  We develop exterior calculus approaches for partial differential equations on
radial manifolds. We introduce numerical methods that approximate with spectral
accuracy the exterior derivative $\mathbf{d}$, Hodge star $\star$, and their
compositions. To achieve discretizations with high precision and symmetry, we
develop hyperinterpolation methods based on spherical harmonics and Lebedev
quadrature. We perform convergence studies of our numerical exterior derivative
operator $\overline{\mathbf{d}}$ and Hodge star operator $\overline{\star}$
showing each converge spectrally to $\mathbf{d}$ and $\star$. We show how the
numerical operators can be naturally composed to formulate general numerical
approximations for solving differential equations on manifolds. We present
results for the Laplace-Beltrami equations demonstrating our approach.
