By Mats Gyllenberg, Lars-Erik Persson
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Extra info for Analysis, Algebra and Computers in Mathematical Research
The curl of a tensor will be left to further reading (see Endnotes). That is, ∂ψijk ... Tijk ... ψijk ... = ∂xp∂xs∂xt ... 5. 7) presented in Part II of this text. Application of Eq. 55) to tensor fields of tensor rank greater than r = 1. (x) of rank r with the vector operator ∂i yielding a tensor of rank r - 1 given by Tjk ... = ∂iψijk ... = ∂ψijk... (x), but the results will, in general, be different. A physical interpretation of Eq. 59) defines the divergence of a tensor field at a point P in V as the limiting value of the flux of ψ per unit volume out of S when ∆V →0.
Tijk... If s represents a symmetry operator, that is, the transformation matrix of a symmetry operation valid for a particular point group, then the generalized transformation laws become Tijk... 7) Tijk... 4).
The maximum inherent symmetry possible for fourth-rank tensors is given by Tmn = Tnm (m, n = 1 - 6) . This eliminates the 15 coefficients below the leading diagonal symmetry line as independent components. 4. 1: The transformation laws for polar tensors up through the rank 4 showing tensor rank, tensor subscript notation, matrix symbolic notation, and the referred matrix text equation. 38), respectively, both of which are sixth-rank tensor properties. The transformation equations for the fifth- and sixth-rank tensor properties mentioned above will be left as additional exercises for the reader.