Recent years have witnessed considerable research activity in the application of digital-computer methods for the determination of the electro-magnetic fields in electrical machinery through the solution of Maxwell's equations, while taking full account of the magnetic saturation. Two distinct numerical approaches are evident in the literature: Finite-Difference Method and Finite-Element Method. The author has presented in the recent years a finite difference formulation for 3-dimensional numerical solutions of the nonlinear electromagnetic field problems in terms of potential functions, and has applied for the analysis of the end-zone fields of aerospace homopolar alternators and solid-rotor induction motors. The present work is directed towards the finite-element formulation for the numerical solution of three-dimensional nonlinear magnetostatic field problems. A variational principle is developed here utilizing the vector potential concept. The approach is based on variational methods in which a corresponding energy functional for the nonlinear case is minimized over the entire region. The minimization is performed by means of the finite-element method and the resultant set of nonlinear algebraic equations is solved through iterative schemes.