I have a clarifying question about sections of vector bundles. Let $\pi:E\to M$ be a smooth vector bundle of rank $n$, and $\sigma:M\to E$ be some map. I want to show that $\sigma$ is a section of $E$.
First, I want to show that $(\pi\circ\sigma)(p)=p$ for all $p\in M$.
Second, I want to show that $\sigma$ is a smooth map. Since $E$ is a smooth vector bundle over $M$ of rank $n$, then there exists an open cover $\{U_i\}$ of $M$ s.t. for all i, there is a diffeomorphism $$\varphi_i:\pi^{-1}(U_i)\to U_i\times\mathbb{R}^n$$ So, in order to show that $\sigma$ is smooth, I want to show that $\sigma|_{U_i}$ is smooth for all $i$ where $$\sigma|_{U_i}:=\varphi_i\circ\sigma:U_i\to U_i\times\mathbb{R}^n.$$ To show that $\sigma|_{U_i}$ is smooth, it's enough to show that $\sigma|_{U_i}$ is smooth on open charts of $M$ i.e. $$(\psi_i\times id)\circ \sigma|_{U_i}\circ \psi_i^{-1}:\hat{U_i}\to \hat{U_i}\times\mathbb{R}^n\text{ is smooth for all }i$$ where, assuming that $\dim(M)=k$, $\psi_i:U\to \hat{U}\subset \mathbb{R}^k$ is a homeomorphism. If the original $U_i$ is big enough, we always can make it smaller to obtain $\psi_i$.
For example:
Consider a tangent bundle of a circle, i.e. $TS^1$, and a map $\sigma:S^1\to TS^1$ where $\sigma(\theta)=(\theta,\frac{\partial}{\partial\theta})$. So, if I want to show that $\sigma$ is a smooth section, then I need to choose an open cover of $S^1$ which will give me a local trivialization of a tangent bundle and write that map in local coordinates.
But, we can use another approach where I will start with some opens $U_i\times\mathbb{R}$ and smooth maps $g_{ij}:U_{i}\cap U_j\to GL(\mathbb{R})$, i.e. so I can use them to glue my tangent bundle. Then if we know each $$s_i:U_i\to U_i\times\mathbb{R}$$ where $s_i(p)=(p,\phi_i(p))$ and that $\phi_j(p)=g_{ij}(p)\phi_i(p)$, then it's enough to check that each $s_i$ is smooth map to obtain that $s$, as a section over tangent bundle, is smooth.
So, my question is basically comparing this two approaches and a general idea how to show that some map is a section.