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Mathematical Techniques and Physical Applications PDF

710 Pages·1971·8.657 MB·English
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MATHEMATICAL TECHNIQUES AND PHYSICAL APPLICATIONS J. Killingbeck LECTURER IN PHYSICS UNIVERSITY OF HULL, HULL, ENGLAND G. H. A. Cole PROFESSOR OF THEORETICAL PHYSICS UNIVERSITY OF HULL, HULL, ENGLAND 0® A C A D E M IC PRESS New York and London 1971 COPYRIGHT © 1971, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London W1X 6BA LIBRARY OF CONGRESS CATALOG CARD NUMBER: 70-163766 PRINTED IN THE UNITED STATES OF AMERICA This is Volume 35 in PURE AND APPLIED PHYSICS A Series of Monographs and Textbooks Consulting Editors: H. S. W. MASSEY AND KEITH A. BRUECKNER A complete list of titles in this series appears at the end of this volume. Preface This book is intended as a senior/graduate level text for students of the physical sciences and, in particular, of physics. A good knowledge of physics (as might reasonably be expected of a good third-year student) can form a basis for studying the mathematics in the book; alternatively, a good foun­ dation in mathematics can form the basis for studying the physical examples of the text. The intention of the authors has been to place roughly equal emphasis on the physics and the mathematics with the goal of enabling the student to approach the theoretical aspects of the current journal literature, once versed in the subject matter of a given paper. This book is essentially a working book. While it can be used simply for general reading purposes, the full benefit will be obtained by the reader who pauses at the end of each section to try at least a few of the exercises. The exercises are chosen to give the reader the manipulative practice which will help him to assimilate more fully the subject matter of the preceding section. Many of the exercises set out material which is supplementary to the main text, and the longer ones are perhaps suitable for tutorial discussions. While many of the problems include hints, and are of the form " show that...it has been considered worthwhile to give a set of solutions and comments. It is hoped that this feature will enhance the usefulness of the text for tutorial and class work. A literature survey is provided for students who wish to study particular topics at greater length. The exercises also have a part to play in the gradual process of instilling a certain point of view in the reader, namely that of looking for connections between the structure of mathematical systems and that of physical systems. A further step is then to link various physical problems together, the same mathematical principles or systems being used for their description. As part of the process of immersing the reader in the material and ideas, certain topics occur in several chapters, and their value is repeatedly stressed. xiv Preface At several points, the word " isomorphism" occurs. The concept of " similar form," that is, isomorphism, is one of the most important in modern mathema­ tics, and most interesting for our purpose is the existence of isomorphisms between physical and mathematical structures. It would not be correct to say that this concept allows a complete codification of physics but it does unify large areas of the subject, and that is a decided advantage. Very often a phenomenon, for example, superconductivity, is described in terms of a simple physical or mathematical model, this model showing the essential features of the behavior studied, with irrelevant details omitted. Here we are dealing with a homomorphism, that is a correspondence between two systems that is not one-to-one in every detail. When physical systems are involved, the word isomorphism cannot really be applied in its exact mathematical sense, since experimental errors introduce an inevitable imprecision in the physical member of the comparison. Questions involving the theory of probability then arise, and some kind of average value is actually used in searching for any isomorphisms. However, we will not embark further on such a discussion here. Chapter 10 deals with some elemen­ tary probability theory. (A more detailed study of the role of probability theory in physics is planned as a further volume.) The approach adopted throughout the text differs from the usual "cookbook" approach; it may also be added that very few books dealing with mathematics for physicists include chapters dealing with the subject matter of Chapters 6 and 7. Before each chapter a summary of its main contents is given. The reader is advised to look through this, particularly if he is going to dip into the book rather than to study it systematically, since the reading of any chapter on its own will be slightly different from the reading of it in serial order. Comment on Notation The intention of this book is to make the reader familiar with a wide range of basic mathematical concepts and methods which are relevant to physical theory. The various chapters deal with different branches of traditional mathematics, each one of which has developed its own notation. We have decided