The question is motivated from the definition of $C^r(\Omega)$ I learned from S.S.Chern's Lectures on Differential Geometry:
Suppose $f$ is a real-valued function defined on an open set $\Omega\subset{\bf R}^m$. If all the $k$-th order partial derivatives of $f$ exist and are continuous for $k\leq r$, then we say $f\in C^r(\Omega)$. Here $r$ is some positive integer.
While in Folland's Introduction to Partial Differential Equations, $C^r(\Omega)$ denotes the space of functions possessing continuous derivatives up to order $r$ on $\Omega$, where $\Omega$ is an open subset of ${\bf R}^m$ and $k$ is a positive integer.
It's trivial to show that these two definitions are equivalent when $m=1$. So here is my question:
Are these two definitions equivalent in the higher dimensions? How to prove it?
Edit: The question was partially answered by Didier here. I do not think it is trivial for me. It boils down to the following one:
In the higher dimension case, say, $f:{\bf R}^n\to {\bf R}$($n\geq 2$), what's the relationship between the higher order partial derivative of $f$ and the high order derivative $f^{(k)}$?
When $1\leq k\leq 2$, $f^{(k)}$ is the gradient and the Hessian matrix respectively. The answer to the above questions in these two cases is clear. I have no idea for $k>2$. I don't know much about tensor, I'm not sure if the question is related to the topic of multilinear algebra.