I'm trying to understand how connections are locally represented, and the definition I have to work with is this:
Let $(x^1,\dots,x^n)$ be local coordinates defined in some chart $U \subset M$ such that $\xi\vert_U$ is trivial, and let $\{s_i\}$ be a frame of sections for $\xi\vert_U$. Then we have $\nabla_Xs_i=\omega^j_i(X)s_j$ (Einstein summation is assumed), where $(\omega^j_i)$ is a matrix of 1-forms.
I thought a concrete example might be helpful; I settled on the Euclidean Connection, which we define as follows.
For $\Omega \subset \mathbb{R}^n$, and $\xi=T\Omega=\Omega \times \mathbb{R}^n$, any section $S$ of $T\Omega$ we can view as $s(p)=(p,f(p))$, where $f:\Omega \rightarrow \mathbb{R}^n$ is a vector function. Then the connection $\nabla_XS(p)=(p,D_pf(X))$ is called the Euclidean Connection on $T\Omega$.
I would like to carry out the computation in the first definition and figure out what the Euclidean Connection looks like in coordinates, but I'm having trouble getting started. Any help would be greatly appreciated.
Anthony, in the case of the trivial bundle, if you take as your frame $s_i$ the standard basis vectors for $\Bbb R^n$, then these are constant vector-valued functions and their derivatives are $0$. So $\omega_i^j = 0$ for all $i,j$. Now, you're free to pick any frame you want and modify this accordingly.