I have some problems with the definition of jets and it would be great if someone could help me here:
In many books it is written, that the $r-th$ order jet $j^r_xf$ of a smooth function $f:M \rightarrow N$ between smooth manifolds 'depends only on the germ of $f$ at x'.
What does it mean?
First I thought this means that all functions in the equivalence class $j^r_xf$ have the same germ at $x$, but this is wrong as the following counterexample shows:
Let
$ \begin{eqnarray} f: \mathbb{R} &\rightarrow& \mathbb{R}^2 \\ x &\mapsto& (x,x) \end{eqnarray}$
and
$ \begin{eqnarray} g: \mathbb{R} &\rightarrow& \mathbb{R}^2 \\ x &\mapsto& (x,\sin(x)) \end{eqnarray} .$
Then $f$ and $g$ have the same first order jet at zero, that is $f,g \in j^1_0f$, but they don't define the same germ at $x$ since there is no neighbourhood of $(0,0)$ in $\mathbb{R}^2$ such that $f$ equals $g$ on that neighbourhood.