germ of smooth functions

If $x$ is a point on a smooth manifold $M$ , then a germ of smooth functions near $x$ is represented by a pair $(U,f)$ where $U\\subseteq M$ is an open neighbourhood of $x$ , and $f$ is a smooth function $U\\rightarrow\\mathbb{R}$ . Two such pairs $(U,f)$ and $(V,g)$ are considered equivalent if there is a third open neighbourhood $W$ of $x$ , contained in both $U$ and $V$ , such that $f|_{W}=g|_{W}$ . To be precise, a germ of smooth functions near $x$ is an equivalence class of such pairs.

In more fancy language: the set $\\mathcal{O}_{x}$ of germs at $x$ is the stalk at $x$ of the sheaf $\\mathcal{O}$ of smooth functions on $M$ . It is clearly an $\\mathbb{R}$ -algebra.

Germs are useful for defining the tangent space $T_{x}M$ in a coordinate-free manner: it is simply the space of all $\\mathbb{R}$ -linear maps $X:\\mathcal{O}_{x}\\rightarrow\\mathbb{R}$ satisfying Leibniz’ rule $X(fg)=X(f)g+fX(g)$ . (Such a map is called an $\\mathbb{R}$ -linear derivation of $\\mathcal{O}_{x}$ with values in $\\mathbb{R}$ .)

单词	GermOfSmoothFunctions
释义	germ of smooth functions If $x$ is a point on a smooth manifold $M$ , then a germ of smooth functions near $x$ is represented by a pair $(U,f)$ where $U\\subseteq M$ is an open neighbourhood of $x$ , and $f$ is a smooth function $U\\rightarrow\\mathbb{R}$ . Two such pairs $(U,f)$ and $(V,g)$ are considered equivalent if there is a third open neighbourhood $W$ of $x$ , contained in both $U$ and $V$ , such that $f\|_{W}=g\|_{W}$ . To be precise, a germ of smooth functions near $x$ is an equivalence class of such pairs. In more fancy language: the set $\\mathcal{O}_{x}$ of germs at $x$ is the stalk at $x$ of the sheaf $\\mathcal{O}$ of smooth functions on $M$ . It is clearly an $\\mathbb{R}$ -algebra. Germs are useful for defining the tangent space $T_{x}M$ in a coordinate-free manner: it is simply the space of all $\\mathbb{R}$ -linear maps $X:\\mathcal{O}_{x}\\rightarrow\\mathbb{R}$ satisfying Leibniz’ rule $X(fg)=X(f)g+fX(g)$ . (Such a map is called an $\\mathbb{R}$ -linear derivation of $\\mathcal{O}_{x}$ with values in $\\mathbb{R}$ .)
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