Tensor unit vector

If a, b are two unit vectors arbitrarily chosen In the ( 6,m) plane, p Is seen from the previous expression to be a two-dimensional tensor with only four components Paaa, pbbb, P abb, Pbaa. This situation will be more thoroughly studied in the following for methyl-(2,4-dlnltrophenyl) a-amlnopropanoate (MAP) crystals. 

The second term in Eq. 2.33 requires taking the divergence of a tensor. This operation, V pVV which produces a vector, is expanded in several coordinate systems in Section A.ll. In noncartesian coordinate systems, since the unit-vector derivatives do not all vanish, the divergence of a tensor produces some unexpected terms. 

The extra terms in the bottom row are a result of nonvanishing unit-vector derivatives. The tensor products of unit vectors (e.g., ezer) are called unit dyads. In matrix form, where the unit vectors (unit dyads) are implied but usually not shown, the velocity-gradient tensor is written as... 

Seeking to find the relationship between stress vectors and tensors, consider Fig. 2.12, which shows an infinitesimally small, arbitrarily oriented surface A whose orientation is defined by the outward-pointing normal unit vector n. As illustrated, the unit vector can be resolved into components nz,nr, and hq,... 

Working in cartesian coordinates, determine the stress vector r on a differential surface whose orientation is represented by a unit vector n = nxtx + nyey + nzez. The stress state is represented by a tensor... 

Carry out all the operations to evaluate the divergence of the stress tensor, V T. Be careful to consider that some unit-vector derivatives do not vanish. Check the results with those provided in the Appendix. 

In cartesian coordinates, the vector-tensor operator can be readily seen by inspection. In other coordinate systems, however, terms like the ones in the third row of the equation above result physically from the fact that control-surface areas vary, and mathematically from the fact that the derivatives of the unit vectors do not all vanish. We have recovered the expression in the previous section, which was developed entirely from vector-tensor manipulations ... 

As discussed in Section A.18, a vector r on a surface whose orientation is described by the unit vector n is determined from the tensor as... 

The explicit identification of the unit vectors helps to understand operations, such as the scalar product of a vector and a tensor as in... 

Assume that a stress tensor T is known at a point. Assume further that there are three orthogonal surfaces passing through the point for which the stress vectors r are parallel to the outward-normal unit vectors n that describe the orientation of the surfaces. In other words, on each of these surfaces the normal stress vector is a scalar multiple of the outward-normal unit vector,... 

Referring to Fig. A.2, assume that the principal coordinates align with z, r, and O. The unit vectors (direction cosines) just determined correspond with the row of the transformation matrix N. Thus, if the principal stress tensor is... 

The metric coefficient in the theory of gravitation [110] is locally diagonal, but in order to develop a metric for vacuum electromagnetism, the antisymmetry of the field must be considered. The electromagnetic field tensor on the U(l) level is an angular momentum tensor in four dimensions, made up of rotation and boost generators of the Poincare group. An ordinary axial vector in three-dimensional space can always be expressed as the sum of cross-products of unit vectors... 

There is no paradox [112] in the use of e(3) as an operator as well as a unit vector. In the same sense [112], there is no paradox in the use of the scalar spherical harmonics as operators. The rotation operators in space are first-rank Toperators, which are irreducible tensor operators, and under rotations, transform into linear combinations of each other. The Toperators are directly proportional to the scalar spherical harmonic operators. The rotation operators, J, of the full rotation group are related to the T operators as follows... 

An alternative and simpler approach to deriving the result in equation (4.12) is to express the polarizability tensor as a general expansion in the two orthogonal unit vectors, u and p, embedded on the principal axes shown in Figure 4.4. Evidently, using Einstein notation, the polarizability can be written as... 

The Raman tensor for a chemical bond oriented along the unit vector u, but attached to a segment oriented along a vector r, can be computed using the analysis given in section 5.3. For a flexible polymer chain, a procedure similar to the Kuhn and Grun analysis of section 7.1.3 can be used to provide a connection between the Raman tensor and the orientation of the end-to-end vector, R. We first express the Raman tensor of an individual segment,... 

Following a deformation specified by the strain tensor, E, the unit vector in the deformed state is related to its initial value by... 

The dynamics of rigid, isolated spheroids was first analyzed for the case of shear flow by Jeffery[95]. When subject to a general linear flow with velocity gradient tensor G, the time rate of change of the unit vector defining the orientation of the symmetry axis of such a particle will have the following general form,... 

Unit vector defining the basis set for a polarization state, (1.54), (1.55) segmental absorption tensor, (5.4). 

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