## You are here

Homegerm of smooth functions

## Primary tabs

# 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 $\mathbb{R}$.)

## Mathematics Subject Classification

53B99*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new image: information-theoretic-distributed-measurement-4.2 by rspuzio

new image: information-theoretic-distributed-measurement-4.1 by rspuzio

new image: information-theoretic-distributed-measurement-3.2 by rspuzio

new image: information-theoretic-distributed-measurement-3.1 by rspuzio

new image: information-theoretic-distributed-measurement-2.1 by rspuzio

Apr 19

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz