Theoretical Physics

Josef Honerkamp Hartmann Romer

Theoretical Physics A Classical Approach

Translated by H. Pollack With 141 Figures and 39 Problems

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Professor Dr. Josef Honerkamp Professor Dr. Hartmann Romer

Albert-Ludwigs-Universitat, Fakultat fUr Physik, Hennann-Herder-StraBe 3, 0-79104 Freiburg, Gennany

Translator:

Howard Pollack

715 South Washington Street, Bloomington, IN 47401, USA

Title of the original German edition: Klassische Theoretische Physik, 3. Auflage (Springer-Lebrbuch) ISBN-13 : 978-3-642-77986-2 © Springer-Verlag Berlin Heidelberg 1986, 1989 and 1993

ISBN-13 : 978-3-642-77986-2 e-ISBN-13 : 978-3-642-77984-8 DOl: 10.1007/978-3-642-77984-8

Library of Congress Cataloging·in-Publication Data. Honerkamp, J. [Klassische Theoretische Physik. English] Theoretical physics: a classical approach / Josef Honerkamp, Hartmann Romer; translated by H. Pollack. p. cm. Translation of: Klassische Theoretische Physik. Includes bibliographical references and index. ISBN·13: 978·3·642-77986-2 (New York: alk. paper). I. Mathematical physics. I. Romer, H. (Hartmann) II. Title. QC20.H5713 1993 530'.1'51-dc20 92·42563

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Preface

This introduction to classical theoretical physics emerged from a course for students in the third and fourth semester, which the authors have given several times at the University of Freiburg (Germany).

The goal of the course is to give the student a comprehensive and coherent overview of the principal areas of classical theoretical physics. In line with this goal, the content, the terminology, and the mathematical techniques of theoret­ical physics are all presented along with applications, to serve as a solid foundation for further courses in the basic areas of experimental and theoretical physics.

In conceiving the course, the authors had four interdependent goals in mind:

• the presentation of a consistent overview, even at this elementary level • the establishment of a well-balanced interactive relationship between phys­

ical content and mathematical methods • a demonstration of the important applications of physics, and • an acquisition of the most important mathematical techniques needed to

solve specific problems.

In relation to the first point, it was necessary to limit the amount of material treated. This introductory course was not intended to preempt a later, primarily theoretical, course. On the other hand, we aimed for a certain completeness in the presentation of the basic principles and concepts of classical theoretical physics, which would serve as a lasting basis for later work. Emphasis was placed on presenting the material clearly and coherently, in the form of a clearly thought out (but not formalistic) introduction to the fundamental concepts and methods. To achieve clarity, the presentations, with few exceptions, go from the general to the particular. The conceptual framework is prepared first and is not developed much further in the examples. Nevertheless, after the structural fundamentals have been clearly explained, the carefully chosen examples play an essential role in each section of this course. Using these examples, the material which we have explained earlier becomes concrete and is demonstrated in a meaningful way.

In addition, we have provided a number of summaries, reviews of earlier material, and tastes of what is to come. This places the subject matter in a larger context, and anticipates further developments, all of which helps to provide a broad perspective on the entire material.

VI Preface

We also demonstrate, in many cases, how particular mathematical concepts and structures appear in different physical fields and contexts with different physical interpretations, for example in our treatment of the elementary results of linear algebra. We deliberately present mathematical concepts in a familiar manner, as they might be introduced in lectures in analysis and linear algebra. In this context, they are already familiar to the student, and this should help the student to recognize them in a physical context. Thus, mathematical knowledge is utilized. We have found that knowledge and understanding of these areas in physics as well as in mathematics have profited from this method.

We cannot talk of an appropriate interaction between physics and math­ematics if physics is seen only as an example of the realization of mathematical structures, or if conceptual exactness is confused with formalistic pedantry. Much is done to combat such a misunderstanding which often arises among students, particularly among talented ones. Physical and mathematical argu­ments are often developed in parallel, and carefully held apart from each other. Wherever possible, the physical origins of mathematical assumptions are re­vealed.

Thus, it is not only from lack of space that mathematical proofs are often avowedly incomplete, or even omitted; rather, this corresponds to our intention. For example, the theory of distributions is developed as far as possible within the conceptual framework of linear algebra, ignoring mathematical subtleties.

