  Diagonal matrix elements of pseudodifferential operators are needed in order
to compute effective Lagrangians and currents. For this purpose the method of
symbols is often used, which however lacks manifest covariance. In this work
the method of covariant symbols, introduced by Pletnev and Banin, is extended
to curved space-time with arbitrary gauge and coordinate connections. For the
Riemannian connection we compute the covariant symbols corresponding to
external fields, the covariant derivative and the Laplacian, to fourth order in
a covariant derivative expansion. This allows to obtain the covariant symbol of
general operators to the same order. The procedure is illustrated by computing
the diagonal matrix element of a nontrivial operator to second order.
Applications of the method are discussed.
