  We give coordinate formula and geometric description of the curvature of the
tensor product connection of linear connections on vector bundles with the same
base manifold. We define the covariant differential of geometric fields of
certain types with respect to a pair of a linear connection on a vector bundle
and a linear symmetric connection on the base manifold. We prove the
generalized Bianchi identity for linear connections and we prove that the
antisymmetrization of the second order covariant differential is expressed via
the curvature tensors of both connections.
