Example of a connection on a vector bundle in terms of a local frame and coordinates on the base. Let $p: E \to M$ be a rank $k$ vector bundle over a smooth manifold $M$.
A connection, or covariant derivative, on $E$ is a real-linear map $\nabla: \Gamma(E) \to \Gamma(\Omega_M^1 \otimes E)$ satisfying the following Leibniz property:
$$ \nabla(f s) = df \otimes s + f \nabla s$$
for $f \in C^{\infty}(M)$ and $s \in \Gamma(E)$.
Let $U \subset M$ be an open subset of $M$ over which $E$ trivializes and let $\phi: \mathbb{R}^k \times U \to \pi^{-1}(U)$ denote a choice of trivialization.
Let $s_1, \dots, s_k$ denote a local frame for $\pi^{-1}(U)$.
In terms of the local frame $s_i$, every section $\sigma$ of $\pi^{-1}(U)$ is of the form
$$f_1 s_1 + \cdots + f_k s_k$$
where the $f_i$ are smooth functions on $M$.
Let us also choose a set of local coordinates $(x_1, \dots, x_n)$ for $M$
so that we can write the $f_i$ and $s_i$ as functions $f_i(x_1, \dots, x_n)$ and $s_i(x_1, \dots, x_n)$, respectively.
Every connection $\nabla$ can be written as $\nabla= d +A$ where $A$ is the connection matrix defined by the property that $A \cdot s_i = \nabla s_i$. The components of $A$ are locally one forms on $M$.
I would like to see an example, in the above coordinates $x_i$ and with respect to the frame $s_i$, of a connection $\nabla$ on $E$ and its associated connection matrix.
I am not looking for an interesting example; just something to help me get a feel for computing connections on vector bundles.
 A: I think that there are two things to say here: Since you are not asking for interesting examples, you have to be aware of the fact that linear connections are a very soft structure, so you can make arbitrary choices to define them.
Indeed, for a local frame $\{s_j\}$ defined on $U\subset M$ you can choose an arbitrary matrix $A=(A^i_j)$ of one-forms $A^i_j\in\Omega^1(U)$ and define
$\nabla(\sum_i f_is_i):=\sum_i(df_i\otimes s_i)+\sum_{i,j}f_iA^j_i\otimes s_j$ which just means
$\nabla_\xi(\sum_i f_is_i)(x):=\sum_i df_i(\xi)(x)s_i(x)+\sum_{i,j}f_i(x)A^j_i(\xi)(x)s_j(x)$.
This also illustrates the second point I want to make, namely that you are trying to go to a setting that is kind of an overkill. A choice of local trivialization of a vector bundle is equivalent to a choice of local frame defined on the same subset. So if you start with a local trivialization, you should work with the frame induced local frame, i.e. with $s_j$ characterized by the fact that $\phi\circ s_j$ is the constant function $e_j$, the $j$th unit vector. In the above language, this just means that the $f_i$ are the components of the function $\phi\circ s:U\to\mathbb R^k$. The formula above then gives you the components of $\phi\circ \nabla_\xi s$
$df_i(\xi)+\sum_jf_jA^j_i(\xi)$. Of course you can then covert this into expressions for a different frame, which causes complications but in my opinion does not lead to additional insight. Of course you can then further write things in terms of local coordinates, but sections have values in $E$, so they can only be expresses in local coordinates after a choice of trivialization (or of a second frame).
