  We demonstrate the effectivity of the covariant background field method by
means of an explicit calculation of the 3-loop beta-function for a pure
Yang-Mills theory. To maintain manifest background invariance throughout our
calculation, we stay in coordinate space and treat the background field
non-perturbatively. In this way the presence of a background field does not
increase the number of vertices and leads to a relatively small number of
vacuum graphs in the effective action. Restricting to a covariantly constant
background field in Fock-Schwinger gauge permits explicit expansion of all
quantum field propagators in powers of the field strength only. Hence, Feynman
graphs are at most logarithmically divergent. At 2-loop order only a single
Feynman graph without subdivergences needs to be calculated. At 3-loop order 24
graphs remain. Insisting on manifest background gauge invariance at all stages
of a calculation is thus shown to be a major labor saving device. All
calculations were performed with Mathematica in view of its superior pattern
matching capabilities. Finally, we describe briefly the extension of such
covariant methods to the case of supergravity theories.
