I am studying differential geometry with the help of Taubes’s book. But I meet difficulty when read the part about flat connection.
In the chapter about flat connection ，they say that for a trivial principal bundle$P$= $M \times G$,where G a lie group. $\omega=g^{-1}dg$(at the point $g \in G$ )defines a flat connection on $P$, here $dg$ denotes the pullback form via the projection map$\pi:P \rightarrow G$.
But I have no idea how to calculate $g^{-1}dg (v)$,here $v$ the vector field at $g \in G$.
Thanks you for your help!