Cartesian Isotropic Tensors for Beginners
DOI:
https://doi.org/10.31349/RevMexFisE.22.020214Keywords:
fourth-rank, tensor, isotropicAbstract
In this paper, we show how to find the isotropic tensors from rank one to four and suggest a way to calculated higher orders following one of the methods exposed here. We describe two methods for calculating the isotropic tensors from rank one to four, almost step by step. An explicit representation of the components of the isotropic tensor from rank one to four is given.
References
R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, New York, 2003), pp. 331-334
R. C. Powell, Symmetry, Group Theory, and the Physical Properties of Crystals (Springer, Heidelberg, 2010), pp. 331-334. https://doi.org/10.1007/978-1-4419-7598-0
J. F. Nye, Physical Properties of Crystals, Their Representations by Tensors and Matrices (Clarendon, Oxford, 2010), pp. 331-334
G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists: A Comprensive Guide, 7th ed. (Academic, New York, 2013), pp. 209, 218
I. Fridtjov, Continuum Mechanics (Springer, Heidelberg, 2008), pp. 47, 275. https://doi.org/10.1007/978-3-540-74298-2
E. S. Jatirian-Foltides et al., About the calculation of the second-order susceptibility χ(2) tensorial elements for crystals using group theory, Rev. Mex. Fis. E 62 (2016) 5
P. G. Hodge, On isotropic cartesian tensors, Am. Math. Mon. 68 (1961) 793. https://doi.org/10.2307/2311997
H. Jeffreys, Cartesian Tensors (Cambridge, London, 1931), pp. 66-70
D. Chandrasekharaiah and L. Debnath, Continuum Mechanics (Academic, New York, 1994), pp. 331-334. https://doi.org/10.1016/C2009-0-21209-8
H. Jeffreys, On isotropic tensors, Proc. Camb. Phil. Soc. 73 (1973) 173, https://doi.org/10.1017/S0305004100047587
E. A. Kearsley and J. T. Fong, Linearly independent sets of isotropic cartesian tensors of ranks up to eight, J. Res. Nat. Bureau Standards 79B (1975) 49, https://doi.org/10.6028/JRES.079B.005
A. Hysa and D. Karriqi, Physical applications of isotropic tensor, Int. J. Inf. Res. Rev. 3 (2016) 1986
M. Moakher, Fourth-order cartesian tensors: old and new facts, notions and applications, Q. Jl Mech. Appl. Math 61 (2008) 181, https://doi.org/10.1093/qjmam/hbm027
K. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Enginering, 3rd ed. (Cambridge, New York, 2006), p. 953
A. Alejo-Molina, H. Hardhienata, and K. Hingerl, Simplified bond-hyperpolarizability model of second harmonic generation, group theory and Neumann’s principle, J. Opt. Soc. Am. B 31 (2014) 526, https://doi.org/10.1364/JOSAB.31.000526
A. Alejo-Molina, K. Hingerl, and H. Hardhienata, Model of third harmonic generation and electric field induced optical second harmonic using simplified bond-hyperpolarizability model, J. Opt. Soc. Am. B 32 (2015) 562, https://doi.org/10.1364/JOSAB.32.000562
A. Alejo-Molina and H. Hardhienata, Isotropic and anisotropic parts for the susceptibility tensor calculated using simplified bond-hyperpolarizability model, J. Phys.: Conf. Ser. 1057 (2018) 012001, https://doi.org/10.1088/1742-6596/1057/1/012001
J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 2021), p. 556
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Copyright (c) 2025 O. Palillero-Sandoval, R. Carrada-Legaria, Y. E. Bravo-Garc´ıa, E. Reynoso-Lara, A. Alejo-Molina
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