Functions of Many Variables

It is probably obvious that in most cases the dependence between quantities is much more complicated than that of a function of a single variable. Although this chapter is concerned with functions of many variables, we will illusdate these by considering a function / of two variables x and y. The extension to further variables is then sdaightforward. 

In mathematical notation the function referred to above would be denoted as / (x, y). In the example where this is defined as 

The Born-Oppenheimer approximation states the vibrational and rotational energies of a molecule can be separated and the individual terms added. The overall energy E can thus be considered as a function of the vibrational quantum number v (taking values 0, 1, 2,. ..) and the rotational quantum number J (which independently takes values 0, 1, 2.). For a diatomic molecule it can be approximated by the function 

Selection rules tell us which transitions between rotational and vibrational levels are allowed. The values of v may increase or decrease by 1. Similarly the values of J may independently increase or decrease by 1. For a transition between vibrational levels v and v, where v — v = 1, we have 

Since J can take only integral values these energy levels are equally spaced. The resulting branch of the spectrum is known as the R branch. 

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Once identified, the costs and/or savings are placed into their appropriate categories and quantified for subsequent analysis. Equipment cost is a function of many variables, one of the most significant of which is capacity. Other important variables include operating temperature and/or pressure conditions, and degree of equipment sophistication. Preliminary estimates are often made using simple cost-capacity relationships that are vahd when the other variables are confined to a narrow range of values. 

In this context it is the separation of solids from water by forcing the water through a porous filter media. The objective is typically to reduce the level of TDS in the water and often to reduce both the size of the particle remaining and the turbidity of the water. Filtration efficiency and quality is a function of many variable factors, although filtration is usually carried out at relatively low velocities, where velocity and pressure drop are directly related to each other. Typically a sand filter will remove a high percentage of particles above a diameter of 20 to 30 pm, whereas dual or multimedia filtration is required to remove particles down to a diameter of 10 to 20... 

The actual task of finding the correct free concentrations [M and [L] is undertaken by the Solver. The Solver is a very powerful tool in Excel. It can be employed to maximise and minimise functions of many variables and to find solutions to functions of many variables. The Solver can be found in the Tools menu. If it is not there, it has to be installed as an Add-In, also found... 

Two central problems remain. One is that one needs the potential which governs the motion. In many-atom systems, even if the motion is confined to the ground electronic state, this potential is a function of the spatial configuration of all the atoms. It is therefore a function of many variables, so its analytical form is far from obvious, nor do we necessarily want to know it everywhere. Indeed, we really only want it at each point along the actual trajectory of the system (so that the forces can be computed and thereby the next point to which the system will move to can be determined). Such an approach has been implemented [25] and applied to many-atom systems, and an extension to a multi-electronic state dynamics will be important... 

In relating properties of molecules to their structure, three-dimensional shape is frequently of great importance. Three-dimensional shape is a function of many variables the nature and number of atoms composing the molecule and the nature of the chemical bonding pattern— which atoms are connected to which—are obvious factors. However, the situation can be more subtle than that. Even in cases in which the atomic composition of two molecules is the same and in which the chemical bonding pattern is the same, key differences in three-dimensional shape can arise. 

Note that, in principle, geometry optimization could be a separate chapter of this text. In its essence, geometry optimization is a problem in applied mathematics. How does one find a minimum in an arbitrary function of many variables [Indeed, we have already discussed that... 

An example is the heat distortion test (ASTM D 648-45 T), which is reproducible and of practical use but yields little information of value to the materials scientist. The reason for this lies in the fact that the quantity measured in the test is a function of many variables whereas the materials scientist usually seeks measures which are functions of... 

Eq(10-16)is known to conform closely with experimental observations. The chief difficulties in applying Eq (10-16) are due to determination of the constants C and Kv. These are functions of many variables including temperature, packing voids, and particle size. 

Applications of Darcy s Law—It is clear that the permeability constant is a function of many variables, of which particle-size and porosity are obviously the most important. Darcy s law, however, has important applications to ground-water hydrology where methods of classical... 

Nonderivative minimization methods are generally easy to implement and avoid derivative computations, but their realized convergence properties are rather poor. They may work well in special cases when the function is quite random in character or the variables are essentially uncorrelated. In general, the computational cost, dominated by the number of function evaluations, can be excessively high for functions of many variables and can far outweigh the benefit of avoiding derivative calculations. 

D. P. O Leary, Math. Prog., 23, 20 (1982). A Discrete Newton Algorithm for Minimizing a Function of Many Variables. 

In this book it is always to be understood that in taking the partial derivative of g with respect to x, then y is held constant, and in taking the partial derivative of g with respect to y, then x is held constant. Since we will be dealing later with functions of many variables, it is impractical to indicate the variables held constant as subscripts. If we take the differential of the... 

Although combustion efficiency is a function of many variables (the most important being combustion... 

The composition of subsurface waters is a complex function of many variables, including ... 

For this FID ky =k = 0, so that t otk. Consequently, yi (t) in (5.4.12) defines a cross-section pikjc) in k space along the kx-axis. This cross-section is the Fourier transform of P(x), and P(x) is a ID projection of the 3D distribution Mo(x, y, z) of longitudinal magnetization onto the x-axis corresponding to the integral of Mo(x, y, z) over the space dimensions y and z (cf. Fig. 1.1.5). In the mathematical sense, a projection is just the integral over a function of many variables, so that fewer variables remain. Fourier transformation over t leads to... 

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