Tensor rotational

For some purposes, it is convenient to express the constitutive equations for an inelastic material relative to the unrotated spatial configuration, i.e., one which has been stretched by the right stretch tensor U from the reference configuration, but not rotated by the rotation tensor R. The referential constitutive equations of Section 5.4.2 may be translated into unrotated terms, using the relationships given in the Appendix. 

A proper orthogonal tensor represents a rigid-body rotation, and R is called the material rotation tensor. It has the properties... 

Figure 5 Definition of the polar angles in the principal axis system of the diffusion-rotation tensor.

Symmetric tensor that results when a deformation gradient tensor is factorized into a rotation tensor followed or preceded by a symmetric tensor. 

If S is the lattice-invariant deformation tensor and R the rigid-body rotation tensor, the total shape deformation tensor, E, producing the invariant plane can be expressed as... 

When we rotate a contravariant nxl column vector (for position, velocity, momentum, electric field, etc.) we premultiply it by an n x n rotation tensor R. When, instead, we transform the coordinate system in which such vectors are defined, then the coordinate system and, for example, the V operator are covariant 1 x n row vectors, which are transformed by the tensor R 1 that is the reciprocal of R. A "dot product" or inner product a b must be the multiplication of a row vector a by a column vector b, to give a single number (scalar) as the result. This will be expanded further in the discussion of special relativity (Section 2.13) and of crystal symmetry (Section 7.10). 

Formally, if one has the experimental values of the dielectric tensor e, the magnetic permeability tensor /jl, and the optical rotation tensors p and p for the substrate, one can construct first the optical matrix M, then the differential propagation matrix A, and C, which, to repeat, is the x component of the wavevector of the incident wave. Once A is known, the law of propagation (wave equation) for the generalized field vector ift (the components of E and H parallel to the x and y axes) is specified by Eq. (2.15.18). Experimentally, one travels this path backwards. 

We now turn to the nuclear spin-rotation tensor c/ for the Mi nucleus in equation (8.155). This contains a purely nuclear contribution, arising from the third term of the first-order expression (8.153), and a purely electronic contribution coming from the third term of the second-order expression (8.154). To be specific ... 

If x, y and z are the principal axes of the molecule, the shielding and nuclear spin rotation tensors for the Mi nucleus are diagonal. The xx components, for example, are... 

In addition to the optical rotation tensor fi, the gyration tensor is often used as the basis of computing optical rotations, since it is more straightforward to define working equations for it in the frequency domain. The relation to the OR tensor is... 

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