  We study generalized heat kernel coefficients, which appear in the trace of
the heat kernel with an insertion of a first-order differential operator, by
using a path integral representation. These coefficients may be used to study
gravitational anomalies, i.e. anomalies in the conservation of the stress
tensor. We use the path integral method to compute the coefficients related to
the gravitational anomalies of theories in a non-abelian gauge background and
flat space of dimensions 2, 4, and 6. In 4 dimensions one does not expect to
have genuine gravitational anomalies. However, they may be induced at
intermediate stages by regularization schemes that fail to preserve the
corresponding symmetry. A case of interest has recently appeared in the study
of the trace anomalies of Weyl fermions.
