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...
Mathematics is the basic language in physics and engineering. It is an essential tool which first and second year students have to master as soon as possible. A lack of competence in mathematics is the main reason for failure and drop out in the beginning periods of study.
This textbook offers an accessible and highly approved approach which is characterized by the combination of the textbook with a detailed study guide available online at extras.springer.com. This study guide divides the whole learning task into small units which the student is very likely to master successfully. Thus he or she is asked to read and study a limited section of the textbook and to return to the study guide afterwards. Working with the study guide his or her learning results are controlled, monitored and deepened by graded questions, exercises, repetitions and finally by problems and applications of the content studied. Since the degree of difficulties is slowly rising the students gain confidence and experience their own progress in mathematical competence thus fostering motivation. Furthermore in case of learning difficulties he or she is given supplementary explanations and in case of individual needs supplementary exercises and applications. So the sequence of the studies is individualised according to the individual performance and needs and can be regarded as full tutorial course.
More than that the study guide aims to satisfy two objectives simultaneously: firstly it enables students to make effective use of the textbook and secondly it offers advice on the improvement of study skills. Empirical studies have shown that the student's competence for using written information has improved significantly by using this study guide.
