  A novel high-order numerical scheme is proposed to compute the covariant
derivative, particularly for divergence and curl, on any curved surface. The
proposed scheme does not require the construction of a curved axis or metric
tensor, which would deteriorate the accuracy of the covariant derivative and
prevent its application to complex surfaces. As an application, the
Helmholtz-Hodge decomposition (HHD) is adapted in the context of the Galerkin
method for displaying the irrotational, incompressible, and harmonic components
of vectors on curved surfaces.
