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2008•The Quarterly Journal of Mechanics and Applied MathematicsOpen access

Linear invariants of a Cartesian tensor

Faiz Ahmad, M. A. Rashid

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Abstract

The number of linear invariants under SO(3) as well as SO(2) of a Cartesian tensor of an arbitrary rank is studied. A linear form is defined in terms of elements of a tensor. It is established that the number of linear invariants of a tensor of rank n under SO(3) equals the dimension of the space of isotropic tensors of rank n. Formulas for the number of invariants in the two cases are also derived. For the elasticity tensor, our analysis confirms the results of Norris.

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The number of linear invariants under SO(3) as well as SO(2) of a Cartesian tensor of an arbitrary rank is studied. A linear form is defined in terms of elements of a tensor. It is established that the number of linear invariants of a tensor of rank n under SO(3) equals the dimension of the space of isotropic tensors of rank n. Formulas for the number of invariants in the two cases are also derived. For the elasticity tensor, our analysis confirms the results of Norris.

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Available abstract

The number of linear invariants under SO(3) as well as SO(2) of a Cartesian tensor of an arbitrary rank is studied. A linear form is defined in terms of elements of a tensor. It is established that the number of linear invariants of a tensor of rank n under SO(3) equals the dimension of the space of isotropic tensors of rank n. Formulas for the number of invariants in the two cases are also derived. For the elasticity tensor, our analysis confirms the results of Norris.

Key concepts: Cartesian tensor, Mathematics, Tensor (intrinsic definition), Invariants of tensors, Rank (graph theory), Symmetric tensor, Tensor density, Cartesian coordinate system

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