Here, again, the many examples we use are significant. We use not only dry, highly idealized systems, chosen for their easy treatment, such as the simple pendulum, but rather the manifold of physical phenomena, including examples from applied branches of physics like geophysics and physical chemistry. We discuss the examples as completely as possible, with particular emphasis on the physical interpretations of the results obtained. Thus, the connection is made between the physical situation, the mathematical formulation and discussion, and the intuitive physical results. It is here that the close relationship between mathematical deduction and intuitive interpretation, which is the essence of theoretical physics, emerges clearly.

These thoroughly discussed examples also serve the last primary goal: they demonstrate the value of mathematical-technical dexterity in the solution of problems.

This technical and methodological knowledge represents, so to speak, the tools of the trade. Familiarity with these techniques does not come from just listening to the lectures or reading, or even following the individual steps in the argument, it must also proceed from individual practice. It is essential for progress towards mastery that the student learns to use the equations, to find possible methods of solution, to go through the calculations in a problem, to interpret a result in its physical meaning, and to examine its plausibility himself or herself.

This is naturally the purpose of the exercises which always accompany an introductory theoretical course. For reasons of space, we have given only a small

Preface VII

collection of 39 worked-through homework problems. Comprehensive collec­tions of such exercises already exist in great numbers.

We should offer one word of explanation as to why this presentation of the fundamental principles of physics is limited to classical physics and thus leaves out modern, important, and "exciting" areas like relativity and quantum mechanics.

First, in the opinions of the authors, the addition ofthis material would have made it impossible to present the course in two semesters - without simultan­eously losing sight of the goal of active mastery of the basics as well as an overview of the entire subject matter.

Furthermore, the classical fields of physics have the advantage that they work within the realm of phenomena more easily accessible to immediate intuitive observation. The interaction between formal deduction and intuitive interpretation, which is tremendously important in theoretical physics, is best practised within the framework of classical physics. Only with a greater sense of security can the student then progress into a realm where intuitive understand­ing is less forthcoming.

We attempted to avoid unnecessary one-sidedness in the selection of mater­ial in the areas of classical physics represented. Thus, for example, statistical mechanics and thermodynamics, as well as the fundamentals of fluid mechanics, have received treatment here, based on their importance particularly for applied physics. As we have already stated, students should receive a sound foundation of knowledge as an initial preparation for further research in fields like quantum mechanics, relativity theory, fluid dynamics, analytical mechanics, irreversible thermodynamics, or the theory of dynamical systems.

Finally, we wish to thank all those who have contributed to the publication of this book. In particular, we want to name Mrs. H. Kranz, Mrs. E. Rupp, Mrs. E. Ruf, and Mrs. W. Wanoth, who wrote out the long, difficult manuscript and never lost patience during the countless corrections.

We thank Mrs. I. Weber and Mrs. B. Miiller for drawing the figures. We also express our gratitude to the participants in our course "Introduction to theoretical physics", in which this concept was first tested, for their many suggestions: also to those who took care of the accompanying exercises, above all Dr. H.C. Oettinger and Mr. R. Seitz, as well as Mr. P. Biller, Dr. H. Hess, Dr. M. Marcu, Mr. J. Miiller, Mr. G. Mutschler, and Dr. A. Saglio de Simonis. Mr. A. Geidel, Dr. H. Simonis, Mr. F.K. Schmatzer, and Mr. M. Zahringer gave us valuable assistance in proofreading.

Freiburg, June 1993 J. Honerkamp . H. Romer

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Newtonian Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Space and Time in Classical Mechanics. . . . . . . . . . . . . . . . . . 5 2.2 Newton's Laws .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 A Few Important Force Laws. . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 The Energy of a Particle in a Force Field. . . . . . . . . . . . . . . . 21

