Let M be a manifold and let nabla be a linear connection on the tangent bundle of M.  If X and Y are vector fields on M, then nabla_X(Y) is a vector field on M, called the directional covariant derivative of Y in the direction X with respect to the connection nabla.  If alpha is a differential 1-form, then nabla_X(alpha) is the 1-form defined by

Let E -> M be a vector bundle and let nabla be a connection on E.  If X is a vector field on M and Z a section of E, then nabla_X(Z) is a section of E, called the directional covariant derivative of the section Z in the direction X with respect to the connection nabla.  The definition of the directional covariant derivative operator nabla_X is extended to tensor fields on the fibers of E as above.

Let E -> M be a vector bundle, let nabla1 be a linear connection on the tangent bundle of M and nabla2 be a connection on E.  Let T be a mixed tensor on E, say, for example, T = U &t V, where U is a tensor field on M and V is a tensor field on the fibers of E. (In general T will be a sum of such tensor products).  Then the directional covariant derivative of T in the direction X with respect to the connections nabla_1 and nabla_2 is nabla_X(T) = nabla1_X(U) &t V + U &t nabla2_X(V).

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form DirectionalCovariantDerivative(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-DirectionalCovariantDerivative.