Mathematics for Physicists and Engineers

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  • Mathematics for Physicists and Engineers

  • Klaus Weltner Wolfgang J. Weber Jean GrosjeanPeter Schuster

    Mathematics for Physicistsand Engineers

    Fundamentals and Interactive Study Guide

    123

  • Prof. Dr. Klaus WeltnerUniversity of FrankfurtInstitute for Didactic of PhysicsMax-von-Laue-Strae 160438 Frankfurt/MainGermanyWeltner@em.uni-frankfurt.de

    Wolfgang J. WeberUniversity of FrankfurtComputing CenterGrneburgplatz 160323 FrankfurtGermanyweber@rz.uni-frankfurt.de

    Dr. Peter SchusterAm Holzweg 3065843 SulzbachGermany

    Prof. Dr. Jean GrosjeanSchool of Engineeringat the University of BathEngland

    This title was originally published by Stanley Thornes (Publisher) Ltd, 1986, entitled Mathematics forEngineers and Scientists by K. Weltner, J. Grosjean, F. Schuster and W.J. Weber.

    Cartoons in the study guide by Martin Weltner.

    ISBN 978-3-642-00172-7 e-ISBN 978-3-642-00173-4DOI 10.1007/978-3-642-00173-4Springer Dordrecht Heidelberg London New York

    Library of Congress Control Number: 2009928636

    Springer-Verlag Berlin Heidelberg 2009This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevant protectivelaws and regulations and therefore free for general use.The publisher and the authors accept no legal responsibility for any damage caused by improper use ofthe instructions and programs contained in this book and the CD. Although the software has been testedwith extreme care, errors in the software cannot be excluded.

    Typesetting and Production: le-tex publishing services GmbH, Leipzig, GermanyCover design: eStudio Calamar S.L., Spain/Germany

    Printed on acid-free paper

    Springer is part of Springer Science+Business Media (www.springer.com)

  • Main Authors of the International Version

    Prof. Dr. Klaus Weltner has studied physics at the Technical University Hannover(Germany) and the University of Bristol (England). He graduated in plasma physicsand was professor of physics and didactic of physics at the universities Osnabrck,Berlin, Frankfurt and visiting professor of physics at the Federal University of Bahia(Brazil).

    Prof. Dr. Jean Grosjean was Head of Applied Mechanics at the School of Engineer-ing at the University of Bath (England).

    Wolfgang J. Weber has studied mathematics at the universities of Frankfurt (Ger-many), Oxford (England) and Michigan State (USA). He is currently responsiblefor the training of computer specialists at the computing center at the University ofFrankfurt.

    Dr.-Ing. Peter Schuster was lecturer at the School of Engineering at the Universityof Bath (England). Different appointments in the chemical industry.

    v

  • Preface

    Mathematics is an essential tool for physicists and engineers which students mustuse from the very beginning of their studies. This combination of textbook and studyguide aims to develop as rapidly as possible the students ability to understand andto use those parts of mathematics which they will most frequently encounter. Thusfunctions, vectors, calculus, differential equations and functions of several variablesare presented in a very accessible way. Further chapters in the book provide thebasic knowledge on various important topics in applied mathematics.

    Based on their extensive experience as lecturers, each of the authors has acquireda close awareness of the needs of first- and second-years students. One of their aimshas been to help users to tackle successfully the difficulties with mathematics whichare commonly met. A special feature which extends the supportive value of themain textbook is the accompanying study guide. This study guide aims to satisfytwo objectives simultaneously: it enables students to make more effective use of themain textbook, and it offers advice and training on the improvement of techniqueson the study of textbooks generally.

    The study guide divides the whole learning task into small units which the stu-dent is very likely to master successfully. Thus he or she is asked to read and studya limited section of the textbook and to return to the study guide afterwards. Learn-ing results are controlled, monitored and deepened by graded questions, exercises,repetitions and finally by problems and applications of the content studied. Since thedegree of difficulties is slowly rising the students gain confidence immediately andexperience their own progress in mathematical competence thus fostering motiva-tion. In case of learning difficulties he or she is given additional explanations and incase of individual needs supplementary exercises and applications. So the sequenceof the studies is individualised according to the individual performance and needsand can be regarded as a full tutorial course.

    The work was originally published in Germany under the title Mathematik frPhysiker (Mathematics for physicists). It has proved its worth in years of actualuse. This new international version has been modified and extended to meet theneeds of students in physics and engineering.

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  • viii Preface

    The CD offers two versions. In a first version the frames of the study guide arepresented on a PC screen. In this case the user follows the instructions given on thescreen, at first studying sections of the textbook off the PC. After this autonomousstudy he is to answer questions and to solve problems presented by the PC. A secondversion is given as pdf files for students preferring to work with a print version.

    Both the textbook and the study guide have resulted from teamwork. The au-thors of the original textbook and study guides were Prof. Dr. Weltner, Prof.Dr. P.-B. Heinrich, Prof. Dr. H. Wiesner, P. Engelhard and Prof. Dr. H. Schmidt.The translation and the adaption was undertaken by the undersigned.

    Frankfurt, August 2009 K. WeltnerJ. GrosjeanP. SchusterW. J. Weber

  • Acknowledgement

    Originally published in the Federal Republic of Germany under the title

    Mathematik fr Physikerby the authors

    K. Weltner, H. Wiesner, P.-B. Heinrich, P. Engelhardt and H. Schmidt.

    The work has been translated by J. Grosjean and P. Schuster and adapted to the needsof engineering and science students in English speaking countries by J. Grosjean,P. Schuster, W.J. Weber and K. Weltner.

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  • Contents

    Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

    1 Vector Algebra I: Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Addition of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.2.1 Sum of Two Vectors: Geometrical Addition . . . . . . . . . . . . . 41.3 Subtraction of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Components and Projection of a Vector . . . . . . . . . . . . . . . . . . . . . . . 71.5 Component Representation in Coordinate Systems . . . . . . . . . . . . . . 9

    1.5.1 Position Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5.2 Unit Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5.3 Component Representation of a Vector . . . . . . . . . . . . . . . . . 111.5.4 Representation of the Sum of Two Vectors

    in Terms of Their Components . . . . . . . . . . . . . . . . . . . . . . . . 121.5.5 Subtraction of Vectors in Terms of their Components . . . . . 13

    1.6 Multiplication of a Vector by a Scalar . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Magnitude of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    2 Vector Algebra II: Scalar and Vector Products . . . . . . . . . . . . . . . . . . . . 232.1 Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    2.1.1 Application: Equation of a Line and a Plane . . . . . . . . . . . . . 262.1.2 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.3 Commutative and Distributive Laws . . . . . . . . . . . . . . . . . . . . 272.1.4 Scalar Product in Terms of the Components of the Vectors . 27

    2.2 Vector Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.1 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.2 Torque as a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.3 Definition of the Vector Product . . . . . . . . . . . . . . . . . . . . . . . 322.2.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.5 Anti-Commutative Law for Vector Products . . . . . . . . . . . . . 332.2.6 Components of the Vector Product . . . . . . . . . . . . . . . . . . . . . 34

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  • xii Contents

    3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1 The Mathema

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