Definition 1 (Germ)   Let and be manifolds and . We consider all smooth mappings , where is some open neighborhood of in . We define an equivalence relation on the set of mappings considered, and we put if there is some open neighborhood of with . The equivalence class of a mapping is called the germ of f at x, denoted by or, sometimes, , and we write

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