Mathematical Modeling of Diverse Phenomena

This book is intended for those students, engineers, scientists, and applied mathematicians who find it necessary to formulate models of diverse phenomena. To facilitate the formulation of such models, some aspects of the tensor calculus will be introduced. However, no knowledge of tensors is assumed. The chief aim of this calculus is the investigation of relations that remain valid in going from one coordinate system to another. The invariance of tensor quantities with respect to coordinate transformations can be used to advantage in formulating mathematical models. As a consequence of the geometrical simplification inherent in the tensor method, the formulation of problems in curvilinear coordinate systems can be reduced to series of routine operations involving only summation and differentiation. When conventional methods are used, the form which the equations of mathematical physics assume depends on the coordinate system used to describe the problem being studied. This dependence, which is due to the practice of expressing vectors in terms of their physical components, can be removed by the simple expedient of expressing all vectors in terms of their tensor components. For the benefit of those who have access to digital computers equipped with formula manipulation compilers, the convenience of computerized formulations will be demonstrated. No programming experience is necessary, and the few programming steps required will be explained as they occur. The first chapter is concerned with those aspects of the tensor calculus that are considered necessary for an understanding of later chapters. It is assumed that the reader has a knowledge of elementary vector analysis and matrix operations. The reader may encounter unfamiliar entities such as covariant and contravariant vectors and tensors, and unfamiliar operations such as covariant differentiation. It will be seen, however, that the only operations involved in applying these concepts to practical problems are summation, in accordance with the summation convention, and differentiation. In using tensor methods to formulate mathematical models, considerable insight is obtained and the striking similarity of all formulations of physical systems becomes apparent. This is due to the fact that all such formulations evolve from a fundamental metric which is simply an expression for the square of the distance between two adjacent points on a surface. Hence, in addition to its utility, the method advocated has a definite educational value. The major part of the book is devoted to applications using the theory given in the first chapter. The applications are chosen to demonstrate the feasibility of combining tensor methods and computer capability to formulate problems of interest to students, engineers, and theoretical physicists. Chapter 2 is devoted to aeronautical applications that culminate in the formulation of a mathematical model of an aeronautical system. In Chapter 3, the equations of motion of a particle are formulated in tensor form. The methods described in Chapter 4 can be used to formulate mathematical models involving fluid dynamics. The tensor theory contained in Chapter 1 is required to formulate the cosmological models described in Chapter 5. The final chapter describes how the symbol manipulation language MACSYMA may be used to assist in the formulation of mathematical models. The techniques described in this book represent an attempt to simplify the formulation of mathematical models by reducing the modeling process to a series of routine operations, which can be performed either manually or by computer. This attempt is part of a continuing effort in support of simulation experimentation in the Simulation Sciences Division