_p s$ be the [[$k$-jet|1QE0Z0756D]] of $s$ at $p$, viewed as a map from $M$ to $E$.\nThe *$k$-th jet manifold* of $\\pi$ is the set\n$$J^k \\pi = \\set{ j^k_p s : p \\in M,\\ s \\text{ a local section of } \\pi \\text{ defined near } p }.$$\nIt comes with the maps $\\pi_k \\colon J^k \\pi \\to M$, $j^k_p s \\mapsto p$, and $\\pi_{k,0} \\colon J^k\\pi \\to E$, $j^k_p s \\mapsto s(p)$.\n\nLet $(x^1, \\dots, x^n, u^1, \\dots, u^m)$ be adapted coordinates on an open set of $E$, so that $x^i$ are coordinates on $M$ and $u^\\alpha$ are fiber coordinates.\nThey induce coordinates $(x^i, u^\\alpha, u^\\alpha_I)$ on $J^k\\pi$, where $I$ runs over multi-indices with $1 \\le \\abs{I} \\le k$ and\n$$u^\\alpha_I(j^k_p s) = \\frac{\\partial^{\\abs{I}} s^\\alpha}{\\partial x^I}(p).$$\nThese charts make $J^k\\pi$ a smooth manifold.\nWith this structure, $\\pi_k \\colon J^k\\pi \\to M$ is called the *$k$-th jet bundle* of $\\pi$.\n\n## Remarks\n\nThe maps $\\pi_k$ and $\\pi_{k,0}$, and more generally the maps $J^k\\pi \\to J^l\\pi$ for $l \u003c k$ that forget the higher derivatives, are smooth fiber bundles.\nIf $E$ is a [[vector bundle|P4DZM975BV]], then $J^k\\pi \\to M$ is again a vector bundle.\n\nExample: for the trivial bundle $\\pi \\colon \\R \\times \\R \\to \\R$, $(x, u) \\mapsto x$, the local sections are graphs of functions $u = s(x)$.\nThe first jet manifold has the global coordinates $(x, u, u_x)$, so $J^1\\pi \\cong \\R^3$, and $j^1_p s = (p, s(p), s'(p))$.\n\nNon-example: the $k$-jets of all smooth maps $M \\to E$, not only of sections, form the larger space $J^k(M, E)$.\nA jet of a map $f$ with $\\pi \\circ f \\ne \\id$ near $p$ is not in $J^k\\pi$, because the definition admits only sections.\n",
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