Cauchy-Difference Conservative Vector Fields for Dimension Two and Three

Summary

Let G be an abelian group and X a vector space over the rationals. If Φ: G → X its 1st Cauchy difference is the function K₂Φ:G² → X defined by K₂Φ(x ₁, x₂) = Φ(x₁ + x₂) − Φ(x₁) − Φ(x₂) and in general for n = 2, 3, … the (n − 1)th Cauchy difference of Φ is the function, K _nΦ: Gⁿ → X, defined by

$$K_{n}\Phi(x_{1},\cdot\cdot\cdot,\ x_{n})={\mathop \sum^n\limits_{r=1}}(-1)^{n-r}{\sum \limits_{\mid J\mid=r}}\Phi(x_{j})$$

where ϕ ≠ ⊆ I _n = {1, 2, …, n} and x _J = Σ_j∈Jx_J. If Ψ: Gⁿ → X(n = 2, 3, …) then its i-th partial Cauchy difference of order r(r = 2, 3, …), K_r ⁽ⁱ⁾Ψ: G^n+r+1 → X, is its Cauchy difference of order r − 1 with respect to its i-th variable with all the other variables held fixed. For n = 2 we have K₂ ⁽¹⁾Ψ(x₁, x₂, x₃) = Ψ(x₁ + x₂, x₃) − Ψ(x₁, x₃) − Ψ(x ₂, x₃) and K₂ ⁽²⁾Ψ(x₁; x₂, x₃) = Ψ(x₁, x₂ + x₃) − Ψ(x₁, x₂) − Ψ(x₁, x₃).

In this paper if ƒ = < ƒ₁, … ƒ_n >: G ⁿ → xⁿ the solution of K₂ ⁽ⁱ⁾ ƒ _j = K₂ ^(j) ƒ _i (i ≠ j) is given for n = 2 and 3.