In this notes Lecture 2 they have at page 29
One defines $$ \mathcal{E}(M ; E):=\Gamma(E), $$ the space of all smooth sections of $E$ endowed with the following local convex topology. To define it, we choose a cover $\mathcal{U}=\left\{U_{i}\right\}_{i \in I}$ of $M$ by opens which are domains of "total trivializations" of $E$, i.e. both of charts $\left(U_{i}, \kappa_{i}\right)$ for $M$ as well as of trivializations $\tau_{i}:\left.E\right|_{U_{i}} \rightarrow U_{i} \times \mathbb{C}^{p}$ for $E$. This data clearly induce an isomorphism of vector spaces $$ \phi_{i}: \Gamma\left(\left.E\right|_{U_{i}}\right) \rightarrow C^{\infty}\left(\kappa_{i}\left(U_{i}\right)\right)^{p} $$
Where $E$ is a vector bundle.
Suppose that $s$ is a section of this vector bundle. In coordinates $X$ the section is given by $$s(x)=(x^m,c^p(x))$$
Is this isomorphism $\phi$ given by $\phi(s)=c^p(x)$ ?