In this paper we introduce and study some mathematical structures on top of
transitive Lie algebroids in order to formulate gauge theories in terms of
generalized connections and their curvature: metrics, Hodge star operator and
integration along the algebraic part of the transitive Lie algebroid (its
kernel). Explicit action functionals are given in terms of global objects and
in terms of their local description as well. We investigate applications of
these constructions to Atiyah Lie algebroids and to derivations on a vector
bundle. The obtained gauge theories are discussed with respect to ordinary and
to similar non-commutative gauge theories.Comment: 30 pages. Final version. arXiv admin note: substantial text overlap
with arXiv:1109.428