I was reading the following lecture notes: Lecture on Connections (p. 1-2). I am going to paraphrase it a bit, but it said the following
Let $\pi \colon P \to M$ be a principal $G$-bundle and $V$ be a representation of $G$. The associated vector bundle is the vector bundle is given by $E(V) := (P \times V)/G$.
From here the author argued that this construction is abstract, but that the (local) sections have concrete descriptions. They are given by the maps $s: P \to V$ for which $$ s(pg) = g \cdot s(p) $$ for all $g \in G$ and $p \in P$. I have two questions:
How is $E(V) = (P \times V)/G$ exactly defined? Is it that $$(p, v) \sim (g p, g^{-1} \cdot v) \quad \text{or} \quad (p,v) \sim (gp, g \cdot v)?$$ According to Mathworld, it should be the former, but why do we choose to define it that way exactly?
How do the maps $s: P \to V$ give information on the sections of $E(V)$? Shouldn't sections of $E(V)$ be of the form $$ \hat s: M \to E(V)? $$
Thanks in advance for all the help!
Given the $G$-equivariant maps $s: P \to V$, the sections of $E(V)$ are then $\hat s: M \to E(V)$ where ${\hat s(x) := (p, s(p))}$ for some $p \in \pi_{E(V)}^{-1}(x)$.