# IJPAM: Volume 60, No. 4 (2010)

**REPRESENTATIONS THEOREMS FOR SCALAR FUNCTIONS**

IN A 4-DIMENSIONAL EUCLIDEAN SPACE

IN A 4-DIMENSIONAL EUCLIDEAN SPACE

Dipartimento di Matematica ed Informatica

Universitá degli Studi di Cagliari

72, Via Ospedale, Cagliari, 09124, ITALY

e-mail: [email protected]

e-mail: [email protected]

**Abstract.**In a 4-dimensional Euclidean space, representations theorems are obtained here for scalar valued isotropic functions depending on an arbitrary number of scalars, skew-symmetric second order tensors and symmetric second order tensors; at least one of these last ones is assumed to have an eigenvalue with multiplicity 1. The case with at least a non null vector, among the independent variables, has already been treated in literature; so it is here not treated. The result is a finite, but long, set of scalar valued isotropic functions such that every other scalar function of the same variables can be expressed as a function of the elements of this set. The methodology used to obtain this set is directed in trying to use similar representation theorems, already known in literature for the 3-dimensional case.

**Received: **March 30, 2010

**AMS Subject Classification: **15A72

**Key Words and Phrases: **representations theorems, scalar valued isotropic functions, skew-symmetric second order tensors, symmetric second order tensors

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2010

**Volume:** 60

**Issue:** 4