In this paper, we generalize an invariant method of calculations of general connections to that on principal bundles. We introduce certain differential k-forms on principal bundles with values in the tangent bundle of an associative algebra, and express general connections by means of differential 1-forms of this kind. Then curvature forms are differential 2-forms of this kind and satisfy certain generalized Bianchi identities. Moreover, we introduce the notion of exterior covariant derivatives for the k-forms, and exhibit certain generalized Ricci identities.

general connection, gauge transformations, Bianchi identity, Ricci identity, action density.