Table Of ContentUndergraduate Texts in Mathematics
Miklós Laczkovich
Vera T. Sós
Real
Analysis
Series, Functions of Several Variables,
and Applications
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
Series Editors:
Sheldon Axler
San Francisco State University, San Francisco, CA, USA
Kenneth Ribet
University of California Berkeley, CA, USA
Advisory Board:
Colin Adams, Williams College
David A. Cox Amherst College
Pamela Gorkin, Bucknell University
Roger E. Howe, Yale University
Michael E. Orrison, Harvey Mudd College
Lisette G. de Pillis, Harvey Mudd College
Jill Pipher, Brown University
Fadil Santosa, University of Minnesota
Undergraduate Texts in Mathematics are generally aimed at third- and fourth-
year undergraduate mathematics students at North American universities. These
texts strive to provide students and teachers with new perspectives and novel
approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation
ofinterrelations among different aspectsofthesubject.Theyfeature examples that
illustrate key concepts as well as exercises that strengthen understanding.
More information about this series at http://www.springer.com/series/666
ó ó
Mikl s Laczkovich Vera T. S s
(cid:129)
Real Analysis
Series, Functions of Several Variables,
and Applications
123
MiklósLaczkovich Vera T. Sós
Department ofAnalysis Alfréd Rényi Institute of Mathematics
EötvösLorándUniversity—ELTE Hungarian Academy of Sciences
Budapest Budapest
Hungary Hungary
Translated byGergely Bálint
ISSN 0172-6056 ISSN 2197-5604 (electronic)
Undergraduate Texts inMathematics
ISBN978-1-4939-7367-5 ISBN978-1-4939-7369-9 (eBook)
https://doi.org/10.1007/978-1-4939-7369-9
LibraryofCongressControlNumber:2017949163
MathematicsSubjectClassification(2010): MSC2601,MSC26BXX,MSC2801,MSC28AXX
©SpringerScience+BusinessMediaLLC2017
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1stedition:AnalízisIIbyLaczkovichMiklósandT.SósVera,©LaczkovichMiklós,T.SósVera,Nemzeti
TankönykiadóZrt.,Budapest,2007
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Preface
Analysisformsanessentialbasisofbothmathematicsandstatistics,aswellasmost
ofthenaturalsciences.Moreover,andtoaneverincreasingextent,mathematicshas
been used to underpin our understanding of the social sciences. It was Galileo’s
insightthat“Nature’sgreatbookiswritteninthelanguageofmathematics.”Andit
is the theory of analysis (specifically, differentiation and integration) that was
created for the express purpose of describing the universe in the language of
mathematics. Working out the precise mathematical theory took almost 300 years,
withalargeportionofthistimedevotedtocreatingdefinitionsthatencapsulatethe
essence of limit and continuity. This task was neither easy nor self-evident.
In postsecondary education, analysis is a foundational requirement whenever
mathematicsisanintegralcomponentofadegreeprogram.Masteringtheconcepts
of analysis can be a difficult process. This is one of the reasons why introductory
analysis courses and textbooks introduce the material at many different levels and
employvariousmethodsofpresentingthemainideas.Thisbookisnotmeanttobe
a first course in analysis, for we assume that the reader already knows the funda-
mental definitions and basic results of one-variable analysis, as is discussed, for
example, in [7]. In most of the cases we present the necessary definitions and
theorems of one-variable analysis, and refer to the volume [7], where a detailed
discussion of the relevant material can be found.
In this volume we discuss the differentiation and integration of functions of
several variables, infinite numerical series, and sequences and series offunctions.
We place strong emphasis on presenting applications and interpretations of the
results, both in mathematics itself, like the notion and computation of arc length,
area, and volume, and in physics, like the flow of fluids. In several cases, the
applications or interpretations serve as motivation for formulating relevant mathe-
maticaldefinitionsandinsights.InChapter8wepresentapplicationsofanalysisin
apparently distant fields of mathematics.
It is important to see that although the classical theory of analysis is now more
than100yearsold,theresultsdiscussedherestillinspireactiveresearchinabroad
spectrumofscientificareas.Duetothenatureofthebookwecannotdelveintosuch
v
vi Preface
matters with any depth; we shall mention only a small handful of unsolved
problems.
Many of the definitions, statements, and arguments of single-variable analysis
can be generalized to functions of several variables in a straightforward manner,
andwe occasionallyomit theproofofatheoremthatcanbeobtainedbyrepeating
the analogous one-variable proof. In general, however, the study of functions of
several variables is considerably richer than simple generalizations of one-variable
theorems. In the realm offunctions of several variables, new phenomena and new
problemsarise,andtheinvestigationsoftenlead tootherbranchesofmathematics,
suchasdifferentialgeometry,topology,andmeasuretheory.Ourintentistopresent
therelevantdefinitions,theorems,andtheirproofsinfulldetail.However,insome
cases the seemingly intuitively obvious facts about higher-dimensional geometry
and functions of several variables prove remarkably difficult to prove in full gen-
erality. When this occurs (for example, in Chapter 5, during the discussion of the
so-calledintegraltheorems)withresultsthataretooimportantforeitherthetheory
or its applications, we present the facts, but not the full proofs.
