there are two definitions of the first Chern class that I don't know how to relate - hints and references are both welcome.
So, first approach: say that I have a complex vector bundle $E\to M$; I can pass to the frame bundle $F(E)\to M$, where I have a $U(n)-$action, for some $n$. Then I consider the Chern-Weil map \begin{equation} S({\frak u}(n)^*)^{U(n)}\to H^*_{U(n)}(F(E))\simeq H^*(M) \end{equation} and pick the image of the invariant polynomial $tr$, the trace. (up to some constant)
Second approach: I pick a connection $D:\Gamma(E)\to \Gamma(E\otimes TM)$, with $\Gamma(\cdot)$ representing the space of sections, and consider its curvature. Work locally on some $U$ and pick a section $s$ which generates the bundle: we get $D(s)=\theta(s)\cdot s$, with $\theta(s)$ a $1-$form, and apparently the first Chern class is proportional to $d\theta(s)$.
I am a bit confused: the very idea of connection seems to be absent from the first approach, so either there's a natural choice of connection, or I get the same result no matter which one I pick. Moreover, it would seem to me that also the choice of generator $s$ matters.
Honestly, I don't have much of a clue about where to start tackling the problem - the two things seem quite unrelated.