Some Combinatorial Aspects of Differential Operation Composition on the Space

Ω(R) d −→ Ω(R) d −→ Ω(R) d −→ Ω(R), where Ωi(R) is the space of differential forms of degree i = 0, 1, 2, 3 on the space R over the ring of functions A = {f : R → R | f ∈ C(R)}. In the consideration, which follows, we give definitions of the first-order differential operations. Let us notice that one-dimensional spaces Ω0(R) and Ω3(R) are isomorphic to A and let φ0 : Ω (R) → A, φ3 : Ω (R) → A be the corresponding isomorphisms. Next, the set of vector functions B = {f =(f1, f2, f3) : R → R | f1, f2, f3 ∈ C(R)}, over the ring A, is three-dimensional. It is isomorphic to Ω1(R) and Ω2(R). Let φ1 : Ω (R) → B, φ2 : Ω (R) → B be the corresponding isomorphisms. In that case, the compositions φ 0 ◦φ3 : Ω (R) → Ω(R) and φ 1 ◦φ2 : Ω (R) → Ω(R) are isomorphisms of the corresponding spaces of differential forms. The first-order differential operations are defined via the operator of the exterior differentiation d of differential forms in the following form: