  A method which uses a generalized tensorial $\zeta$-function to compute the
renormalized stress tensor of a quantum field propagating in a (static) curved
background is presented. The starting point of the method is the direct
computation of the functional derivatives of the Euclidean one-loop effective
action with respect to the background metric. This method, when available,
gives rise to a conserved stress tensor and produces the conformal anomaly
formula directly. It is proven that the obtained stress tensor agrees with
statistical mechanics in the case of a finite temperature theory. The
renormalization procedure is controlled by the structure of the poles of the
stress-tensor $\zeta$ function. The infinite renormalization is automatic and
is due to a ``magic'' cancellation of two poles. The remaining finite
renormalization involves conserved geometrical terms arising by a certain
residue. Such terms renormalize coupling constants of the geometric part of
Einstein's equations (customary generalized through high-order curvature
terms). The method is checked on particular cases (closed and open Einstein`s
universe) finding agreement with other approaches. The method is also checked
considering a massless scalar field in the presence of a conical singularity in
the Euclidean manifold (i.e. Rindler spacetimes/large mass black hole
manifold/cosmic string manifold). There, the method gives rise to the stress
tensor already got by the point-splitting approach for every coupling with the
curvature regardless of the presence of the singular curvature. Comments on the
measure employed in the path integral, the use of the optical manifold and
different approaches to renormalize the Hamiltonian are made.
