  We derive a Cartesian componentwise description of the covariant derivative
of tangential tensor fields of any degree on general manifolds. This allows to
reformulate any vector- and tensor-valued surface PDE in a form suitable to be
solved by established tools for scalar-valued surface PDEs. We consider
piecewise linear Lagrange surface finite elements on triangulated surfaces and
validate the approach by a vector- and a tensor-valued surface Helmholtz
problem on an ellipsoid. We experimentally show optimal (linear) order of
convergence for these problems. The full functionality is demonstrated by
solving a surface Landau-de Gennes problem on the Stanford bunny. All tools
required to apply this approach to other vector- and tensor-valued surface PDEs
are provided.
