Mathematics for Physicists and Engineers: Fundamentals and Interactive Study Guide

Mathematicsisanessentialtoolforphysicistsandengineerswhichstudentsmust usefromtheverybeginningoftheirstudies. Thiscombinationoftextbookandstudy guideaimstodevelopasrapidlyaspossiblethestudents’abilitytounderstandand tousethosepartsofmathematicswhichtheywillmostfrequentlyencounter. Thus functions,vectors,calculus,differentialequationsandfunctionsofseveralvariables arepresentedin averyaccessible way. Furtherchaptersinthe bookprovidethe basicknowledgeonvariousimportanttopicsinappliedmathematics. Basedontheirextensiveexperienceaslecturers,eachoftheauthorshasacquired acloseawarenessoftheneedsof rst-andsecond-yearsstudents. Oneoftheiraims hasbeentohelpuserstotacklesuccessfullythedif cultieswithmathematicswhich are commonlymet. A special feature which extendsthe supportivevalue of the maintextbookistheaccompanying“studyguide”. Thisstudyguideaimstosatisfy twoobjectivessimultaneously:itenablesstudentstomakemoreeffectiveuseofthe maintextbook,anditoffersadviceandtrainingontheimprovementoftechniques onthestudyoftextbooksgenerally. Thestudyguidedividesthewholelearningtaskintosmallunitswhichthes- dentisverylikelytomastersuccessfully. Thusheorsheisaskedtoreadandstudy alimitedsectionofthetextbookandtoreturntothestudyguideafterwards. Lea- ingresultsarecontrolled,monitoredanddeepenedbygradedquestions,exercises, repetitionsand nallybyproblemsandapplicationsofthecontentstudied. Sincethe degreeofdif cultiesisslowlyrisingthestudentsgaincon denceimmediatelyand experiencetheirownprogressinmathematicalcompetencethusfosteringmoti- tion. Incaseoflearningdif cultiesheorsheisgivenadditionalexplanationsandin caseofindividualneedssupplementaryexercisesandapplications. Sothesequence ofthestudiesisindividualisedaccordingtotheindividualperformanceandneeds andcanberegardedasafulltutorialcourse. TheworkwasoriginallypublishedinGermanyunderthetitle“Mathematikfür Physiker”(Mathematicsforphysicists). Ithasproveditsworthinyearsofactual use. Thisnew internationalversionhasbeenmodi edand extendedto meet the needsofstudentsinphysicsandengineering. vii viii Preface TheCDofferstwoversions. Ina rstversiontheframesofthestudyguideare presentedonaPCscreen. Inthiscasetheuserfollowstheinstructionsgivenonthe screen,at rststudyingsectionsofthetextbookoffthePC. Afterthisautonomous studyheistoanswerquestionsandtosolveproblemspresentedbythePC. Asecond versionisgivenaspdf lesforstudentspreferringtoworkwithaprintversion. Boththetextbookandthestudyguidehaveresultedfromteamwork. The- thors of the original textbook and study guides were Prof. Dr. Weltner, Prof. Dr. P. -B. Heinrich,Prof. Dr. H. Wiesner,P. EngelhardandProf. Dr. H. Schmidt. Thetranslationandtheadaptionwasundertakenbytheundersigned. Frankfurt,August2009 K. Weltner J. Grosjean P. Schuster W. J. Weber Acknowledgement OriginallypublishedintheFederalRepublicofGermanyunderthetitle MathematikfürPhysiker bytheauthors K. Weltner,H. Wiesner,P. -B. Heinrich,P. EngelhardtandH. Schmidt. TheworkhasbeentranslatedbyJ. GrosjeanandP. Schusterandadaptedtotheneeds ofengineeringandsciencestudentsinEnglishspeakingcountriesbyJ. Grosjean, P. Schuster,W. J. WeberandK. Weltner. ix Contents Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 VectorAlgebraI:ScalarsandVectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1 ScalarsandVectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 2 AdditionofVectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. 2. 1 SumofTwoVectors:GeometricalAddition . . . . . . . . . . . . . 4 1. 3 SubtractionofVectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1. 4 ComponentsandProjectionofaVector . . . . . . . . . . . . . . . . . . . . . . . 7 1. 5 ComponentRepresentationinCoordinateSystems. . . . . . . . . . . . . . 9 1. 5. 1 PositionVector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1. 5. 2 UnitVectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1. 5. 3 ComponentRepresentationofaVector . . . . . . . . . . . . . . . . . 11 1. 5. 4 RepresentationoftheSumofTwoVectors inTermsofTheirComponents. . . . . . . . . . . . . . . . . . . . . . . . 12 1. 5. 5 SubtractionofVectorsinTermsoftheirComponents. . . . . 13 1. 6 MultiplicationofaVectorbyaScalar. . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 7 MagnitudeofaVector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 VectorAlgebraII:ScalarandVectorProducts. . . . . . . . . . . . . . . . . . . . 23 2. 1 ScalarProduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2. 1. 1 Application:EquationofaLineandaPlane. . . . . . . . . . . . . 26 2. 1. 2 SpecialCases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2. 1. 3 CommutativeandDistributiveLaws. . . . . . . . . . . . . . . . . . . . 27 2. 1. 4 ScalarProductinTermsoftheComponentsoftheVectors. 27 2. 2 VectorProduct. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2. 2. 1 Torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2. 2. 2 TorqueasaVector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2. 2. 3 De nitionoftheVectorProduct. . . . . . . . . . . . . . . . . . . . . . .