not to attempt to make the notation uniform throughout the book, for two principal reasons. First, each branch of mathematics has developed its notation historically in the context of the problems which it studies, and the notation of one branch may not be well-adapted for use in the study of prob­ lems belonging to another branch. (As a simple example, the notation/" for a second derivative is economical in the study of second-order differential equations, but this notation cannot be easily extended (nto discuss general «th derivatives, which are best represented by the symbol f\) Second, when the student pursues his studies further, he will encounter this range of notations in the existing literature, and will have to learn to cope with it. For example, students often complain that the works of German authors on group theory are difficult because they use Gothic script symbols. This is essentially a problem of symbol recognition which has to be overcome by the student himself, but in almost all cases a little thought about the context makes clear what the symbol means mathematically, even if the reader cannot pronounce it! In the present text, a difference in the style or the thickness of the printed symbols is used to distinguish between scalars and vectors, matrices and matrix elements, groups and group elements, etc., and the context should make this clear. We do not think it necessary to give a complete glossary here, but point out as specific examples that the{ 1first derivative of a function / is denoted by the several symbols df/dx,f, Df9f\ and that volume integrals are denoted only rarely by multiple integral signs but usually by a dx or dx after the integrand. Vector Analysis 1.1. Scalars, Tensors, and Vectors In the mathematical description of physical phenomena several types of quantity occur that are essentially different. The simplest type of quantity is specified by only a number (or magnitude) and is called a scalar. Examples of scalar quantities are mass, temperature, density, and energy. There are more general physical quantities that are incompletely specified by a magnitude alone, but which also require the assignment of at least one direction. Such general physical quantities are represented by tensors. A tensor that has a single magnitude and a single direction is called a vector or tensor of rank one, A direction in three-dimensional space is defined by two independent cosines, and on including the magnitude, we see that three numbers are required to define a vector. Examples of a vector are displacements of position in three dimensions, velocity, force, and electric-field. Some physical quantities must be described by tensors of rank greater than one; e.g., the stress in a deformable medium and the electrical conductivity of an anisotropic solid or plasma. The tensors required are then of the second rank, and are specified by nine numbers in a three-dimensional space. If the electrical field is described by the three Cartesian components Ex, Ey, Ez, and the current by components Ix, Iy 9Iz, then the conductivity tensor Gt jis defined by the set of linear equations = G Ix xx^x ^xyEy + ^xz^z iy = (7yXEx -f GyyEy "h (Ty^ G (l) h = °zxEx + VzyEy + zz^z The amount of information required for the specification of a physical 2 Vector Analysis system will be greater if time is included as an extra variable coordinate. In relativistic theory, which involves a four-dimensional space-time system, a vector is defined by four numbers, and as analogy with Eq. (1) makes clear, a second-rank tensor requires sixteen numbers for its specification. Very often the tensors that describe physical systems have special symmetry properties; e.g., some second-rank tensors are symmetric: °ij = (Tjl (2) while some others are antisymmetric: (Jij = -(Tji (3) Thus, to describe gravitation according to general relativity theory, a sym­ metric space-time tensor g{ j is used; twelve of the sixteen numbers required to specify the tensor are equal in pairs. Tensors of rank greater than two are also used in physics, e.g. in the theories of anisotropic effects such as elasticity and piezoelectricity in crystal lattices, stress-strain relationships in amorphous materials, and the Einstein theory of gravitation (for the curvative tensor of rank four, see Chapter 3). Tensor calculus, sometimes called the absolute differential calculus, is the mathematical theory of relationships that are independent of choice of co­ ordinate axes; this theory applies equally to any number of dimensions. The fact that relations between tensors are independent of any particular co­ ordinate system is of great interest to physicists. A basic principle is that physical laws should be independent of the reference frame chosen, and this means that the tensor calculus provides a natural formalism for the mathe­ matical expression of physical laws. The indifference of physical laws or mathematical relationships to a choice of coordinate system is called covari- ance, and the postulate of covariance of physical laws is one of the most basic principles of theoretical physics. As a simple example, the distance