2.4.1 Line Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.2 Work and Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Several Interacting Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.6 Momentum and Momentum Conservation. . . . . . . . . . . . . . . 34 2.7 Angular Momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.8 The Two-Body Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.9 The Kepler Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.10 Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.10.1 Relative Motion in the Scattering Process. . . . . . . . . . 60 2.10.2 The Center of Mass System

and the Laboratory System. . . . . . . . . . . . . . . . . . . . . . 64 2.11 The Scattering Cross-Section. . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.12 The Virial Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.13 Mechanical Similarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.14 Some General Observations About the Many-Body Problem. : 81 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3. Lagrangian Methods in Classical Mechanics. . . . . . . . . . . . . . . . . . 87 3.1 A Sketch of the Problem and Its Solution

in the Case of a Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.2 The Lagrangian Method of the First Type . . . . . . . . . . . . . . . 89 3.3 The Lagrangian Method of the Second Type . . . . . . . . . . . . . 97 3.4 The Conservation of Energy in Motions

Which are Limited by Constraints. . . . . . . . . . . . . . . . . . . . . . 104 3.5 Non-holonomic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.6 Invariants and Conservation Laws . . . . . . . . . . . . . . . . . . . . . 117 3.7 The Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3.7.1 Lagrange's Equations and Hamilton's Equations. . . . . 123

X Contents

3.7.2 Aside on the Further Development of Theoretical Mechanics and the Theory of Dynamical Systems . . . . 128

3.8 The Hamiltonian Principle of Stationary Action. . . . . . . . . . . 133 3.8.1 Functionals and Functional Derivatives. . . . . . . . . . . . 133 3.8.2 Hamilton's Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.8.3 Hamilton's Principle for Systems

with Holonomic Constraints. . . . . . . . . . . . . . . . . . . . . 139 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4. Rigid Bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.1 The Kinematics of the Rigid Body. . . . . . . . . . . . . . . . . . . . . . 145 4.2 The Inertia Tensor and the Kinetic Energy of a Rigid Body. . 152

4.2.1 Definition and Elementary Properties of the Inertia Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4.2.2 Calculation of Inertia Tensors. . . . . . . . . . . . . . . . . . . . 157 4.3 The Angular Momentum of a Rigid Body, Euler's Equations. . 160 4.4 The Equations of Motion for the Eulerian Angles. . . . . . . . . . 167 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5. Motion in a Noninertial System of Reference. . . . . . . . . . . . . . . . . . 179 5.1 Fictitious Forces in Noninertial Systems. . . . . . . . . . . . . . . . . 179 5.2 Foucault's Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6. Linear Oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.1 Linear Approximations About a Point of Equilibrium. . . . . . 190 6.2 A Few General Remarks About Linear Differential Equations. 192 6.3 Homogeneous Linear Systems with One Degree of Freedom

and Constant Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 6.4 Homogeneous Linear Systems with n Degrees of Freedom

and Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.4.1 Normal Modes and Eigenfrequencies . . . . . . . . . . . . . . 200 6.4.2 Examples of the Calculation of Normal Modes . . . . . . 204

6.5 The Response of Linear Systems to External Forces. . . . . . . . 211 6.5.1 External Oscillating Forces. . . . . . . . . . . . . . . . . . . . . . 211 6.5.2 Superposition of External Harmonic Forces. . . . . . . . . 214 6.5.3 Periodic External Forces. . . . . . . . . . . . . . . . . . . . . . . . 215 6.5.4 Arbitrary External Forces. . . . . . . . . . . . . . . . . . . . . . . 216

Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

7. Classical Statistical Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.1 Thermodynamic Systems and Distribution Functions. . . . . . . 224 7.2 Entropy.......................................... 229 7.3 Temperature, Pressure, and Chemical Potential. . . . . . . . . . . 233

7.3.1 Systems with Exchange of Energy. . . . . . . . . . . . . . . . . 234 7.3.2 Systems with an Exchange of Volume. . . . . . . . . . . . . . 238 7.3.3 Systems with Exchanges of Energy and Particles. . . . . 239

Contents XI

7.4 The Gibbs Equation and the Forms of Energy Exchange. . . . 241 7.5 The Canonical Ensemble and the Free Energy. . . . . . . . . . . . 245 7.6 Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 7.7 Material Constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 7.8 Changes of State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

7.8.1 Reversible and Irreversible Processes . . . . . . . . . . . . . . 260 7.8.2 Adiabatic and Non-adiabatic Processes . . . . . . . . . . . . 264 7.8.3 The Joule-Thomson Process. . . . . . . . . . . . . . . . . . . . . 269

