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MATHEMATICS OF PHYSICS AND ENGINEERING r '** Wee, j±g BORN EDWARD K. BLUM SERGEY v. LOTOTSKY MATHEMATICS OF PHYSICS AND ENGINEERING This page is intentionally left blank MATHEMATICS OF PHYSICS AND ENGINEERING EDWARD K. BLUM SERGEY v LOTOTSKY University of Southern California, USA YJ? World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. About the cover: The pyramid design by Martin Herkenhoff represents the pyramid of knowledge built up in levels over millenia by the scientists named, and others referred to in the text. MATHEMATICS OF PHYSICS AND ENGINEERING Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-256-621-X Printed in Singapore by B & JO Enterprise To Lori, Debbie, Beth, and Amy. To Kolya, Olya, and Lya. This page is intentionally left blank Preface What is mathematics of physics and engineering? An immediate answer would be "all mathematics that is used in physics and engineering", which is pretty much ... all the mathematics there is. While it is nearly impossible to present all mathematics in a single book, many books on the subject seem to try this. On the other hand, a semester-long course in mathematics of physics and engineering is a more well-defined notion, and is present in most univer sities. Usually, this course is designed for advanced undergraduate students who are majoring in physics or engineering, and who are already familiar with multi-variable calculus and ordinary differential equations. The basic topics in such a course include introduction to Fourier analysis and partial differential equations, as well as a review of vector analysis and selected top ics from complex analysis and ordinary differential equations. It is therefore useful to have a book that covers these topics — and nothing else. Besides the purely practical benefits, related to the reduction of the physical di mensions of the volume the students must carry around, the reduction of the number of topics covered has other advantages over the existing lengthy texts on engineering mathematics. One major advantage is the opportunity to explore the connection be tween mathematical models and their physical applications. We explore this connection to the fullest and show how physics leads to mathematical models and conversely, how the mathematical models lead to the discovery of new physics. We believe that students will be stimulated by this inter play of physics and mathematics and will see mathematics come alive. For example, it is interesting to establish the connection between electromag- netism and Maxwell's equations on the one side and the integral theorems of vector calculus on the other side. Unfortunately, Maxwell's equations vii viii Mathematics of Physics and Engineering are often left out of an applied mathematics course, and the study of these equations in a physics course often leaves the mathematical part somewhat of a mystery. In our exposition, we maintain the full rigor of mathemat ics while always presenting the motivation from physics. We do this for the classical mechanics, electromagnetism, and mechanics of continuous medium, and introduce the main topics from the modern physics of rel ativity, both special and general, and quantum mechanics, topics usually omitted in conventional books on "Engineering Mathematics." Another advantage is the possibility of further exploration through prob lems, as opposed to standard end-of-section exercises. This book offers a whole chapter, about 30 pages, worth of problems, and many of those prob lems can be a basis of a serious undergraduate research project. Yet another advantage is the space to look at the historical developments of the subject. Who invented the cross product? (Gibbs in the 1880s, see page 3.) Who introduced the notation i for the imaginary unit sf^ll (Euler in 1777, see page 79.) In the study of mathematics, the fact that there are actual people behind every formula is often forgotten, unless it is a course in the history of mathematics. We believe that historical background material makes the presentation more lively and should not be confined to specialized history books. As far as the accuracy of our historical passages, a disclaimer is in order. According to one story, the Russian mathematician ANDREI NiKOLAEVlCH KOLMOGOROV (1903-1987) was starting as a history major, but quickly switched to mathematics after being told that historians require at least five different proofs for each claim. While we tried to verify the historical claims in our presentation, we certainly do not have even two independent proofs for most of them. Our historical comments are only intended to satisfy, and to ignite, the curiosity of the reader. An interesting advice for reading this, and any other textbook, comes from the Russian physicist and Nobel Laureate LEV DAVIDOVICH LANDAU (1908-1968). Rephrasing what he used to say, if you do not understand a particular place in the book, read again; if you still do not understand after five attempts, change your major. Even though we do not intend to force a change of major on our readers, we realize that some places in the book are more difficult than others, and understanding those places might require a significant mental effort on the part of the reader. While writing the book, we sometimes followed the advice of the German mathematician CARL GUSTAV JACOB JACOBI (1804-1851), who used to say: "One should always generalize." Even though we tried to keep abstract Preface IX constructions to a minimum, we could not avoid them altogether: some ideas, such as the separation of variables for the heat and wave equations, just ask to be generalized, and we hope the reader will appreciate the benefits of these generalizations. As a consolation to the reader who is not comfortable with abstract constructions, we mention that everything in this book, no matter how abstract it might look, is nowhere near the level of abstraction to which one can take it. The inevitable consequence of unifying mathematics and physics, as we do here, is a possible confusion with notations. For example, it is customary in mathematics to denote a generic region in the plane or in space by G, from the German word Gebiet, meaning "territory." On the other hand, the same letter is used in physics for the universal gravitational constant; in our book, we use G to denote this constant (notice a slight difference between G and G). Since these two symbols never appear in the same formula, we hope the reader will not be confused. We are not including the usual end-of-section exercises, and instead incorporate the exercises into the main presentation. These exercises act as speed-bumps, forcing the reader to have a pen and pencil nearby. They should also help the reader to follow the presentation better and, once solved, provide an added level of confidence. Each exercise is rated with a super-script A, B, or C; sometimes, different parts of the same exercise have different ratings. The rating is mostly the subjective view of the authors and can represent each of the following: (a) The level of difficulty, with C being the easiest; (b) The degree of importance for general understanding of the material, with C being the most important; (c) The aspiration of the student attempting the exercise. Our suggestion for the first reading is to understand the question and/or conclusion of every exercise and to attempt every C-rated exercise, especially those that ask to verify something. The problems are at the very end, in the chapter called "Further Developments," and are not rated. These problems provide a convenient means to give brief extensions of the subjects treated in the text (see, for example, the problems on special relativity). A semester-long course using this book would most likely emphasize the chapters on complex numbers, Fourier analysis, and partial differential equations, with the chapters on vectors, mechanics, and electromagnetic theory covered only briefly while reviewing vectors and vector analysis. The chapters on complex numbers and Fourier analysis are short enough to be covered more or less completely, each in about ten 50-minute lectures. The chapter on partial differential equations is much longer, and, beyond