Vector, axial Cartesian

Fig. 7. The nature of information concerning the mean orientation and dynamics of an internuclear vector r, which can be obtained from RDC analysis. Upon diagonali-zation of the Cartesian dipolar interaction tensor R, described in the text, the mean vector orientation, r, will be described by the Euler angles a and /3. The eigenvalues will correspond to the axial and rhombic order parameters which describe the amplitude of motion. If the motion is asymmetric, as reflected in a nonzero rhombic order parameter, then the principal direction of asymmetry is described by the Euler angle y.

The unit vector components of the classical magnetic fields Ba>, Ba and B<3> in vacuo are all axial vectors by definition, and it follows that their unit vector components must also be axial in nature. In matrix form, they are, in the Cartesian basis... 

In the case of a scalar field, the irreducible matrix D is a unit matrix, and drops out of. I1. For rotation through an angle S9t about the Cartesian axis ek, the rotational submatrix of the Lorentz matrix is given by Xkx = ()Hkekl]x], where el]k is the totally antisymmetric Levi-Civita tensor. For the one-electron Schrodinger field f, Noether s theorem defines three conserved components of a spatial axial vector,... 

The use of redundant coordinates requires extensive modification of the lattice dynamical procedure. It is, however, often worth the additional complication to use redundancies if this facilitates the formulation of symmetry coordinates. When the Wigner projection operator (Wigner, 1931) is used to build such symmetry coordinates, it is necessary to first understand the results of the application of all symmetry operations of the applicable group to the displacement coordinates chosen. This is indeed relatively straightforward for the direction cosine displacement coordinates and therein lies their principal value. These coordinates transform like axial vectors in contrast to cartesian coordinates, which transform like polar vectors. 

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... 

It is important to emphasize at this point that the expression for the rotation w.r.t. the y-axis requires a negative sign as a result of the assumption that translations are positive in the sense of the Cartesian coordinates and also the respect of the right-hand mle for the coordinate system. By superposing the spectral solutions for abeam element under axial, flexural, and torsion deformations, the displacement solution vector is given by ... 

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