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Higher-order vector basis functions are described that provide a linear normal and quadratic tangential representation of a vector quantity on rectangular or triangular cells. These functions are used to represent the surface current density on a variety of scatterers, including perfectly conducting plates, spheres, and cone-spheres. Results are compared with lower-order rooftop basis functions that provide a constant normal and linear tangential vector representation. Results suggest that the higher-order basis functions provide improved accuracy for a given number of unknowns but that special functions incorporating edge singularities are needed for a robust treatment of scatterers with corners or edges.