(cid:36)(cid:77)(cid:66)(cid:86)(cid:69)(cid:74)(cid:80)(cid:1)(cid:36)(cid:66)(cid:79)(cid:86)(cid:85)(cid:80)(cid:1)(cid:116)(cid:1)(cid:34)(cid:79)(cid:74)(cid:85)(cid:66)(cid:1)(cid:53)(cid:66)(cid:67)(cid:66)(cid:68)(cid:68)(cid:80) Mathematical Analysis II TT XX EE TT II NN (cid:52)(cid:70)(cid:68)(cid:80)(cid:79)(cid:69)(cid:1)(cid:38)(cid:69)(cid:74)(cid:85)(cid:74)(cid:80)(cid:79) 123 UU UNITEXT – La Matematica per il 3+2 Volume 85 Forfurthervolumes: http://www.springer.com/series/5418 · Claudio Canuto Anita Tabacco Mathematical Analysis II Second Edition ClaudioCanuto AnitaTabacco DepartmentofMathematicalSciences DepartmentofMathematicalSciences PolitecnicodiTorino PolitecnicodiTorino Torino,Italy Torino,Italy UNITEXT–LaMatematicaperil3+2 ISSN2038-5722 ISSN2038-5757(electronic) ISBN978-3-319-12756-9 ISBN978-3-319-12757-6(eBook) DOI10.1007/978-3-319-12757-6 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014952083 ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart ofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,re- citation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefex- cerptsinconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurpose ofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. 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CoverDesign:SimonaColombo,GiochidiGrafica,Milano,Italy FilesprovidedbytheAuthors SpringerisapartofSpringerScience+BusinessMedia(www.springer.com) To Arianna, Susanna and Chiara, Alessandro, Cecilia Preface The purpose of this textbook is to present an array of topics that are found in the syllabus of the typical second lecture course in Calculus, as offered in many universities. Conceptually, it follows our previous book Mathematical Analysis I, published by Springer, which will be referred to throughout as Vol. I. While the subject matter knownas ‘Calculus 1’concerns realfunctions of real variables,andassuchismoreorlessstandard,thechoicesforacourseon‘Calculus 2’ can vary a lot, and even the way the topics can be taught is not so rigid. Due to this larger flexibility we tried to cover a wide range of subjects, reflected in the fact that the amount of content gathered here may not be comparable to the numberofcreditsconferredtoasecondCalculuscoursebythecurrentprogramme specifications. The reminders disseminated in the text render the sections more independent from one another, allowing the reader to jump back and forth, and thus enhancing the book’s versatility. The succession of chapters is what we believe to be the most natural. With the first three chapters we conclude the study of one-variable functions, begun in Vol. I, by discussing sequences and series of functions, including power series and Fourierseries.Thenwepasstoexaminemultivariableandvector-valuedfunctions, investigating continuity properties and developing the corresponding integral and differential calculus (over open measurable sets of Rn first, then on curves and surfaces). In the final part of the book we apply some of the theory learnt to the study of systems of ordinary differential equations. Continuing alongthe same strandofthought of Vol.I, we wantedthe present- ation to be as clear and comprehensible as possible. Every page of the book con- centrates on very few essential notions, most of the time just one, in order to keepthereaderfocused.Fortheorems’statements,wechosetheformthathastens an immediate understanding and guarantees readability at the same time. Hence, they are as a rule followed by several examples and pictures; the same is true for the techniques of computation. The large number of exercises, gathered according to the main topics at the end of each chapter, should help the student test his improvements. We provide the solution to all exercises, and very often the procedure for solving is outlined. VIII Preface Some graphical conventions are adopted: definitions are displayed over grey backgrounds, while statements appear on blue; examples are marked with a blue vertical bar at the side; exercises with solutions are boxed (e.g., 12. ). This second edition is enriched by two appendices, devoted to differential and integralcalculus,respectively.Therein,theinterestedreadermayfindtherigorous explanation of many results that are merely stated without proof in the previous chapters, together with useful additional material. We completely omitted the proofs whose technical aspects prevail over the fundamental notions and ideas. These may be found in other, more detailed, texts, some of which are explicitly suggested to deepen relevant topics. AllfigureswerecreatedwithMATLABTM andeditedusingthefreely-available package psfrag. This volume originates from a textbook written in Italian, itself an expanded version of the lecture courses on Calculus we have taught over the years at the Politecnico di Torino. We owe much to many authors who wrote books on the subject: A. Bacciotti and F. Ricci, C. Pagani and S. Salsa, G. Gilardi to name a few. We have also found enduring inspiration in the Anglo-Saxon-flavouredbooks by T. Apostol and J. Stewart. Special thanks aredue to Dr. SimonChiossi, for the careful andeffective work of translation. Finally, we wish to thank Francesca Bonadei – Executive Editor, Mathem- atics and Statistics, Springer Italia – for her encouragement and support in the preparation of this textbook. Torino, August 2014 Claudio Canuto, Anita Tabacco Contents 1 Numerical series............................................... 1 1.1 Round-up on sequences ...................................... 1 1.2 Numerical series............................................. 4 1.3 Series with positive terms .................................... 9 1.4 Alternating series............................................ 16 1.5 The algebra of series ......................................... 19 1.6 Exercises ................................................... 21 1.6.1 Solutions ............................................. 24 2 Series of functions and power series............................ 33 2.1 Sequences of functions ....................................... 34 2.2 Properties of uniformly convergentsequences.................... 37 2.2.1 Interchanging limits and integrals ....................... 38 2.2.2 Interchanging limits and derivatives...................... 39 2.3 Series of functions ........................................... 41 2.4 Power series ................................................ 44 2.4.1 Algebraic operations................................... 52 2.4.2 Differentiation and integration .......................... 