Differential equation for a functional integral

We propose a new method for calculating functional integrals in cases where the averaged (integrated) functional depends on functions of more than one variable. The method is analogous to that used by Feynman in the one-dimensional case (quantum mechanics). We consider the integration of functionals that depend on functions of two variables and are symmetric under rotations about a point in the plane. We assume that the functional integral is taken over functions defined in a finite spatial domain (in a disc of radius $r$). We obtain a differential equation describing change in the functional as the radius $r$ increases.