Scalar quantities

Onsager relation implies that measurement of one of these effects is sufficient to detemiine the coupling for both. The coefficient L is proportional to the heat conductivity coefficient and is a single scalar quantity in... 

Almost everyone has a concept of pressure from weather reports of tlie pressure of the atmosphere around us. In this context, high pressure is a sign of good weather while very low pressures occur at the eyes of cyclones and hurricanes. In elementary discussions of mechanics, hydrostatics of fluids and the gas laws, most scientists leam to compute pressures in static systems as force per unit area, often treated as a scalar quantity. They also leam that unbalanced pressures cause fluids to flow. Winds are the flow of the atmosphere from regions of high to low... 

The electric moments are examples of tensor properties the charge is a rank 0 tensor (which i the same as a scalar quantity) the dipole is a rank 1 tensor (which is the same as a vectoi with three components along the x, y and z axes) the quadrupole is a rank 2 tensor witl nine components, which can be represented as a 3 x 3 matrix. In general, a tensor of ran] n has 3" components. 

The third integral vanishes because the derivative of the dipole operator itself p = Zi e rj + Za Za e Ra with respect to the coordinates of atomic centers, yields an operator that contains only a sum of scalar quantities (the elementary charge e and the... 

Fig. 16. Nonstaggered grid system showing the velocity and scalar control volumes (cv). The scalar quantities are calculated at the nodes (eg, P, N, S, E, W)...

Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simphfied here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity... 

Q and R are the state and control weighting matrices and are always square and symmetric. J is always a scalar quantity. 

In general, for a second-order system, when Q is a diagonal matrix and R is a scalar quantity, the elements of the Riccati matrix P are... 

Forces are vector quantities and the potential energy t/ is a scalar quantity. For a three-dimensional problem, the link between the force F and the potential U can be found exactly as above. We have... 

We need to be clear about the various coordinates, and about the difference between the various vector and scalar quantities. The electron has position vector r from the centre of mass, and the length of the vector is r. The scalar distance between the electron and nucleus A is rp, and the scalar distance between the electron and nucleus B is tb- I will write / ab for the scalar distance between the two nuclei A and B. The position vector for nucleus A is Ra and the position vector for nucleus B is Rb. The wavefunction for the molecule as a whole will therefore depend on the vector quantities r, Ra and Rb-It is an easy step to write down the Hamiltonian operator for the problem... 

The force / and displacement s are vector quantities, and equation (2.5) indicates that the vector dot product of the two gives a scalar quantity. The result of this operation is equation (2.6)... 

REDEFINITION OF THE LENGTH AND MASS DIMENSIONS 1.6.1. Vector and scalar quantities... 

It is important to recognise the differences between scalar quantities which have a magnitude but no direction, and vector quantities which have both magnitude and direction. Most length terms are vectors in the Cartesian system and may have components in the X, Y and Z directions which may be expressed as Lx, Ly and Lz. There must be dimensional consistency in all equations and relationships between physical quantities, and there is therefore the possibility of using all three length dimensions as fundamentals in dimensional analysis. This means that the number of dimensionless groups which are formed will be less. 

There are strict limitations to the application of the analogy between momentum transfer on the one hand, and heat and mass transfer on the other. Firstly, it must be borne in mind that momentum is a vector quantity, whereas heat and mass are scalar quantities. Secondly, the quantitative relations apply only to that part of the momentum transfer which arises from skin friction. If form drag is increased there is little corresponding increase in the rates at which heat transfer and mass transfer will take place. 

The rate of a chemical reaction (the chemical flux ), 7ch, in contrast to the above processes, is a scalar quantity and, according to the Curie principle, cannot be coupled with vector fluxes corresponding to transport phenomena, provided that the chemical reaction occurs in an isotropic medium. Otherwise (see Chapter 6, page 450), chemical flux can be treated in the same way as the other fluxes. 

In this case where U and Y are scalar quantities, COB must also be scalar.1 In fact, if we make an association between Eq. (4-7) and what we have learned in Chapter 2, COB is our ubiquitous transfer function. We can rewrite Eq. (4-7) as... 

If a vector it is a function of a single scalar quantity s, the curve traced as a function of s by its terminus, with respect to a fixed origin, can be represented as shown in Fig. 8. Within the interval As the vector AR = R2 - Ri is in the direction of the secant to the curve, which approaches the tangent in the limit as As - 0. Tins argument corresponds to that presented in Section 2.3 and illustrated in Fig. 4 of that section. In terms of unit vectors in a Cartesian coordinate system... 

The eigenvalue problem can be described in matrix language as follows. Given a matrix ff, determine the scalar quantities X and the nonzero vectors U which satisfy simultaneously file equation... 

To provide a mathematical description of a particle in space it is essential to specify not only its mass, but also its position (perhaps with respect to an arbitrary origin), as well as its velocity (and hence its momentum). Its mass is constant and thus independent of its position and velocity, at least in the absence of relativistic effects. It is also independent of the system of coordinates used to locate it in space. Its position and velocity, on the other hand, which have direction as well as magnitude, are vector quantities. Their descriptions depend on the choice of coordinate system. In this chapter Heaviside s notation will be followed, viz. a scalar quantity is represented by a symbol in plain italics, while a vector is printed in bold-face italic type. 

The scalar product of the vector operator V and a vector A yields a scalar quantity, the divergence of A. Thus,... 

Here, /3 and / are constants known as the Bohr magneton and nuclear magneton, respectively g and gn are the electron and nuclear g factors a is the hyperfine coupling constant H is the external magnetic field while I and S are the nuclear and electron spin operators. The electronic g factor and the hyperfine constant are actually tensors, but for the hydrogen atom they may be treated, to a good approximation, as scalar quantities. 

The spin Hamiltonian for the hydrogen atom will be used to determine the energy levels in the presence of an external magnetic field. As indicated in Section II.A, the treatment may be simplified if it is recognized that the g factor and the hyperfine constant are essentially scalar quantities in this particular example. An additional simplification results if the z direction is defined as the direction of the magnetic field. For this case H = Hz and Hx = Hv = 0 hence,... 

In the previous section the g value was considered as a scalar quantity, which was indeed a good approximation since the unpaired electron on the hydrogen atom occupies a spherically symmetric s orbital. If the unpaired electron exhibits p or d character the electron possesses both spin and orbital angular momentum. As a result the spin is not quantized exactly along the direction of the external field and the g value becomes a tensor... 

The hyperfine constant a in Eq. (1) was also taken to be a scalar quantity for the hydrogen atom however, it is in general a tensor because of the various directional interactions in a paramagnetic species. The hyperfine term in the spin Hamiltonian is more correctly written as S-a-I, where a is the hyperfine coupling tensor. 

Our finding that linewidth anisotropy in biomolecular EPR spectra can be described by a statistical theory in which the random variables that cause the broadening are fully correlated, does not only make analysis by simulation practical it also holds a message on the nature of the ultimate source of the broadening if the three principal elements of the p-tensor are fully correlated, then they should find their cause in a single, scalar quantity. 

The motions of the individual fluid parcels may be overlooked in favor of a more global, or Eulerian, description. In the case of single-phase systems, convective transport equations for scalar quantities are widely used for calculating the spatial distributions in species concentrations and/or temperature. Chemical reactions may be taken into account in these scalar transport equations by means of source or sink terms comprising chemical rate expressions. The pertinent transport equations run as... 

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