Recently I've been trying to understand the geometric formulation of gauge field theory. The mantra I've been hearing is that a gauge field is a connection for a principal bundle where the structure group corresponds to the gauge group. Fields which are charged under the gauge group form sections of associated vector bundles. John Baez's book has a slight different take, however, in which he essentially defines the gauge field as the connection for a G-bundle, which is vector bundle obtained by gluing together trivial bundles subject to G-identifications on the overlaps, what need then for the principal bundle? I'm also grappling with the idea that a choice of gauge corresponds to a local trivialization. The idea seems to be that sufficiently small regions of the bundle are diffeomorphic to the product [itex]U \times F[/itex] where F is the typical fiber. I don't quite see how the different gauge possibilities arise out of this, however. Thanks.