Our explicit intent isto presentthe material gradually, and todevelop precision
basedonintuitionwiththehelpofwell-designedexamples.Masteringthismaterial
demands full student involvement, and to this end we have included about 600
exercises. Someof these areroutine, butseveral ofthem are problems that call for
anincreasinglydeepunderstandingofthemethodsandresultsdiscussedinthetext.
The most difficult exercises require going beyond the text to develop new ideas;
these are marked by ð(cid:2)Þ. Hints and/or complete solutions are provided for many
exercises, and these are indicated by (H) and (S), respectively.
Budapest, Hungary Miklós Laczkovich
February 2017 Vera T. Sós
Contents
1 Rp !R functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Real Functions of Several Variables and Their Graphs . . . . . . . . . 3
1.3 Convergence of Point Sequences. . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Basics of Point Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.8 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.9 Higher-Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.10 Applications of Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.11 Appendix: Tangent Lines and Tangent Planes . . . . . . . . . . . . . . . 63
2 Functions from Rp to Rq. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.1 Limits and Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.2 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.3 Differentiation Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.4 Implicit and Inverse Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3 The Jordan Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.1 Definition and Basic Properties of the Jordan Measure . . . . . . . . . 95
3.2 The measure of a Few Particular Sets . . . . . . . . . . . . . . . . . . . . . 106
3.3 Linear Transformations and the Jordan Measure. . . . . . . . . . . . . . 115
3.4 Appendix: The Measurability of Bounded Convex Sets . . . . . . . . 119
4 Integrals of Multivariable Functions I . . . . . . . . . . . . . . . . . . . . . . . 123
4.1 The Definition of the Multivariable Integral . . . . . . . . . . . . . . . . . 123
4.2 The Multivariable Integral on Jordan Measurable Sets . . . . . . . . . 128
4.3 Calculating Multivariable Integrals . . . . . . . . . . . . . . . . . . . . . . . 135
4.4 First Appendix: Proof of Theorem 4.12 . . . . . . . . . . . . . . . . . . . . 146
4.5 Second Appendix: Integration by Substitution
(Proof of Theorem 4.22). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
vii
viii Contents
5 Integrals of Multivariable Functions II. . . . . . . . . . . . . . . . . . . . . . . 155
5.1 The Line Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.2 Conditions for the Existence of the Primitive Function . . . . . . . . . 163
5.3 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.4 Surface and Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.5 Integral Theorems in Three Dimension . . . . . . . . . . . . . . . . . . . . 187
6 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.1 Basics on Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.2 Operations on Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.3 Absolute and Conditionally Convergent Series. . . . . . . . . . . . . . . 202
6.4 Other Convergence Criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6.5 The Product of Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.6 Summable Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
6.7 Appendix: On the History of Infinite Series . . . . . . . . . . . . . . . . . 227
7 Sequences and Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 229
7.1 The Convergence of Sequences of Functions . . . . . . . . . . . . . . . . 229
7.2 The Convergence of Series of Functions . . . . . . . . . . . . . . . . . . . 239
7.3 Taylor Series and Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . 249
7.4 Abel Summation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
7.5 Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
7.6 Further Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
7.7 First Appendix: The Cauchy–Hadamard Formula . . . . . . . . . . . . . 292
7.8 Second Appendix: Complex Series . . . . . . . . . . . . . . . . . . . . . . . 295
7.9 Third Appendix: On the History of the Fourier Series. . . . . . . . . . 297
8 Miscellaneous Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
8.1 Approximation of Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
8.2 Approximation of Definite Integrals. . . . . . . . . . . . . . . . . . . . . . . 311
8.3 Parametric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
8.4 Sets with Lebesgue Measure Zero and the Lebesgue Criterion
for Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
8.5 Two Applications of Lebesgue’s Theorem . . . . . . . . . . . . . . . . . . 343
8.6 Some Applications of Integration in Number Theory . . . . . . . . . . 346
8.7 Brouwer’s Fixed-Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 352
8.8 The Peano Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
9 Hints, Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
Notation.. .... .... .... .... ..... .... .... .... .... .... ..... .... 383
References.... .... .... .... ..... .... .... .... .... .... ..... .... 385
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 387
Functions of Several Variables
Functions of several variables are needed in order to describe complex processes.
Adetailedmeteorologicalreliefmapindicatingthetemperatureasitchangesduring
the day needs four variables: three coordinates of the place (longitude, latitude,
altitude) and one coordinate of the time. The mathematical description of complex
systems, e.g., the motion of gases or fluids, may need millions of variables.
Ifasystemdependsonpparameters,thenwecandescribeaquantitydetermined
by the system using a function that assigns the value of the quantity to the
sequences of length p that characterize the state of the system.
Wesaythatf isafunctionofpvariablesifeveryelementofthedomainoff is
asequence oflength p.Forexample,if we assign toevery date (year, month,day)
thecorrespondingdayoftheweek,thenweobtainafunctionofthreevariables,for
which f (2016, July, 18) = Monday.
Inthesequelwewillmainlyconsiderfunctionsthatdependonsequencesofreal
parameters.
ix
