Taylor development on manifolds and Manifolds of differentiable Mappings?

Question

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..

Thanks

Answer 1

You don't need to define the Taylor development for functions $M \to N$ since you're working in the coordinate charts he mentions in the definition. For example if the coordinate chart about $p$ is $\phi : U \subset M \to \mathbb R^m$ and likewise $\psi : V \subset N \to \mathbb R^n$ about $f(p)$ then he is talking about the Taylor polynomials of the maps $$\psi \circ f \circ\phi^{-1}, \psi \circ g \circ \phi^{-1} : \phi(U) \to \mathbb R^n.$$ These are just functions between Euclidean spaces, so you can take the component-wise Taylor expansion as usual.