Vector and scalar quantities

It is important to recognise the differences between sca/ar 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. 

It should be noted in this respect that a torque is obtained as a product of a force in the X-direction and an arm of length Ly, say, in a direction at right-angles to the T-direction. Thus, the dimensions of torque are MLxLyT , which distinguish it from energy. 

Of particular interest in fluid flow is the distinction between shear stress and pressure (or pressure difference), both of which are defined as force per unit area. For steady-state 

The force F acting on the fluid in the X (axial)-direction has dimensions MLxT and hence  

For a pipe of radius r and length /, the dimensions of r/l are L Lr and hence AP/R) r/l) is a dimensionless quantity. The role of the ratio rjl would not have been establisted had the lengths not been treated as vectors. It is seen in Chapter 3 that this conclusion is consistent with the results obtained there by taking a force balance on the fluid. 

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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... 

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

As can be seen from this introduction, there are many vector and scalar quantities that are important to correlated product state measurements. In order to aid the reader in keeping track of the quantities important to these measurements, Table 2 defines and summarizes all the scalar and vector quantities used in this chapter. 

By definition, an isotropic system cannot support a vector quantity associated with it. Therefore, the vectorial flows can only be related to the vector forces. The scalar reaction rates can be functions of the scalar forces and the trace of the dyadic, but not the vector forces. According to the Curie-Prigogine principle, vector and scalar quantities interact only in an anisotropic medium. This principle has important consequences in chemical reactions and transport processes taking place in living cells. 

According to the Curie-Prigogine principle, vector and scalar quantities interact only in an anisotropic medium. This principle as originally stated by Curie in 1908 is quantities whose tensorial characters differ by an odd number of ranks cannot interact in an isotropic medium. Consider a flow J,- with tensorial rank m. The value of m is zero for a scalar, it is unity for a vector, and it is two for a dyadic. If a conjugate force Xj also has a tensorial rank m, than the coefficient Ly is a scalar, and is consistent with the isotropic character of the system. The coefficients Ly are determined by the isotropic medium they need not vanish, and hence the flow J, and the force Xj can interact or couple. If a force Xj has a tensorial rank different from m by an even integer k, then Ly has a tensor at rank k. In this case, Lij Xj is a tensor product. Since a tensor coefficient Ly of even rank is also consistent with the isotropic character of the fluid system, the Ly is not zero, and hence J,- andXy can interact. However, for a force Xj whose tensorial rank differs from m by an odd integer k, Ly has a tensorial rank of k. A tensor coefficient Ly of odd rank implies an anisotropic character for the system. Consequently, such a coefficient vanishes for an isotropic system, and J, and Xj do not interact. For example, if k is unity, then Ly would be a vector. 

However, A and T are scaler quantities, and because Lu is only a property of the medium, in an isotropic medium it must also be a scalar. Because there is no way to define a direction for the vector heat flow, Lu = 0 is required. There can be no coupling between vector and scalar irreversible flow processes in isotropic media.10,11 This is known as the Curie-Prigogine principle. 

In order to qualify properly as a vector, a quantity must obey the rules of vector algebra (scalar quantities obey the rules of arithmetic). Consequently, we need to describe and define these rules before we can solve problems in chemistry involving vector quantities. Linear algebra is the field of mathematics that provides us with the notation and rules required to work with directional quantities. 

A moment in mechanics is generally defined as Uj = Qd, where Uj is the jth moment, about a specified line or plane a of a vector or scalar quantity Q (e.g., force, weight, mass, area), d is the distance from Q to the reference line or plane, and j is a number indicating the power to which d is raised. [For example, the first moment of a force or weight about an axis is defined as the product of the force and the distance of the fine of action of the force jfrom the axis. It is commonly known as the torque. The second moment of the force about the same axis (i.e., i = 2) is the moment of inertia.] If Q has elements Qi, each located a distance di from the same reference, the moment is given by the sum of the individual moments of the elements ... 

Equation (B.3) is recognized as the well-known Pauli equation that describes the motion of an electron in an electromagnetic field. A and V are respectively known as the vector and scalar potentials of the electromagnetic field. The quantity V X A=cmlA=B represents the magnetic field strength. In order to... 

Before we can start with the discussion of time-dependent perturbation theory in the form of response theory, we need to introduce an alternative formulation of quantum mechanics, called the interaction or Dirac representation. In general, several representations of the wavefunctions or state vectors and of the operators of quantum mechanics are equivalent, i.e. valid, as long as the expectation values of operators ( 0 I d I o) or inner products of the wavefunctions ( o n) are always the same. Measurable quantities and thus the physics are contained in the expectation values or inner products, whereas operators and wavefunctions are mathematical constructs used in a particular formulation of the theory. One example of this was already discussed in Section 2.9 on gauge transformations of the vector and scalar potentials. In the present section we want to look at a transformation that is related to the time dependence of the wavefunctions and operators. 

Each term in Eq. (9.2-8) is a product of a scalar quantity (the component) and a unit vector. A scalar quantity can be positive, negative, or zero but has no specific direction in space. The unit vector i points in the direction of the positive x axis, the unit vector j points in the direction of the positive y axis, and the unit vector k points in the direction of the positive z axis. Eigure 9.2 shows the vector r , the Cartesian axes, the unit vectors, and the Cartesian components of the vector. 

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... 

A few comments on the layout of the book. Definitions or common phrases are marked in italic, these can be found in the index. Underline is used for emphasizing important points. Operators, vectors and matrices are denoted in bold, scalars in normal text. Although I have tried to keep the notation as consistent as possible, different branches in computational chemistry often use different symbols for the same quantity. In order to comply with common usage, I have elected sometimes to switch notation between chapters. The second derivative of the energy, for example, is called the force constant k in force field theory, the corresponding matrix is denoted F when discussing vibrations, and called the Hessian H for optimization purposes. 

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)... 

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. 

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,... 

The sum and scalar product of two three-dimensional vectors are similar to those quantities in two dimensions, as seen from the following relationships ... 

The value of the dot product is a measure of the coalignment of two vectors and is independent of the coordinate system. The dot product therefore is a true scalar, the simplest invariant which can be formed from the two vectors. It provides a useful form for expressing many physical properties the work done in moving a body equals the dot product of the force and the displacement the electrical energy density in space is proportional to the dot product of electrical intensity and electrical displacement quantum mechanical observations are dot products of an operator and a state vector the invariants of special relativity are the dot products of four-vectors. The invariants involving a set of quantities may be used to establish if these quantities are the components of a vector. For instance, if AiBi forms an invariant and Bi are the components of a vector, then Az must be the components of another vector. 

In order to describe second-order nonlinear optical effects, it is not sufficient to treat (> and x<2) as a scalar quantity. Instead the second-order polarizability and susceptibility must be treated as a third-rank tensors 3p and Xp with 27 components and the dipole moment, polarization, and electric field as vectors. As such, the relations between the dipole moment (polarization) vector and the electric field vector can be defined as ... 

Figure 1,7b shows the corresponding process in which a jet of liquid flows into the tank. In this case, the rate of addition of mass Af is simply the mass flow rate. If the x-component of the jet s velocity is vx then the rate of flow of x-momentum into the tank is Mvx. Note that the mass flow rate Af is a scalar quantity and is therefore always positive. The momentum is a vector quantity by virtue of the fact that the velocity is a vector. 

Note The atomic units are used throughout the paper in the adopted notation P denotes the square or rectangular matrix, P stands for the row vector, and P represents the scalar quantity. 

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