Let $P:\Gamma(\Bbb R^n, E_0) \rightarrow \Gamma(\Bbb R^n, E_1)$ be a differential operator where $E_0$, $E_1$ are trivial vector bundles, with the standard bundle metric over $\Bbb R^n$.
In page 28 of the notes, the author writes,
We can write $P = \sum_{|\alpha| \le k } A^\alpha(x) D^\alpha $ with some smooth matrix-valued functions $A^\alpha$ and a composition of vector fields.
What does this mean explicitly? Why is this a composition of vector fields?