Tensor coordinate transformation

For the example in Figure 2.14 it would be possible to perform the coordinate transformation analytically by introducing cylindrical coordinates. However, in general, geometries are too complex to be described by a simple analytical transformation. There are a variety of methods related to numerical curvilinear coordinate transformations relying on ideas of tensor calculus and differential geometry [94]. The fimdamental idea is to establish a numerical relationship between the physical space coordinates and the computational space curvilinear coordinates The local basis vectors of the curvilinear system are then given as... 

The coefficients /, g, and h are unique for each second-harmonic signal and depend on the three susceptibility tensors. We normalize the relative values of the tensor components to = 1- The task is then to determine the complex values of the other 14 tensor components (see Table 9.2). A sufficient number of 8 independent measurements is provided by the p- and s--polarized components of the reflected and transmitted second-harmonic signals for the two orientations of the sample shown in Figure 9.17. The change in sample orientation corresponds to a coordinate transformation that reverses the... 

If j = c2, then the z axis—the direction of propagation—is called the optic axis or the c axis. In this analysis we have tacitly assumed that the coordinate transformation to principal axes diagonalizes both the real and imaginary parts of the dielectric tensor. 

The antisymmetric tensor is generally not observable in NMR experiments and is therefore ignored. The symmetric tensor is now diagonalized by a suitable coordinate transformation to orient into the principal axis system (PAS). After diagonalization there are still six independent parameters, the three principal components of the tensor and three Euler angles that specify the PAS in the molecular frame. 

An important property of symmetric tensors is that there are three invariants, which are independent of any coordinate transformation. These invariants are... 

The approach to finding the transformation metric factors can be found in most books that discuss vector-tensor analysis (an excellent reference is Malvern [257]). For orthogonal coordinate transformations, metric factors are given generally as... 

Many materials properties are anisotropic they vary with direction in the material. When anisotropic materials properties are characterized, the values used to represent the properties must be specified with respect to particular coordinate axes. If the material remains fixed and the properties are specified with respect to some new set of coordinate axes, the properties themselves must remain invariant. The way in which the properties are described will change, but the properties themselves (i.e., the material behavior) will not. The components of tensor quantities transform in specified ways with changes in coordinate axes such transformation laws distinguish tensors from matrices [6]. 

It is often convenient to select the coordinate system for which the only nonzero elements of the property tensor lie on its diagonal. This is the eigensystem. To find the eigensystem, the general rules for transformation of a tensor must be identified. The transformation of Ohm s law (Eq. 1.24) illustrates the way in which the material properties tensor xold transforms to xnew and serves to demonstrate the general rule for transforming rank-two tensors ... 

Tensor Matrix operator that transforms one vector function into another all tensorial functions and entities must transform properly according to laws of coordinate transformation and retain both formal and operational invariance. 

Pq, is expressed in Cartesian coordinates. These polar tensors T), can be derived from experimental intensities by elementary coordinate transformation. If the axes x, y, and z are chosen such that the bonds are oriented along one of the axes, then the derivatives can be used to interpret the changes of the electron clouds during a vibration. Besides, considering the definitions of the axes, it is possible to transfer atomic polar tensors between similar molecules and to estimate their intensities (Person and Newton, 1974 Person and Overend, 1977). 

Thus, if A solves an eigenvalue equation of the form (E2.8), then a coordinate transformation can be found which will yield a diagonalized form of the matrix. The matrix A is itself a tensor (see Section E5.2) which can be constructed by taking the product of a column and a row vector... 

The Wigner rotations describe the coordinate transformations from the principal axis frame (P ) in which the tensor describing the interaction X is diagonal, via a molecule-fixed frame (C) and the rotor-fixed frame (R) to the laboratory frame (L) as illustrated in an ORTEP representation in Fig. 1. 

The flux tensor components Q Q, and are the products of the coordinate transformation metrics by the corresponding Cartesian quantities + uyQ y + USXM,... 

Since the quantities Qij (k) are analytic functions of k the same is true for the frequencies Qi k) as can be seen also from eqn (5.26). This, of course, is not surprising since (see also Chs. 3 and 4) the resonances of the tensor (w, k) occur at the so-called mechanical exciton frequencies which are analytic functions of k, regardless of the model being used. The vectors L<1> and L<2> are orthogonal. Therefore, if the coordinate axes x and y are parallel to the vectors L 2) and L 2 1 the tensor is transformed to diagonal form with nonzero components ... 

