I want to prove that the order of contact of two functions $f, g: M \to N$ between manifolds is well defined.
Definition. Let $f, g : M^{n} \to N^{m}$ be smooth functions. We say $f$ has order of contact $k$ at $p \in M$ with $g$, and write $f \sim_{k} g$ at $p \in M$ if there exist charts $\varphi: U \to \mathbb{R}^n$ and $\psi: V \to \mathbb{R}^m$ around $p \in M$ and $f(p) \in N$ such that $f(p) = g(p)$ and all the derivatives of $f$ and $g$ in the charts up to order $k$ coincide at $p$, that is:
$$\frac{\partial^{|\alpha|}}{\partial r^{\alpha}}(r^j \circ \psi \circ f \circ \varphi^{-1})(\varphi(p)) = \frac{\partial^{|\alpha|}}{\partial r^{\alpha}}(r^j \circ \psi \circ g \circ \varphi^{-1})(\varphi(p))$$
for every multi-index $\alpha = (\alpha_1, \cdots, \alpha_n)$ (where $|\alpha| = \alpha_1 + \cdots + \alpha_n$) and every $1 \leq j \leq m$, where we define:
$$\frac{\partial^{|\alpha|}}{\partial r^{\alpha}} \doteq \frac{\partial^{|\alpha|}}{\partial r^{\alpha_1} \cdots \partial r^{\alpha_n}}$$
and $r^j: \mathbb{R}^m \to \mathbb{R}$ is the projection onto the $j$-th coordinate and $\frac{\partial}{\partial r^{k}}$ is just the ordinary (Euclidean) $k$-th partial derivative.
I'm trying to prove this definition does not depend on charts, that is, if the above is satisfied in charts $\varphi = (x^1, \cdots, x^n), \psi = (y^1, \cdots, y^m)$, then it's also satisfied in any other charts $\tilde{\varphi} = (\tilde{x}^1, \cdots, \tilde{x}^n), \tilde{\psi} = (\tilde{y}^1, \cdots, \tilde{y}^m)$. I've been able to prove it is well defined if $k = 1$ or $k = 2$ by using the chain rule in the form of the following formulas (where I'm using Einstein notation):
$$\frac{\partial}{\partial y^i} = \frac{\partial x^{\ell}}{\partial y^i} \frac{\partial}{\partial x^{\ell}}$$
$$\frac{\partial^2}{\partial y^{i} \partial y^j} = \frac{\partial^2 x^{\ell}}{\partial y^i \partial y^j} \frac{\partial}{\partial x^{\ell}} + \frac{\partial x^{\ell}}{\partial y^j} \frac{\partial x^k}{\partial y^i} \frac{\partial}{\partial x^k}$$
and I think the general case has to be done by induction, but I'm having trouble actually using the inductive hypothesis, since I'd have to obtain a relation between the $k$-th partial derivatives in relation to different charts (which in turn I'd also have to use induction to obtatin, but I'm failing to see the pattern in order to think of the formula I'd want to prove for this general case). I'd really appreciate some help on this! Thanks in advance.
It's not really necessary to write down an explicit formula. Instead one can note that every term in the expression for $\frac{\partial^p f^j}{\partial y^{i_1}\cdots\partial y^{i_p}}$ is of the form $$ a\frac{\partial^q f^j}{\partial x^{k_1}\cdots\partial x^{k_q}} $$ where $a\in C^\infty M$ does not depend on $f$ and $q\le p$. Taking another partial derivative (w.r.t. $x$ or $y$)of such a term will produce more terms of the same form, and the order of the partial derivatives of $f$ will increase by at most $1$.