Some Basic Calculus

Simply speaking, a functional maps a function into a number. An example is the area under a curve, which is a functional of the function that defines the curve between two points. An arbitrary functional, which we call Cl, can be 

As in the theory of functions, a calculus exists for functionals. This calculus provides the tools necessary to develop and implement density functional theory. We begin with the discussion of expansions of functionals, which plays an important role in developing models within DFT and in deriving perturbation expansions. Analogous to the Taylor Series expansion for a function, a functional can be expanded about a reference function. This expansion, called a Volterra expansion, exists provided the functional has functional derivatives to any order and provided the last term in the infinite expansion has limit zero. Assuming these conditions, the Volterra expansion of ft[p] about a reference function, po, is given by 

The definition of the operator Ap(r,) - Ap(r ) implies that for = 0 the fol-lowine reladonshio holds 

Isolating like powers of X. gives the order-by-order corrections to fl[p]. When applying the above perturbation expansion, we assume that to any order of the perturbation expansion all lower order solutions are exactly known. This greatly simplifies the resulting order-by-order corrections to The above perturbation expansion is completely general and can also apply to wavefunc-tion funaional theory. 

Functional derivatives are easily performed using the following expression  

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By applying some basic calculus, we can integrate this equation into a new formula (it is not necessary for you to know how this was derived) ... 

This chapter is primarily about wave mechanics since that is the most convenient way to introduce undergraduates to quantum mechanics using calculus. An equivalent form called matrix mechanics will be discussed briefly in a later chapter. Consider again the 1923 paper by De Broglie and the experiments that validated the particle-wave duality in Chapter 10. One might well ask that if there really is some wave that describes the behavior of particles, then is there an equation that the wave obeys Even today it is difficult to say what the wave is, but it may help to find an equation it obeys. Step back a moment to some basic calculus ... 

The evaluation of matrix elements for exphcitly correlated Gaussians (46) and (49) can be done in a very elegant and relatively simple way using matrix differential calculus. A systematic description of this very powerful mathematical tool is given in the book by Magnus and Neudecker [105]. The use of matrix differential calculus allows one to obtain compact expressions for matrix elements in the matrix form, which is very suitable for numerical computations [116,118] and perhaps facilitates a new theoretical insight. The present section is written in the spirit of Refs. 116 and 118, following most of the notation conventions therein. Thus, the reader can look for information about some basic ideas presented in these references if needed. 

Recognizing that many chemistry students do not have a strong background in physics, I have introduced most of the chapters with some essential physics, concerning waves, mechanics, and electrostatics. I have also tried to keep the mathematical level at a minimum, consistent with a proper understanding of what is necessary. Basic calculus and an understanding of the properties of elementary trigonometical and exponential functions are assumed but I have not used complex numbers. Each chapter ends with some simple problems. 

In this Chapter, we quickly review some basic definitions and concepts from thermodynamics. We then provide a brief description of the first and second laws of thermodynamics. Next, we discuss the mathematical consequences of these laws and cover some relevant theorems in multivariate calculus. Finally, free energies and their importance are introduced. 

In this appendix some important mathematical methods are briefly outlined. These include Laplace and Fourier transformations which are often used in the solution of ordinary and partial differential equations. Some basic operations with complex numbers and functions are also outlined. Power series, which are useful in making approximations, are summarized. Vector calculus, a subject which is important in electricity and magnetism, is dealt with in appendix B. The material given here is intended to provide only a brief introduction. The interested reader is referred to the monograph by Kreyszig [1] for further details. Extensive tables relevant to these topics are available in the handbook by Abramowitz and Stegun [2]. 

Mainardi, F., Fractional calculus some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum mechanics, Carpinteri, A. and Mainardi, F., Eds., Springer, Wien, 1997, pp. 291-348. 

It is the nature of the subject that makes its presentation rather formal and requires some basic, mainly conceptual knowledge in mathematics and physics. However, only standard mathematical techniques (such as differential and integral calculus, matrix algebra) are required. More advanced subjects such as complex analysis and tensor calculus are occasionally also used. Furthermore, also basic knowledge of classical Newtonian mechanics and electrodynamics will be helpful to more quickly understand the concise but short review of these matters in the second chapter of this book. 

