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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 Mathematikfur 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 MathematikfurPhysiker 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...

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