  Spectral methods are an efficient way to solve partial differential equations
on domains possessing certain symmetries. The utility of a method depends
strongly on the choice of spectral basis. In this paper we describe a set of
bases built out of Jacobi polynomials, and associated operators for solving
scalar, vector, and tensor partial differential equations in polar coordinates
on a unit disk. By construction, the bases satisfy regularity conditions at r=0
for any tensorial field. The coordinate singularity in a disk is a prototypical
case for many coordinate singularities. The work presented here extends to
other geometries. The operators represent covariant derivatives, multiplication
by azimuthally symmetric functions, and the tensorial relationship between
fields. These arise naturally from relations between classical orthogonal
polynomials, and form a Heisenberg algebra. Other past work uses more specific
polynomial bases for solving equations in polar coordinates. The main
innovation in this paper is to use a larger set of possible bases to achieve
maximum bandedness of linear operations. We provide a series of applications of
the methods, illustrating their ease-of-use and accuracy.
