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Isotropic Tensor


A tensor which has the same components in all rotated coordinate systems. All rank-0 tensors (scalars) are isotropic, but no rank-1 tensors (vectors) are. The unique rank-2 isotropic tensor is the Kronecker delta, and the unique rank-3 isotropic tensor is the permutation symbol (Goldstein 1980, p. 172).

The number of isotropic tensors of rank 0, 1, 2, ... are 1, 0, 1, 1, 3, 6, 15, 36, 91, 232, ... (OEIS A005043). These numbers are called the Motzkin sum numbers and are given by the recurrence relation

 a_n=(n-1)/(n+1)(2a_(n-1)+3a_(n-2))
(1)

with a(0)=1 and a(1)=0. Closed forms for a_n are given by

a_n=-1/2(-3)^(n+1)(1/2; n+1)_2F_1(-n-1,1/2;1/2;-1/3)
(2)
=((-1)^n3^(n+1)sqrt(pi))/(4Gamma(n+2))_2F^~_1(1/2,-1-n;1/2-n;-1/3).
(3)

The terms have the generating function

G(x)=1/(2x)(1-sqrt((1-3x)/(1+x)))
(4)
=sum_(n=0)^(infty)a_nx^n
(5)
=1+x^2+x^3+3x^4+6x^5+15x^6+....
(6)

Starting at rank 5, syzygies play a role in restricting the number of isotropic tensors. In particular, syzygies occur at rank 5, 7, 8, and all higher ranks.


See also

Kronecker Delta, Permutation Tensor, Scalar, Syzygy, Tensor, Vector

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References

Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1980.Jeffreys, H. and Jeffreys, B. S. "Isotropic Tensors." §3.03 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 87-89, 1988.Kearsley, E. A. and Fong, J. T. ""Linearly Independent Sets of Isotropic Cartesian Tensors of Ranks up to Eight." J. Res. Nat. Bureau Standards 79B, 49-58, 1975.Sloane, N. J. A. Sequence A005043/M2587 in "The On-Line Encyclopedia of Integer Sequences."Smith G. F. "On Isotropic Tensors and Rotation Tensors of Dimension m and Order n." Tensor, N. S. 19, 79-88, 1968.

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Isotropic Tensor

Cite this as:

Weisstein, Eric W. "Isotropic Tensor." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IsotropicTensor.html

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