Considerable advances in the formal development of DFT as well as in its practical application have been made these last few years. In this chapter, we will not attempt to provide a comprehensive review of the subject, but rather provide an introduction to DFT, covering many of the important attributes that pertain to chemical problems. In the next section, we provide some of the basic calculus associated with functionals. Then some early models used in DFT are presented, followed by a formal presentation of DFT and the formalism used in practical applications. Finally, in the last section, we discuss the ability of DFT to compete with WFT in electronic structure calculations. A more detailed and comprehensive discussion of DFT can be found in the 1989 book by Parr and Yang. In what follows, we assume the Born-Oppenheimer nonrelativistic approximation and use atomic units throughout. Since the nuclei are assumed to be fixed in space, we will not explicitly show this dependence in equations and variables. 

In thermodynamics, the state of a system is specified in terms of macroscopic state variables such as volume V, pressure p, temperature T, mole numbers of the chemical constituents N, which are self-evident. The two laws of thermodynamics are founded on the concepts of energy U, and entropy S, which, as we shall see, are functions of state variables. Since the fundamental quantities in thermodynamics are functions of many variables, thermodynamics makes extensive use of calculus of many variables. A brief summary of some basic identities used in the calculus of many variables is given in Appendix 1.1 (at the end of this chapter). Functions of state variables, such as U and S, are called state functions. 

A complete study of quantum mechanics (QM) would require a knowledge of higher mathematics and a course in theoretical physics, but the derivation of some basic equations for the calculation of molecular energies requires no more than a little algebra and calculus [1]. 

We now consider probability theory, and its applications in stochastic simulation. First, we define some basic probabihstic concepts, and demonstrate how they may be used to model physical phenomena. Next, we derive some important probability distributions, in particular, the Gaussian (normal) and Poisson distributions. Following this is a treatment of stochastic calculus, with a particular focus upon Brownian dynamics. Monte Carlo methods are then presented, with apphcations in statistical physics, integration, and global minimization (simulated annealing). Finally, genetic optimization is discussed. This chapter serves as a prelude to the discussion of statistics and parameter estimation, in which the Monte Carlo method will prove highly usefiil in Bayesian analysis. 

An understanding of optimization techniques does not require complex mathematics. We require as background only basic tools from multivariable calculus and linear algebra to explain the theory and computational techniques and provide you with an understanding of how optimization techniques work (or, in some cases, fail to work). 

This volume is addressed mainly to anyone interested in the life sciences. There are, however, a few minimal prerequisites, such as elementary calculus and thermodynamics. A basic knowledge of statistical thermodynamics would be useful, but for understanding most of this book (except Chapter 9 and some appendices), there is no need for any knowledge of statistical mechanics. 

William M. Jackson, University of California, Davis The kind of description that Joseph Francisco talked about is what we actually tried at University of California, Davis, with an NSF grant. It was in physical sciences and not just in chemistry. What we found was that just by putting freshmen in the research groups and laboratories of professors, students whose average grade point levels were Cs were raised in the basic core courses—chemistry, physics, and math and calculus were raised to Bs. They were doing research, and then we had some other interventions, but it was an observation that I had made before I got to the Davis campus. 

This book presents in a popular manner the elements of game theory—the mathematical study of conflict situations whose purpose is to work out recommendations for a rational behaviour of each of the participants of a conflict situation. Some methods for solving matrix games are given. There are but few proofs in the book, the basic propositions of the theory being illustrated by worked examples. Various conflict situations are considered. To read the book, it is sufficient to be familiar with the elements of probability theory and those of calculus. 

Mathematical Level. Generally, the principles of chemical analysis developed here are based on college algebra. Some of the concepts presented require basic differential and integral calculus. 

Some of the basic principles of the calculus of molecular chaos may be illustrated in a more direct way by the detailed consideration of the expansion of a perfect gas. 

The chosen modeling approach is Colored Petri Nets as implemented in CPN tools. The model construction has confirmed the expressional power of CPN. All basic mechanisms and procedures can be modeled with sufficient level of detail and exactness. However some limits were identified. The most exphcit one can be found in Subset-091 Safety Requirements for the Technical Interoperability of ETCS in Levels 1 2. This document gives very concrete values to be satisfied for safety assessments. These levels are standard SIL4 orders such as 10 dangerous failures/hour. Even though, these values can be integrated in the model, their in-depth analysis is harsh due to the lack of analytical calculus tools in the used software. 

This textbook assumes that the reader has completed a basic first-year university course, including univariate calculus and Unear algebra. Multivariate calculus, set theory, and numerical methods are useful for understanding some of the concepts. 

In the second form of Eq. (5.2) the notation has been changed to emphasize that the interest is in the derivative at some point x and Ax is some change in X around that point. If only the first order term in Eq. (5.2) is kept, this leads to the basic single sided derivative definition that is typically learned from Calculus of... 

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