A review on octupolar tensors. (English) Zbl 1531.81185
Summary: In its most restrictive definition, an octupolar tensor is a fully symmetric traceless third-rank tensor in three space dimensions. So great a body of works have been devoted to this specific class of tensors and their physical applications that a review would perhaps be welcome by a number of students. Here, we endeavour to place octupolar tensors into a broader perspective, considering non-vanishing traces and non-fully symmetric tensors as well. A number of general concepts are recalled and applied to either octupolar and higher-rank tensors. As a tool to navigate the diversity of scenarios we envision, we introduce the octupolar potential, a scalar-valued function which can easily be given an instructive geometrical representation. Physical applications are plenty; those to liquid crystal science play a major role here, as they were the original motivation for our interest in the topic of this review.
MSC:
| 81T32 | Matrix models and tensor models for quantum field theory |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 35G20 | Nonlinear higher-order PDEs |
| 76A15 | Liquid crystals |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
Keywords:
third-rank tensors; harmonic tensors; generalized eigenvectors; generalized eigenvalues; octupolar potential; multipoles; generalized liquid crystal phasesReferences:
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