over the earth's surface between Lands End and John o'Groat's surely does not depend on whether we use Cartesian or spherical polar coordinates for its measure­ ment. The change from one set of coordinate axes to another involves trans­ formations, and it is possible for a relationship to be covariant for some transformations and not covariant for others. One of the tasks of theoretical physics is to frame physical laws so that they are covariant under the most general transformations; e.g., the equations of classical mechanics must be modified to a relativistic form before they are covariant with respect to the Lorentz transformations of relativity theory. The full tensor formalism, which applies to the most general changes of axes, is treated in Chapter 3. In this chapter we deal with the traditional vector analysis. 1.2. Scalar, Vector, and Tensor Fields 3 The transformations with respect to which the Cartesian vector and tensor relationships are covariant are those involving translations and rotations between Cartesian sets of axes. This facet of conventional vector theory is usually not stressed but is mentioned here because of the way in which it fits into the more general tensor calculus. Having gained familiarity with the transformation properties of the usual vectors and tensors, the reader should find it easier to proceed to the more general types of transformation properties dealt with in Chapter 3. EXERCISES 1. Suppose that the conductivity tensor of Eq. (1) has the components <*ij = 0 0' # j)> <*xx = Vyy = ° z z =° What can be said about the vectors E and I ? 2. Consider a second-rank tensor in n dimensions. If it is symmetric, how many independent numbers are needed to specify it ? In how many dimen­ sions is the number of nonzero independent components o i } of an anti­ symmetric second-rank tensor equal to the number of components of a vector ? 1.2. Scalar, Vector, and Tensor Fields Probably the most frequently encountered examples of a scalar and a vector in elementary physics are the mass and velocity of a moving particle. If, how­ ever, the temperature within a solid is considered (temperature being a scalar quantity), we have a scalar field, i.e., a set of values of a scalar that must be assigned throughout a continuous region of space; furthermore the field may be time dependent if heat is supplied to the solid. A solid in which the electrical conductivity changes from point to point provides an example of a tensor field. A fluid flowing along a tube of varying cross section is another example of a continuous system; in this case, if we specify the fluid velocity at each point, we obtain a vector field, which may be time dependent if the pressure difference across the tube is varied. Thus a field involves extension and con­ tinuity in space and time. A useful graphical representation of a scalar field is that used in geo­ graphical survey maps, in which contour lines of constant altitude are drawn. (The altitude is, of course, a scalar field.) If the scalar field variable is denoted by cj), a surface or curve of constant $ is called an isotimic surface or curve. Regions in which (f) changes most rapidly have the isotimic surfaces most closely packed together (see Fig. 1.1). 4 Vector Analysis Fig. 1.1. Isotimic surfaces displayed as contours of constant </>. The contours are more closely spaced the greater the rate of change of (f>. Thus the rate of change of <f> in the direction of the line A is greater than that in the direction B. It is sometimes useful to treat a component of a vector field as a scalar field in its own right; e.g., a plot of the horizontal component of the earth's magnetic field can be represented as a scalar field. This procedure generates a scalar field from a vector field. If we give at every point in a scalar field the direction in which (j) changes most rapidly, and also the magnitude of the maximum rate of change, then we generate a vector field from the scalar field. (This procedure is investigated in Sec. 1.5A.) Graphical representation of a vector field V is complicated because both magnitude and direction must be specified at every point in the field region. Instead of lines of constant V, we often use field lines that are everywhere tangential to the vector-field direction; further, the number of these lines pas­ sing through unit area perpendicular to the field at any point is proportional to the vector-field magnitude at that point. This kind of representation of a vector field will be familiar to those readers who have encountered " lines of force " in the elementary theory of electricity and magnetism. From their mode of construction, the lines representing the vector field cannot cross, since this would yield a nonunique field direction at the point of intersection (but see Exercise 4). If we form a tube whose surface is composed of field lines (see Fig. 1.2), then the field magnitudes and tube cross section at different points are related by 6 = |V(2)| dS2 - |V(1)| dSt = 0 (4) if we invoke the no-crossing idea and suppose that no field lines start or end

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