Higher Mathematics for Physics and Engineering · Hiroyuki Shima Tsuneyoshi Nakayama Higher Mathematics for Physics and Engineering 123 Dr.HiroyukiShima,AssistantProfessor Dr.TsuneyoshiNakayama,Professor DepartmentofAppliedPhysics ToyotaPhysicalandChemicalResearchInstitute HokkaidoUniversity Aichi480-1192,Japan Sapporo060-8628,Japan [email protected] [email protected] ISBN978-3-540-87863-6 e-ISBN978-3-540-87864-3 DOI10.1007/b138494 SpringerHeidelbergDordrechtLondonNewYork LibraryofCongressControlNumber:2009940406 (cid:2)c Springer-VerlagBerlinHeidelberg2010 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Coverdesign:eStudioCalamarSteinen Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To our friends and colleagues Preface Owing to the rapid advances in the physical sciences and engineering, the de- mand for higher-level mathematics is increasing yearly. This book is designed foradvancedundergraduatesandgraduatestudentswhoareinterestedinthe mathematical aspects of their own fields of study. The reader is assumed to have a knowledge of undergraduate-level calculus and linear algebra. There are any number of books available on mathematics for physics and engineering but they all fall into one of two categories: the one emphasizes mathematicalrigorandtheexpositionofdefinitionsortheorems,whereasthe other is concerned primarily with applying mathematics to practical prob- lems.Webelievethatneitheroftheseapproachesaloneisparticularlyhelpful to physicists and engineers who want to understand the mathematical back- ground of the subjects with which they are concerned. This book is different in that it provides a short path to higher mathematics via a combination of these approaches. A sizable portion of this book is devoted to theorems and definitions with their proofs, and we are convinced that the study of these proofs,whichrangefromtrivialtodifficult,isusefulforagraspofthegeneral ideaofmathematical logic.Moreover,severalproblemshavebeenincludedat the end of each section, and complete solutions for all of them are presented in the greatest possible detail. We firmly believe that ours is a better peda- gogical approach than that found in typical textbooks, where there are many well-polished problems but no solutions. This book is essentially self-contained and assumes only standard under- graduate preparation such as elementary calculus and linear algebra. The first half of the book covers the following three topics: real analysis, func- tional analysis, and complex analysis, along with the preliminaries and four appendixes. Part I focuses on sequences and series of real numbers of real functions, with detailed explanations oftheir convergence properties.Wealso emphasize the concepts of Cauchy sequences and the Cauchy criterion that determine the convergence of infinite real sequences. Part II deals with the theory of the Hilbert space, which is the most important class of infinite vec- torspaces.ThecompletenesspropertyofHilbertspacesallowsonetodevelop VIII Preface various types of complex orthonormal polynomials, as described in the mid- dle of Part II. An introduction to the Lebesgue integration theory, a subject of ever-increasing importance in physics, is also presented. Part III describes the theory of complex-valued functions of one complex variable. All relevant elements including analytic functions, singularity, residue, continuation, and conformal mapping are described in a self-contained manner. A thorough un- derstanding of the fundamentals treated is important in order to proceed to more advanced branches of mathematical physics. In the second half of the volume, the following three specific topics are discussed: Fourier analysis, differential equations, and tensor analysis. These three are the most important subjects in both engineering and the physical sciences,buttheirrigorousmathematicalstructureshavehardlybeencovered inordinarytextbooks.Weknowthatmathematicalrigorisoftenunnecessary forpracticaluse.However,theblindusageofmathematicalmethodsasatool may lead to a lack of understanding of the symbiotic relationship between mathematics and the physical sciences. We believe that readers who study the mathematical structures underlying these three subjects in detail will ac- quire a better understanding of the theoretical backgrounds associated with their own fields. Part IV describes the theory of Fourier series, the Fourier transform, and the Laplace transform, with a special emphasis on the proofs of their convergence properties. A more contemporary subject, the wavelet transform, is also described toward the end of Part IV. Part V deals with or- dinary and partial differential equations. The existence theorem and stability theory for solutions, which serve as the underlying basis for differential equa- tions,aredescribedwithrigorousproofs.PartVIisdevotedtothecalculusof tensorsintermsofbothCartesianandnon-Cartesiancoordinates,alongwith the essentials of differential geometry. An alternative tensor theory expressed in terms of abstract vector spaces is developed toward the end of Part VI. The authors hope and trust that this book will serve as an introductory guide for the mathematical aspects of the important topics in the physical sciences and engineering. Sapporo, Hiroyuki Shima November 2009 Tsuneyoshi Nakayama Contents 1 Preliminaries .............................................. 1 1.1 Basic Notions of a Set.................................... 1 1.1.1 Set and Element .................................. 1 1.1.2 Number Sets...................................... 3 1.1.3 Bounds .......................................... 3 1.1.4 Interval .......................................... 4 1.1.5 Neighborhood and Contact Point.................... 5 1.1.6 Closed and Open Sets.............................. 7 1.2 Conditional Statements .................................. 9 1.3 Order of Magnitude...................................... 10 1.3.1 Symbols O, o, and ∼............................... 10 1.3.2 Asymptotic Behavior .............................. 11 1.4 Values of Indeterminate Forms ............................ 12 1.4.1 l’Hoˆpital’s Rule ................................... 12 1.4.2 Several Examples.................................. 13 Part I Real Analysis 2 Real Sequences and Series ................................. 17 2.1 Sequences of Real Numbers ............................... 17 2.1.1 Convergence of a Sequence ......................... 17 2.1.2 Bounded Sequences................................ 18 2.1.3 Monotonic Sequences .............................. 19 2.1.4 Limit Superior and Limit Inferior.................... 21 2.2 Cauchy Criterion for Real Sequences ....................... 25 2.2.1 Cauchy Sequence.................................. 25 2.2.2 Cauchy Criterion.................................. 26 2.3 Infinite Series of Real Numbers............................ 29 2.3.1 Limits of Infinite Series ............................ 29 2.3.2 Cauchy Criterion for Infinite Series .................. 31