# 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 $U\beta \x8a\x86M$ is an open neighbourhood of $x$, and $f$ is a smooth function^{} $U\beta \x86\x92\mathrm{\beta \x84\x9d}$. 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 ${\mathrm{\pi \x9d\x92\u037a}}_{x}$ of germs at $x$ is the stalk at $x$ of the sheaf $\mathrm{\pi \x9d\x92\u037a}$ of smooth functions on $M$. It is clearly an $\mathrm{\beta \x84\x9d}$-algebra.

Germs are useful for defining the tangent space^{} ${T}_{x}\beta \x81\u2019M$ in a coordinate-free manner: it is simply the space of all $\mathrm{\beta \x84\x9d}$-linear maps $X:{\mathrm{\pi \x9d\x92\u037a}}_{x}\beta \x86\x92\mathrm{\beta \x84\x9d}$ satisfying Leibnizβ rule $X\beta \x81\u2019(f\beta \x81\u2019g)=X\beta \x81\u2019(f)\beta \x81\u2019g+f\beta \x81\u2019X\beta \x81\u2019(g)$. (Such a map is called an $\mathrm{\beta \x84\x9d}$-linear derivation of ${\mathrm{\pi \x9d\x92\u037a}}_{x}$ with values in $\mathrm{\beta \x84\x9d}$.)

Title | germ of smooth functions |
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Canonical name | GermOfSmoothFunctions |

Date of creation | 2013-03-22 13:05:08 |

Last modified on | 2013-03-22 13:05:08 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 4 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 53B99 |