  Working within the framework of the covariant perturbation theory, we obtain
the coincidence limit of the heat kernel of an elliptic second order
differential operator that is applicable to a large class of quantum field
theories. The basis of tensor invariants of the curvatures of a gravity and
gauge field background, to the second order, is derived, and the form factors
acting on them are obtained in two integral representations. The results are
verified by the functional trace operation, by the short proper time
(Schwinger-DeWitt) expansions, as well as by the computation of the Green
function for the two-dimensional scalar field model.
