#### Title

Cauchy-Difference Conservative Vector Fields for Dimension Two and Three

#### Document Type

Article

#### Publication Date

1-1-1994

#### Abstract

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(x1, 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, KnΦ: Gn → X, defined by where ϕ ≠ ⊆In = {1, 2, …, n} and xJ = Σj∈JxJ. If Ψ: Gn → X(n = 2, 3, …) then its i-th partial Cauchy difference of order r(r = 2, 3, …), Kr(i)Ψ: Gn+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 K21Ψ(x1, x2, x3) = Ψ(x1+ x2, x3) − Ψ(x1, x3) − Ψ(x2, x3) and K22Ψ(x1; x2, x3) = Ψ(x1, x2+ x3) − Ψ(x1, x2) − Ψ(x1, x3). In this paper if ƒ = < ƒ1, … ƒn > : Gn → xn the solution of K2(i)ƒj = K2(j)ƒi (i ≠ j) is given for n = 2 and 3. © 1994, Birkhäuser Verlag, Basel. All rights reserved.

#### Publication Title

Results in Mathematics

#### Recommended Citation

Heuvers, K.
(1994).
Cauchy-Difference Conservative Vector Fields for Dimension Two and Three.
*
Results in Mathematics,
26*(3), 298-305.
http://doi.org/10.1007/BF03323052

Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/4573