  We propose a direct correspondence between the classical evolution equations
of 5-d supergravity and the renormalization group (RG) equations of the dual
4-d large $N$ gauge theory. Using standard Hamilton-Jacobi theory, we derive
first order flow equations for the classical supergravity action $S$, that take
the usual form of the Callan-Symanzik equations, including the corrections due
to the conformal anomaly. This result gives direct support for the
identification of $S$ with the quantum effective action of the gauge theory. In
addition we find interesting new relations between the beta-functions and the
counterterms that affect the 4-d cosmological and Newton constant.
