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  "body": "\n## Definition\n\nLet $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]], let $p \\in M$ and let $k \\ge 0$ be an integer.\nLet $f$ and $g$ be [[smooth maps|0V33BNKZ9S]] into $N$, each defined on an open neighbourhood of $p$ in $M$.\nThen $f$ and $g$ have the *same $k$-jet at $p$* if both conditions hold.\n\n1. $f(p) = g(p)$.\n2. Take a chart $x$ of $M$ around $p$ and a chart $y$ of $N$ around $f(p)$. Then $y \\circ f \\circ x^{-1}$ and $y \\circ g \\circ x^{-1}$ have equal [[partial derivatives|09EGC6F8PT]] of every order $1, \\dots, k$ at $x(p)$.\n\nThis is an equivalence relation.\nThe *$k$-jet of $f$ at $p$*, written $j^k_p f$, is the equivalence class of $f$.\nThe set of all $k$-jets at $p$ of maps from $M$ to $N$ is written $J^k_p(M, N)$.\n\n## Remarks\n\nCondition 2 does not depend on the charts, by the chain rule.\nIndeed, the derivatives up to order $k$ of a composite depend only on those of its factors.\nThe jet $j^k_p f$ depends only on the [[germ|7SWSRGDT4A]] of $f$ at $p$.\nFor $M = \\R^n$ and $N = \\R$, the $k$-jet of $f$ at $p$ is the same data as its [[Taylor polynomial|0ZZ4RHTR9X]] of degree $k$ at $p$.\n\nExample: on $\\R$, $f(x) = \\sin x$ and $g(x) = x$ have the same $2$-jet at $0$.\nIndeed $f(0) = g(0) = 0$, $f'(0) = g'(0) = 1$ and $f''(0) = g''(0) = 0$.\n\nNon-example: $f(x) = e^x$ and $g(x) = 1 + x$ have the same $1$-jet at $0$ but not the same $2$-jet.\nCondition 2 fails for order $2$, since $f''(0) = 1$ and $g''(0) = 0$.\n",
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