Vector quantities

Vector quantities, such as a magnetic field or the gradient of electron density, can be plotted as a series of arrows. Another technique is to create an animation showing how the path is followed by a hypothetical test particle. A third technique is to show flow lines, which are the path of steepest descent starting from one point. The flow lines from the bond critical points are used to partition regions of the molecule in the AIM population analysis scheme. 

A vector quantity is indicated by bold italic type and its magnitude by italic type. 

The Einstein coefficients are related to the wave functions j/ and of the combining states through the transition moment R , a vector quantity given by... 

In the examples shown in Figures 4.18(a)-4.18(g) all the molecules clearly have a charge asymmetry and, therefore, a non-zero dipole moment. Since a dipole moment has magnitude and direction it is a vector quantity and, if we wish to emphasize this, we use the vector symbol /x, whereas if we are concerned only with the magnitude we use the symbol /r. 

For an electron having orbital and spin angular momentum there is a quantum number j associated with the total (orbital + spin) angular momentum which is a vector quantity whose magnitude is given by... 

Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components i , and v.. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. 

Momentum Balance Since momentum is a vector quantity, the momentum balance is a vector equation. Where gravity is the only body force acting on the fluid, the hnear momentum principle, apphed to the arbitraiy control volume of Fig. 6-3, results in the following expression (Whitaker, ibid.). 

Depending on the structure of the optical probe, components of vector quantities (velocity field, displacement field) and their signs can be distinguished in measurements, ensuring directional sensitivity. 

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... 

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. 

Force. The action of one body on another. This action will cause a change in the motion of the first body unless counteracted by an additional force or forces. A force may be produced either by actual contact or remotely (gravitation, electrostatics, magnetism, etc.). Force is a vector quantity. 

Velocity. A measure of the instantaneous rate of change of position in space with respect to time. Velocity is a vector quantity. 

Conservation of Momentum. If the mass of a body or system of bodies remains constant, then Newton s second law can be interpreted as a balance between force and the time rate of change of momentum, momentum being a vector quantity defined as the product of the velocity of a body and its mass. 

While the modified energy equation provides for calculation of the flowrates and pressure drops in piping systems, the impulse-momenlum equation is required in order to calculate the reaction forces on curved pipe sections. I he impulse-momentum equation relates the force acting on the solid boundary to the change in fluid momentum. Because force and momentum are both vector quantities, it is most convenient to write the equations in terms of the scalar components in the three orthogonal directions. 

The time rate of change of position of a body. It is a vector quantity having direction as well as magnitude. 

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)... 

To evaluate the terms in equation (10.38) we relate e, the energy of a particular molecule to its momentum / , (not to be confused with pressure p). Momentum is a vector quantity that is related to its components along the three coordinate axes by... 

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. 

In practical terms, this can lead to the possibility of using both mass dimensions as fundamentals, thereby achieving similar advantages to those arising from taking length as a vector quantity. This subject is discussed in more detail by Huntley 5. ... 

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. 

Force and velocity are however both vector quantities and in applying the momentum balance equation, the balance should strictly sum all the effects in three dimensional space. This however is outside the scope of this text and the reader is referred to more standard works in fluid dynamics. 

The PPP-MO method is capable of calculating not only the magnitude of the dipole moment change on excitation, but it can also predict the direction of the electron transfer. The vector quantity that expresses the magnitude and direction of the electronic transition is referred to as the transition dipole moment. For example, the direction of the transition dipole moment of azo dye 15f as calculated by the PPP-MO method is illustrated in Figure 2.15. 

Why, then, is the magnetisation density used The answer is that the magnetisation density is important for certain approximations which are usually made in analysing neutron scattering experiments. In the standard polarised neutron diffraction (PND) experiment [5], only one parameter is measured - the so-called flipping ratio . It is impossible to determine a vector quantity like the magnetisation density from a single number, unless some assumptions are made. The assumptions usually made are ... 

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. 

A second question arises for those who understand the importance of dimensional analysis, a subject that is treated briefly in Appendix II. If A and B are both vector quantities with, say, dimensions of length, how can their cross product result in a vector C, presumably with dimensions of length The answer is hidden in the homogeneous equations developed above [Eqs. (IS) to (20)]. The constant a was set equal to unity. However, in this case it has the dimension of reciprocal length. In other words, C = aABsirtd is the length of the vector C. In general, a vector such as C which represents the cross product of two ordinary vectors is an areal vector with different symmetry properties from those of A and B. 

The term scalar field is used to describe a region of space in which a scalar function is associated with each point. If there is a vector quantity specified at each point, the points and vectors constitute a vector field. 

Here, u, v, and w are the components of the velocity vector in the x, y, and z directions, respectively. Note that velocity is treated as a vector quantity, so that the vector sum ui + vj + wk (where i, j, and k are the unit vectors in the x, y, and z directions) represents both the direction and magnitude of the fluid velocity at a particular position and time. The symbol P represents fluid pressure, p is the fluid viscosity, p is the fluid density, and the F parameters are the components of a body force acting on the fluid in the x, y, and z directions. (A body force is a force that acts on the fluid as a result of its mass rather than its surface area gravity is the most common body force.)... 

The Navier-Stokes equations have a complex form due to the necessity of treating many of the terms as vector quantities. To understand these equations, however, one need only recognize that they are not mass balances but an elaboration of Newton s second law of motion for a flowing fluid. Recall that Newton s second law states that the vector sum of all the forces acting on an object ( F) will be equal to the product of the object s mass (m) and its acceleration (a), or XF = ma. Now consider the first of the three Navier-Stokes equations listed above, Eq. (10). The object in this case is a differential fluid element, that is, a small cube of fluid with volume dx dy dz and mass p(dx dy dz). The left-hand side of the equation is essentially the product of mass and acceleration for this fluid element (ma), while the right-hand side represents the sum of the forces... 

There have been a number of past attempts to unify hardness measurements but they have not succeeded. In several cases, hardness numbers have been compared with scalar properties that is, with cohesive energies (Plendl and Gielisse, 1962) or bulk moduli (Cohen, 1988). But hardness is not based on scalar behavior, since it involves a change of shape and is anisotropic. Shape changes (shears) are vector quantities requiring a shear plane, and a shear direction for their definition. In this book, the fact that plastic... 

The direction of polarity of a polar bond is symbolized by a vector quantity ... 

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