Definition 1 (Germ)   Let $M$ and $N$ be manifolds and $x \in M$ . We consider all smooth mappings $f: U_f \to N$ , where $U_f$ is some open neighborhood of $x$ in $M$ . We define an equivalence relation on the set of mappings considered, and we put $f \underset{x}{\sim} g$ if there is some open neighborhood $V$ of $x$ with $f|_V = g|_V$ . The equivalence class of a mapping $f$ is called the germ of f at x, denoted by $\overline{f}$ or, sometimes, $germ_x f$ , and we write $$\overline{f}:(M,x) \to (N,f(x))$$

Remark 1   Germs arise naturally in differential topolgy. It is very convenient when dealing with derivatives at the point $x$ , as every mapping in a germ will have the same derivative values and properties in $x$ , and hence can be identified for such purposes: every mapping in a germ gives rise to the same tangent vector of $M$ at $x$ .