In Christian Bar's "Geometric Wave Equations" notes it has this definition of a differential operator. I know what $\frac{\partial f}{\partial x^i}$ means when $f:M\rightarrow \mathbb{R}^n$ is a smooth function. But I don't understand what is meant by $\frac{\partial v}{\partial x^i}$ when $v:M\rightarrow E$ is a smooth section (M and E being manifolds).
Any help is appreciated, cheers.
You are choosing a local trivialisation of $E, F$. So when you restrict a section $v:M\to E$ to $U$ it is of the form
$$x\mapsto (x,(v_1(x),...,v_p(x)))$$ with $v_1,...,v_p$ functions $U\to\Bbb K$. Concretely if $\iota: E\lvert _U\to U\times \Bbb K^p$ is the trivialisation then $v_i$ is the $i$-th component of $\iota\circ v$. Now you can apply the partial differentials of the coordinates of $U$ to get a map $U\to\Bbb K^p$, then apply the matrices $A(x)$ at each point and sum it all up to get a map $p:U\to \Bbb K^q$. Now if $\kappa : F\lvert_U\to U\times \Bbb K^q$ is the local trivialisation of $F$ apply $\kappa^{-1}$ to $u\mapsto (u, p(v))$ to get a local section of $F$.
