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The need to evaluate electric and magnetic fields arises frequently in the design of electrical equipment. Analysis by the finite element method (FEM) is usually employed for this purpose, being most efficiently carried out by solving first for the node potentials, thus obtaining a vector of approximate potential values on a discrete finite set of field points. In this paper it is shown how a linear operator may be set up to yield, at each node, an estimate for the potential function derivative along a prescribed direction in space. The operation consists in taking, for each node, a weighed average of the previously calculated potential values at its neighbouring nodes. The possibility of achieving high accuracy with a simple method, is attributed to its taking advantage of potential function properties inherent in their being solutions of certain field equations. The method is applicable, in principle, to 2-D and 3-D problems and is developed explicitly here for the plane 2-D and axisymmetric cases, using triangular elements with linear basis functions. Results for a typical axisymmetric magnetic field having a known analytic solution are presented and compared with exact values.