53 2.5 Analytic functions ........................................... 56 2.6 Power series in C ............................................ 60 2.7 Exercises ................................................... 60 2.7.1 Solutions ............................................. 64 3 Fourier series .................................................. 75 3.1 Trigonometric polynomials.................................... 76 3.2 Fourier Coefficients and Fourier series.......................... 79 3.3 Exponential form............................................ 88 3.4 Differentiation .............................................. 89 3.5 Convergence of Fourier series ................................. 90 3.5.1 Quadratic convergence ................................. 90 X Contents 3.5.2 Pointwise convergence.................................. 93 3.5.3 Uniform convergence................................... 95 3.5.4 Decay of Fourier coefficients ............................ 96 3.6 Periodic functions with period T >0........................... 96 3.7 Exercises ................................................... 98 3.7.1 Solutions .............................................100 4 Functions between Euclidean spaces ...........................111 4.1 Vectors in Rn ...............................................111 4.2 Matrices ...................................................114 4.3 Sets in Rn and their properties................................120 4.4 Functions: definitions and first examples........................126 4.5 Continuity and limits ........................................130 4.5.1 Properties of limits and continuity.......................137 4.6 Curves in Rm ...............................................138 4.7 Surfaces in R3 ..............................................142 4.8 Exercises ...................................................145 4.8.1 Solutions .............................................147 5 Differential calculus for scalar functions........................155 5.1 First partial derivatives and gradient...........................155 5.2 Differentiability and differentials...............................160 5.2.1 Mean Value Theorem and Lipschitz functions .............165 5.3 Second partial derivatives and Hessian matrix...................168 5.4 Higher-order partial derivatives................................170 5.5 Taylor expansions; convexity..................................171 5.5.1 Convexity ............................................173 5.6 Extremal points of a function; stationary points .................174 5.6.1 Saddle points .........................................178 5.7 Exercises ...................................................183 5.7.1 Solutions .............................................186 6 Differential calculus for vector-valued functions................201 6.1 Partial derivatives and Jacobian matrix ........................201 6.2 Differentiability and Lipschitz functions ........................202 6.3 Basic differential operators ...................................204 6.3.1 First-order operators...................................204 6.3.2 Second-order operators.................................211 6.4 Differentiating composite functions ............................212 6.4.1 Functions defined by integrals...........................214 6.5 Regular curves ..............................................217 6.5.1 Congruence of curves; orientation........................220 6.5.2 Length and arc length .................................222 6.5.3 Elements of differential geometry for curves...............225 6.6 Variable changes ............................................227 Contents XI 6.6.1 Special frame systems..................................230 6.7 Regular surfaces.............................................236 6.7.1 Changing parametrisation ..............................240 6.7.2 Orientable surfaces ....................................241 6.7.3 Boundary of a surface; closed surfaces....................243 6.7.4 Piecewise-regular surfaces ..............................247 6.8 Exercises ...................................................248 6.8.1 Solutions .............................................251 7 Applying differential calculus ..................................261 7.1 Implicit Function Theorem ...................................261 7.1.1 Local invertibility of a function..........................267 7.2 Level curves and level surfaces ................................268 7.2.1 Level curves ..........................................269 7.2.2 Level surfaces.........................................273 7.3 Constrained extrema.........................................274 7.3.1 The method of parameters..............................277 7.3.2 Lagrange multipliers ...................................278 7.4 Exercises ...................................................285 7.4.1 Solutions .............................................288 8 Integral calculus in several variables ...........................297 8.1 Double integral over rectangles................................298 8.2 Double integrals over measurable sets ..........................304 8.2.1 Properties of double integrals ...........................313 8.3 Changing variables in double integrals..........................317 8.4 Multiple integrals............................................322 8.4.1 Changing variables in triple integrals.....................328 8.5 Applications and generalisations...............................330 8.5.1 Mass, centre of mass and moments of a solid body.........330 8.5.2 Volume of solids of revolution...........................332 8.5.3 Integrals of vector-valued functions ......................335 8.5.4 Improper multiple integrals.............................335 8.6 Exercises ...................................................337 8.6.1 Solutions .............................................343 9 Integral calculus on curves and surfaces .......................367 9.1 Integrating along curves......................................368 9.1.1 Centre of mass and moments of a curve ..................374 9.2 Path integrals...............................................375 9.3 Integrals over surfaces........................................377 9.3.1 Area of a surface ......................................381 9.3.2 Centre of mass and moments of a surface.................383 9.4 Flux integrals...............................................383 9.5 The Theorems of Gauss, Green, and Stokes .....................385