  We investigate a particular realization of generalized q-differential
calculus of exterior forms on a smooth manifold based on the assumption that
the N-th power (N>2) of exterior differential is equal to zero. It implies the
existence of cyclic commutation relations for the differentials of first order
and their generalization for the differentials of higher order. Special
attention is paid to the cases N=3 and N=4. A covariant basis of the algebra of
such q-grade forms is introduced, and the analogues of torsion and curvature of
higher order are considered. We also study a graded exterior calculus on a
generalized Clifford algebra.
