Germ
Germs are an Equivalence class of an Equivalence relation, rather than the usual way of denoting equiv classes we use [math][f]_p[/math] for the germ of [ilmath]f[/ilmath] at [ilmath]p[/ilmath]
Contents
Definition
A germ is an equivalence class of an equivalence relation defined as follows:
Given a point [ilmath]p\in\mathbb{R}^n[/ilmath], we define an equivalence relation on the [math]C^\infty[/math] functions defined in some neighbourhood of [ilmath]p[/ilmath] as:
[math](f,U)\equiv(g,V)[/math] if:
- For some [math]W\subseteq U\cap V[/math] (where [math]W[/math] is open) that [math]x\in W\implies f(x)=g(x)[/math] - that is [ilmath]f[/ilmath] and [ilmath]g[/ilmath] agree when restricted to [ilmath]W[/ilmath]
The equivalence class of [ilmath](f,U)[/ilmath] is the germ of [ilmath]f[/ilmath] at [ilmath]p[/ilmath] - See notation below before writing [math][(f,U)][/math] for germs
Where:
- [ilmath]U[/ilmath] is a neighbourhood of [ilmath]p[/ilmath]
- [math]f:U\rightarrow\mathbb{R}[/math] which is [math]C^\infty[/math] (smooth)
- (same for V as U)
- (same for g as f)
Notation
With equiv relations it is customary to write equivalence classes using "[" and "]" around a representative item, however: [math][(f,U)][/math] makes no mention of [ilmath]p[/ilmath] and the neighbourhood is hardly important (as indeed any neighbourhood will do!)
That is to say the equivalence class is purely determined by the point and the function. As such
We denote germs as: [math][f]_p[/math]
