  We revisit a study of local renormalization group (RG) with background gauge
fields incorporated using the AdS/CFT correspondence. Starting with a
$(d+1)$-dimensional bulk gravity coupled to scalars and gauge fields, we derive
a local RG equation from a flow equation by working in the Hamilton-Jacobi
formulation of the bulk theory. The Gauss's law constraint associated with
gauge symmetry plays an important role. RG flows of the background gauge fields
are governed by vector $\beta$-functions, and some interesting properties of
them are known to follow. We give a systematic rederivation of them on the
basis of the flow equation. Fixing an ambiguity of local counterterms in such a
manner that is natural from the viewpoint of the flow equation, we determine
all the coefficients uniquely appearing in the trace of the stress tensor for
$d=4$. A relation between a choice of schemes and a Virial current is
discussed. As a consistency check, these are found to satisfy the integrability
conditions of local RG transformations. From these results, we are led to a
proof of a holographic $c$-theorem by finding out a full family of schemes
where a trace anomaly coefficient is related with a holographic $c$-function.
