I'm trying to study the book "Manifolds of Differentiable Mappings" written by Michor and I came across the following:
He considers two smooth manifolds $M$ and $N$ and define an equivalence relation $\simeq$ on $C^\infty(M,N)\times M$ as follows:
$(f, p)\simeq (g, q)$ if and only if $p=q$ and $f$ and $g$ have the same Taylor development of order $k$ at $p$ in some pair of coordinate charts about $p$ and $f(p)$, respectively.
Can anyone explain me this?
More precisely, how is the Taylor development of order $k$ at $p$ is define for a function $f:M\longrightarrow N$ between manifolds?
References will be welcome..