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This paper presents a general theory of maximally orthogonalized div-and curl-conforming higher order basis functions (HOBFs) over generalized wires, quadrilaterals, and hexahedra. In particular, all elements of such bases, necessary for fast and easy implementation, are listed up to order . Numerical results, given for div-conforming bases applied in an iterative method of moments solution of integral equations, show that the condition number and the number of iterations are a) much lower than in the case of other HOBFs of polynomial type and b) practically not dependent on the applied expansion order.Index Terms-Basis functions, higher order modeling, integral equations, method of moments (MoM), finite-element method (FEM), numerical techniques.