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One approach to eliminating spurious modes in the finite-element solution of the vector wave equation is the use of tangential vector finite elements. With tangential vector finite elements, only the tangential components of the vector field are made continuous across the element boundaries. Edge-elements are the simplest example of tangential vector finite elements. However, edge-elements provide only the lowest-order of accuracy in numerical computations, since in this approach the tangential component of the field is assumed to be constant along each edge of the element. The configurations of the tangential vector finite elements which are of higher-order approximations on two- and three-dimensional tetrahedral elements are presented. The vector-valued basis functions are written explicitly, and the interpolatory meanings of the unknowns are derived.