An important problem of calculus of variations used in constrained differential systems, i.e., the commutation relation of differential operator and variational oprerator, is investigated by means of Frobenius theorem of integrability. Based on analyzing the d-δ commutation relation for the linear stationay differential constrained systems and affine differential constrained systems, the relationship between noncommutator of differentiation and variation and nonholonomicity of the differential constraints is briefly proved by means of Frobenius integrability theory. The commutation relation for non-linear differential constrained systems is also discussed in the paper. Finally, three examples are given to verify the results.