7.9 The Transformation of Heat into Work, the Carnot Efficiency 271 7.10 The Laws of Thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . 278 7.11 The Phenomenological Basis of Thermodynamics. . . . . . . . . . 282

7.11.1 Thermodynamics and Statistical Mechanics. . . . . . . . . 282 7.11.2 The First Law of Thermodynamics . . . . . . . . . . . . . . . . 284 7.11.3 The Second and Third Laws. . . . . . . . . . . . . . . . . . . . . 285 7.11.4 The Thermal and Caloric Equations of State. . . . . . . . 289

7.12 Equilibrium and Stability Conditions. . . . . . . . . . . . . . . . . . . 292 7.12.1 Equilibrium and Stability in Exchange Processes. . . . . 292 7.12.2 Equilibrium, Stability and Thermodynamic Potentials 296

Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

8. Applications of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 8.1 Phase Transformations and Phase Diagrams. . . . . . . . . . . . . 306 8.2 The Latent Heat of Phase Transitions. . . . . . . . . . . . . . . . . . . 309 8.3 Solutions......................................... 317 8.4 Henry's Law, Osmosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

8.4.1 Henry's Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 8.4.2 Osmosis..................................... 322

8.5 Phase Transitions in Solutions . . . . . . . . . . . . . . . . . . . . . . . . 325 8.5.1 Case (2): Miscibility in Only One Phase. . . . . . . . . . . . 325 8.5.2 Case (3): Miscibility in Two Phases. . . . . . . . . . . . . . . . 330

Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

9. Elements of Fluid Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 9.1 A Few Introductory Remarks About Fluid Mechanics. . . . . . 333 9.2 The General Balance Equation. . . . . . . . . . . . . . . . . . . . . . . . 337 9.3 Particular Balance Equations. . . . . . . . . . . . . . . . . . . . . . . . . 341 9.4 Entropy Production, Generalized Forces, and Fluids. . . . . . . 348 9.5 The Differential Equations of Fluid Mechanics. . . . . . . . . . . . 354 9.6 A Few Elementary Applications

of the Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . 359 Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

10. The Most Important Linear Partial Differential Equations of Physics 367 10.1 General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

10.1.1 Types of Linear Partial Differential Equations, the Formulation of Boundary and Initial Value Problems. . . . . . . . . . . . . . . . . . . . . . 367

XII Contents

10.1.2 Initial Value Problems in 1RD . . . . • . . . . . . . . . . . . . . . 371 10.1.3 Inhomogeneous Equations and Green's Functions. . . . 374

10.2 Solutions of the Wave Equation. . . . . . . . . . . . . . . . . . . . . . . 376 10.3 Boundary Value Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

10.3.1 Initial Observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 10.3.2 Examples of Boundary Value Problems. . . . . . . . . . . . 381 10.3.3 The General Treatment of Boundary Value Problems. . 386

10.4 The Helmholtz Equation in Spherical Coordinates, Spherical Harmonics, and Bessel Functions. . . . . . . . . . . . . . . 388 10.4.1 Separation of Variables. . . . . . . . . . . . . . . . . . . . . . . . . 389 10.4.2 The Angular Equations, Spherical Harmonics . . . . . . . 390 10.4.3 The Radial Equation, Bessel Functions. . . . . . . . . . . . . 395 10.4.4 Solutions of the Helmholtz Equation. . . . . . . . . . . . . . 398 10.4.5 Supplementary Considerations. . . . . . . . . . . . . . . . . . . 399

Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40t

11. Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 11.1 The Basic Equations of Electrostatics

and Their First Consequences. . . . . . . . . . . . . . . . . . . . . . . . . 405 11.1.1 Coulomb's Law and the Electric Field. . . . . . . . . . . . . 405 11.1.2 Electrostatic Potential and the Poisson Equation. . . . . 407 11.1.3 Examples and Important Properties

of Electrostatic Fields. . . . . . . . . . . . . . . . . . . . . . . . . . 410 11.2 Boundary Value Problems in Electrostatics,

Green's Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 11.2.1 Dirichlet and Neumann Green's Functions. . . . . . . . . . 413 11.2.2 Supplementary Remarks

on Boundary Value Problems in Electrostatics. . . . . . . 417 11.3 The Calculation of Green's Functions, the Method of Images 420 11.4 The Calculation of Green's Functions,

