Converting Vector and Tensor Equations to Scalar Equations in Spherical Coordinates

We show how to represent a vector field in a spherical shell in terms of three scalar fields, and a second-order tensor field in terms of nine scalar fields. We derive the scalar representations of the most common algebraic and differential operations on vector and second-order tensor fields, and apply the results to obtain scalar formulations of various tensor problems in the continuum mechanics of the Earth's mantle. We give an exhaustive catalogue of all possible equilibrium stress fields in the mantle, we deduce the scalar equations of elastic-gravitational oscillation of a transversely isotropic, radially stratified, spherically symmetrical Earth, and we give the scalar convection equations for a viscous, self-gravitating, spherically symmetrical mantle with radially variable viscosity.