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A novel double higher order Galerkin-type method of moments based on higher order geometrical modeling and higher order current modeling is proposed for surface integral equation analysis of combined metallic and dielectric antennas and scatterers of arbitrary shapes. The technique employs generalized curvilinear quadrilaterals of arbitrary geometrical orders for the approximation of geometry (metallic and dielectric surfaces) and hierarchical divergence-conforming polynomial vector basis functions of arbitrary orders for the approximation of electric and magnetic surface currents within the elements. The geometrical orders and current-approximation orders of the elements are entirely independent from each other, and can be combined independently for the best overall performance of the method in different applications. The results obtained by the higher order technique are validated against the analytical solutions and the numerical results obtained by low-order moment-method techniques from literature. The examples show excellent accuracy, flexibility, and efficiency of the new technique at modeling of both current variation and curvature, and demonstrate advantages of large-domain models using curved quadrilaterals of high geometrical orders with basis functions of high current-approximation orders over commonly used small-domain models and low-order techniques. The reduction in the number of unknowns is by an order of magnitude when compared to low-order solutions.