In a 4-dimensional Euclidean space, representation theorems have been recently obtained for 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. Here the new case is considered where no symmetric tensor has eigenvalues with multiplicity 1, but there is at least one symmetric tensor with two distinct eigenvalues. 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. Similarly, a set of symmetric tensor valued isotropic functions is found such that every other symmetric tensor valued function of the same variables can be expressed as a linear combination, trough scalar coefficients, of the elements of this set. Finally, we obtain also a set of skew-symmetric tensor valued isotropic functions such that every other skew-symmetric tensor valued function of the same variables can be expressed as a linear combination, through scalar coefficients, of the elements of this set.
REPRESENTATION THEOREMS IN A 4-DIMENSIONAL EUCLIDEAN SPACE. A NEW CASE
PENNISI, SEBASTIANO
2013-01-01
Abstract
In a 4-dimensional Euclidean space, representation theorems have been recently obtained for 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. Here the new case is considered where no symmetric tensor has eigenvalues with multiplicity 1, but there is at least one symmetric tensor with two distinct eigenvalues. 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. Similarly, a set of symmetric tensor valued isotropic functions is found such that every other symmetric tensor valued function of the same variables can be expressed as a linear combination, trough scalar coefficients, of the elements of this set. Finally, we obtain also a set of skew-symmetric tensor valued isotropic functions such that every other skew-symmetric tensor valued function of the same variables can be expressed as a linear combination, through scalar coefficients, of the elements of this set.
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