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A generalized equivalence integral equation (GEIE) approach to formulating scattering from essentially convex closed surfaces is proposed. The GEIE approach invokes the generalized surface field equivalence to partially fill the volume originally occupied by the scatterer with judiciously selected materials, as opposed to the conventional replacement of the scatterer by the free space. The type and shape of the material inclusions can be selected to allow for a numerically efficient construction of the modified Green's function. Introduction of impenetrable and lossy materials confines the field interaction along the scatterer surface and reduces the coupling between the distant parts of the scatterer, which essentially makes the impedance matrix banded. The presence of lossy materials also resolves the nonuniqueness problem of the electric and magnetic field integral equations by eliminating the internal resonances. The formulation provides a pathway for developing fast iterative and direct electromagnetic integral equation solvers.