By definition the components of the second-rank Cartesian tensor ax transform under rotation just like the product of coordinates xy (e.q., see Jeffreys, 1961) The motivation for what ensues springs from the observation that the spherical harmonics Ym (0, ft) (where 6, ft) are the polar and azimuthal angles of the unit vector (r/1 r )) can be written in terms of the coordinates (x, y, z) of the vector r, for example,... 

A scalar quantity is invariant under rotation. Other (vector, tensor) quantities transform like products of coordinates, e.g. x2, xy, etc. 

The stress tensor thus allows us to completely describe the state of stress in a continuum in terms of quantities that depend on position and time only, not on the orientation of the surface on which the stress acts. More precisely, the stress tensor should be referred to as a tensor of second order or tensor of second rank because its components transform as squares of the coordinates. We shall, however, simply use the term tensor, since tensors of order higher than second generally are not dealt with in fluid mechanics. We note in passing that a vector is a tensor of first order, its components transforming like the coordinates themselves, and a scalar is a tensor of zeroth order, a scalar being invariant under coordinate transformation. 

Fig. 2. Pictorial representation of the transformation between the EFG and chemical shift tensors produced by the Euler angles a,l3, y) defined in this review. The principal coordinate of the EFG tensor is transformed to that of the chemical shift...

Should one however, not transform the parameter the components of a projective scalar behave under coordinate transformation exactly like the components of an affine scalar. Not transforming the parameter means, so to speak, that we keep our space fixed in a specific state. If some scalar is given, each such state corresponds to a certain component of the scalar. As we shall soon see this applies to all our projective tensors as well. Each state is specifically associated with a certain coordinate system in each tangent space. 

On coordinate transformation the 11 , hehave like the components of an affine connection and the fl j, like the components of an affine tensor. 

Let us now derive phenomenological equations of the kind (5.193) corresponding to the expression (5.205). As has been mentioned before, each flux is a linear function of all thermodynamic forces. However the fluxes and thermodynamic forces that are included in the expression (5.205) for the dissipative function, have different tensor properties. Some fluxes are scalars, others are vectors, and the third one represents a second rank tensor. This means that their components transform in different ways under the coordinate transformations. As a result, it can be proven that if a given material possesses some symmetry, the flux components cannot depend on all components of thermodynamic forces. This fact is known as Curie s symmetry principle. The most widespread and simple medium is isotropic medium, that is, a medium, whose properties in the equilibrium conditions are identical for all directions. For such a medium the fluxes and thermodynamic forces represented by tensors of different ranks, cannot be linearly related to each other. Rather, a vector flux should be linearly expressed only through vectors of thermodynamic forces, a tensor flux can be a liner function only of tensor forces, and a scalar flux - only a scalar function of thermodynamic forces. The said allows us to write phenomenological equations in general form... 

Indeed, the usual coordinate transformation of tensor W (i.e., of the type (c) in Rem. 4) leads to axiality transformation (a). Namely, (b) must be vtilid tilso for the new (starred) coordinate system... 

The <, Uf, Uy can be thought of as polar vector components (as opposed to axial vector components u, Uy, ) and they transform accordingly. When the lattice dynamical problem is treated in terms of the dynamical variable ujtyU ujigUy, Cochran and Pawley have pointed out that the two-molecule interaction force constants 0, (/A , I k ) can be treated as a two-dimensional tensor of dimension six. If S is the cartesian coordinate transformation matrix corresponding to a symmetry transformation, then the six-dimensional transformation matrix is... 

The three Euler angles used to characterize the coordinate transformation from the PAS to the FRAG are principally determined by the molecular electronic structure. Figure 3 shows the CS tensor directions with respect to the local molecular frame of 60CB and the FILAG frame. For C(ar)-C(ar) such as at carbon sites 2, 5, 6, and 9 in 60CB, the 5n component in the C CS tensor is parallel to the C-C bond and the 33 component is perpendicular to the molecular plane. For C(ar)-FI such as at sites 3, 4, 7, and 8, both 5n and 22 components lie on the molecular plane and the 5n component is parallel to the C-FI bond. For C=N such as at site 1, the 33 component is parallel to the C=N bond and the 5n component is perpendicular to the molecular plane of the benzene ring. Therefore, the Euler... 

---