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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] and let $p \\in M$.\nConsider pairs $(U, f)$, where $U$ is an open [[neighbourhood|0ZCY0Z80B6]] of $p$ and $f \\colon U \\to \\R$ is a [[smooth function|0V33BNKZ9S]].\nTwo pairs $(U, f)$ and $(V, g)$ are equivalent if there is an open neighbourhood $W \\subseteq U \\cap V$ of $p$ with $f = g$ on $W$.\nThe *germ at $p$* of $(U, f)$ is its equivalence class, written $[f]_p$.\nThis is the [[germ|VZXF7RNFKS]] of $f$ at $p$ in the general sense, for smooth real functions.\n\nThe set of germs at $p$ is written $C^\\infty_p(M)$ or $\\mathcal{E}_p$.\nSums and products of germs are defined by $[f]_p + [g]_p = [f + g]_p$ and $[f]_p [g]_p = [fg]_p$, with $f + g$ and $fg$ taken on $U \\cap V$.\n\n## Remarks\n\nWith these operations $\\mathcal{E}_p$ is a commutative $\\R$-algebra.\nIt is a [[local ring|MZ02BKPHJA]]: its unique [[maximal ideal|0PV74TR5FP]] $\\mathfrak{m}_p$ consists of the germs with $f(p) = 0$.\nIndeed, if $f(p) \\ne 0$, then $f \\ne 0$ on a neighbourhood of $p$ and $[1/f]_p$ is an inverse of $[f]_p$.\nThe value $f(p)$ and all partial derivatives of $f$ at $p$ in a chart depend only on the germ.\n\nExample: on $M = \\R$, let $f(x) = x^2$ on $\\R$ and $g(x) = x^2$ on $(-1, 1)$.\nThey agree on $W = (-1, 1)$, so $[f]_0 = [g]_0$.\n\nNon-example: let $f = 0$ on $\\R$, and let $h(x) = e^{-1/x^2}$ for $x \\ne 0$ with $h(0) = 0$.\nAll derivatives of $f$ and $h$ at $0$ agree, but $h(x) \\ne 0$ for every $x \\ne 0$.\nSo $f$ and $h$ agree on no neighbourhood of $0$, and $[f]_0 \\ne [h]_0$.\nA germ is thus finer information than the Taylor series at $p$.\n",
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