Expansion in Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . 427 11.5 Localized Charge Distributions, the Multipole Expansion. . . 431 11.6 Electrostatic Potential Energy. . . . . . . . . . . . . . . . . . . . . . . . . 434 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

12. Moving Charges, Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 12.1 The Biot-Savart Law, the Fundamental Equations

of Magnetostatics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 12.1.1 Electric Current Density and Magnetic Fields. . . . . . . 439 12.1.2 The Vector Potential and Ampere's Law. . . . . . . . . . . 443 12.1.3 The SI-System of Units in Electrodynamics. . . . . . . . . 446

12.2 Localized Current Distributions. . . . . . . . . . . . . . . . . . . . . . . 447 12.2.1 The Magnetic Dipole Moment. . . . . . . . . . . . . . . . . . . 447 12.2.2 Force, Potential, and Torque in a Magnetic Field. . . . 450

Contents XIII

13. Time Dependent Electromagnetic Fields. . . . . . . . . . . . . . . . . . . . . . 455 13.1 Maxwell's Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 13.2 Potentials and Gauge Transformations. . . . . . . . . . . . . . . . . . 458 13.3 Electromagnetic Waves in a Vacuum, the Polarization

of Transverse Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 13.4 Electromagnetic Waves, the Influence of Sources. . . . . . . . . . 464 13.5 The Energy of the Electromagnetic Field. . . . . . . . . . . . . . . .. 469

13.5.1 Balance of Energy and the Poynting Vector. . . . . . . . . 469 13.5.2 The Energy Flux of the Radiation Field. . . . . . . . . . . . 472 13.5.3 The Energy of the Electric Field. . . . . . . . . . . . . . . . . . 474 13.5.4 The Energy of the Magnetic Field . . . . . . . . . . . . . . . . 476 13.5.5 Self-Energy and Interaction Energy. . . . . . . . . . . . . . . 478

13.6 The Momentum of the Electromagnetic Field. . . . . . . . . . . . . 480

14. Elements of the Electrodynamics of Continuous Media .. . . . . . . . . 483 14.1 The Macroscopic Maxwell Equations. . . . . . . . . . . . . . . . . . . 483

14.1.1 Microscopic and Macroscopic Fields. . . . . . . . . . . . . . 483 14.1.2 The Average Charge Density

and Electric Displacement. . . . . . . . . . . . . . . . . . . . . . . 485 14.1.3 The Average Current Density

and the Magnetic Field Strength. . . . . . . . . . . . . . . . .. 488 14.2 Electrostatic Fields in Continuous Media. . . . . . . . . . . . . . . . 491 14.3 Magnetostatic Fields in Continuous Media. . . . . . . . . . . . . . . 496 14.4 Plane Waves in Matter, Wave Packets. . . . . . . . . . . . . . . . . . 499

14.4.1 The Frequency Dependence of Susceptibility. . . . . . . . 500 14.4.2 Wave Packets, Phase and Group Velocity. . . . . . . . . . 503

14.5 Reflection and Refraction at Plane Boundary Surfaces. . . . . . 508 14.5.1 Boundary Conditions, the Laws of Reflection

and Refraction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 14.5.2 Fresnel's Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 14.5.3 Special Effects of Reflection and Refraction. . . . . . . . . 513

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 A. The r-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 B. Conic Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 C. Tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 522 D. Fourier Series and Fourier Integrals. . . . . . . . . . . . . . . . . . . . . 526

D.1 Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 D.2 Fourier Integrals and Fourier Transforms. . . . . . . . . . . . 532

E. Distributions and Green's Functions .................... 536 E.1 Distributions................................... 536 E.2 Green's Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

F. Vector Analysis and Curvilinear Coordinates . . . . . . . . . . . . . . 543 F.1 Vector Fields and Scalar Fields. . . . . . . . . . . . . . . . . . . . 544 F.2 Line, Surface, and Volume Integrals. . . . . . . . . . . . . . . . . 544

XIV Contents

F.3 Stokes's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 F.4 Gauss's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 F.5 Applications of the Integral Theorems. . . . . . . . . . . . . . . 550 F.6 Curvilinear Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . 551

Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

Subject Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561