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. . . . . . . . . . . . . . . . . . . . . . .
From the book reviews:
“In this mathematics textbook, you’ll find possibly all the mathematics you’d need for first years of an undergraduate physics course - algebra, geometry, calculus and statistics. ... With the book there is also an online study guide designed to help the student advance quite quickly. ... If taken alone, the textbook is as any other mathematics textbook for physicists, together with the study guide it is an excellent combination, especially if you prefer to study alone ... .” (Kadri Tinn, AstroMadness.com, February, 2015)
