Differential operator between vector bundles

Definition

Let $M$ be a smooth manifold of dimension $n$, and let $\mathbb{K}$ be $\R$ or $\C$. Let $E$ and $F$ be smooth vector bundles over $M$ with fibres $\mathbb{K}^r$ and $\mathbb{K}^s$. Write $\Gamma(E)$ for the space of smooth sections of $E$. Let $m \ge 0$ be an integer. Consider a linear map $P \colon \Gamma(E) \to \Gamma(F)$. It is a differential operator of order at most $m$ if the following holds. Let $(U, x_1, \ldots, x_n)$ be any chart of $M$ over which $E$ and $F$ are trivial. In the trivialisations, there are smooth $A_\alpha \colon U \to \mathbb{K}^{s \times r}$ for $\abs{\alpha} \le m$ with $$(P u)(x) = \sum_{\abs{\alpha} \le m} A_\alpha(x) \, \partial^\alpha u(x) \quad \text{for all } u \in \Gamma(E) \text{ and } x \in U,$$ where $u$ is written on $U$ as a function $U \to \mathbb{K}^r$. The operator has order $m$ if moreover some $A_\alpha$ with $\abs{\alpha} = m$ is not identically zero.