Vectors general properties

Both the permanent dipole and the applied field are vector quantities, and the direction of the induced dipole need not be the same as the direction of the applied field. Hence, we need a more general property than a vector to describe the polarizability. 

Cle72, Gor75], Their values follow from the general properties of the vector coupling of angular momentum operators. 

This is a manifestation of the general property of chain molecules oriented in the electric field by the mechanism of large-scale motion on the average, the three main directions in the molecule coincide the direction of the greatest geometrical length of the molecule (vector h), of the orientational-axial order (responsible for the anisotropy Ti — 72) and of the orientational polar order (determining the total dipole moment n of the molecule). 

The reciprocality condition in Eq. (120) can be used to deduce some general properties of the measuring vectors associated with the shape coordinates and Euler angles. For example, because the measuring vector e 1 = u, is the same for any nucleus a, it follows that... 

Generally, properties of liquid crystals depend on direction, they are tensorial. Some of them (like density in nematics) may be scalar. A scalar is a tensor of rank 0. It has one component in a space of any dimensionality, 1 = 2° = 3 ... = 1. Other properties, like spontaneous polarization P (e.g., in chiral smectic C ) are vectors, i.e., the tensors of rank 1. In the two-dimensional space they have 2 = 2 components, in the 3D space there are 3 = 3 components. For instance in the Cartesian system P = iP + jPy + kP. Such a vector can be written as a row (Px> Py, Pz) or as a coluirm. Properties described by tensor of rank 2 have 2 = 4 components in 2D space and 3 = 9 components in the 3D-space. They relate two vector quantities, such as magnetization M and magnetic induction B, M = xB, where % is magnetic susceptibility. Each of the two vectors has three components and, generally, each component (projection) (a = x,y,z) may depend on each of Bp components (p = x,y,z) ... 

After introduction of these potentials the homogeneous Maxwell equations (2.96) and (2.97) are therefore identically satisfied due to general properties of differential vector calculus (cf. appendix A.l). The dynamical behavior of the potentials is determined by the inhomogeneous Maxwell equations, which in terms of the potentials read... 

So far, only general properties of Lorentz transformations have been investigated but no explicit expression for the transformation matrix A has yet been given. We are now going to derive the transformation matrix A for a Lorentz boost in x-direction in a very clear and elementary fashion. For t = t = 0 the two inertial frames IS and IS shall coincide, and the constant motion of IS relative to IS shall be described by the velocity vector v = vCx, cf. Figure 3.2. Since the y- and z-directions are not affected by this transformation, we explicitly write down the transformation given by Eq. (3.12) (for a = 0) for the relevant subspace... 

Chapter 17 describes the flux of a flowing fluid. You can also define the flux for electric field vectors. Why is it useful to define a flux for electric fields The electric field flux has the important general property that it is independent of... 

We will now turn to show the general properties of the scalar and vector fields presented in the last Section. Our purpose here is not that of systematizing, but just picking some very simple systems where we will compare the fields with each other when relevant, or show their shape and basic topology. 

The fact that the determinant is 1 is another general property of matrices which represent symmetry operations. This ensures that the symmetry operation does not affect the size of any basis vectors on which it operates. 

Realistic intermolecular interaction potentials for mesogenic molecules can be very complex and are generally unknown. At the same time molecular theories are often based on simple model potentials. This is justified when the theory is used to describe some general properties of liquid crystal phases that are not sensitive to the details on the interaction. Model potentials are constructed in order to represent only the qualitative mathematical form of the actual interaction energy in the simplest possible way. It is interesting to note that most of the popular model potentials correspond to the first terms in various expansion series. For example, the well known Maier-Saupe potential JP2 (Sfli )) is just the first nonpolar term in the Legendre polynomial expansion of an arbitrary interaction potential between two uniaxial molecules, averaged over the intermolecular vector r,-, ... 

Generalized acceleration vector in property space General vector or tensor valued function Constant vector in algebraic equation system Element-abundance vector (mole)... 

Generalized velocity vector in property space Bubble velocity vector in sample i measured by PDA (m) Instantaneous diffusion velocity for species c, relative to the local motion of the mixture stream (m/s)... 

Unlike the case of metals and semi-conductors, surface and bulk screening effects in insulators have been little studied. In this section, we will review the general properties of the dielectric constant - its small wave vector and low-frequency limits - and we will put a particular emphasis on local field effects. 

Equations (9.2.4), (9.2.5), (9.2.7), and (9.2.9) are sufficient for describing the motions of a planetary atmosphere. The vector notation used in these expressions is convenient for the study of their general properties, but for most applications it is necessary to write the equations in a specific coordinate system. Eor some calculations, a local rectangular system may suffice. More generally, spherical coordinates are employed with the origin at the center of the planet and the polar axis coincident with fi. hi this coordinate system, the equations take the form... 

The formula for the length i of a vector v satisfies the more general properties of a norm n of u 6 3t. A norm u is a rule that assigns a real scalar, u e 9i, to each vector n 6 such that for every v,we Di, and for every c e 9t, we have... 

The representation of molecular properties on molecular surfaces is only possible with values based on scalar fields. If vector fields, such as the electric fields of molecules, or potential directions of hydrogen bridge bonding, need to be visualized, other methods of representation must be applied. Generally, directed properties are displayed by spatially oriented cones or by field lines. 

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