Table Of ContentMoscow Lectures 7
Boris M. Makarov
Anatolii N. Podkorytov
Smooth
Functions
and Maps
Moscow Lectures
Volume 7
SeriesEditors
LevD.Beklemishev,Moscow,Russia
VladimirI.Bogachev,Moscow,Russia
BorisFeigin,Moscow,Russia
ValeryGritsenko,Moscow,Russia
YulyS.Ilyashenko,Moscow,Russia
DmitryB.Kaledin,Moscow,Russia
AskoldKhovanskii,Moscow,Russia
IgorM.Krichever,Moscow,Russia
AndreiD.Mironov,Moscow,Russia
VictorA.Vassiliev,Moscow,Russia
ManagingEditor
AlexeyL.Gorodentsev,Moscow,Russia
Moreinformationaboutthisseriesathttp://www.springer.com/series/15875
Boris M. Makarov • Anatolii N. Podkorytov
Smooth Functions and Maps
Boris M. Makarov Anatolii N. Podkorytov
St. Petersburg State University St. Petersburg State University
St. Petersburg, Russia St. Petersburg, Russia
1. Translated from the Russian by Natalia Tsilevich. Originally published as Гладкие функции
и отображения by MCCME 2020
2. под ред. О.Л.Виноградова
1st edition: © MCCME Moscow, Москва Издательство МЦНМО 2020
ISSN 2522-0314 ISSN 2522-0322 (electronic)
Moscow Lectures
ISBN 978-3-030-79437-8 ISBN 978-3-030-79438-5 (eBook)
https://doi.org/10.1007/978-3-030-79438-5
Mathematics Subject Classification (2020): 54C30, 57H35, 57В50
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland
AG 2021
This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether
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Inlovingmemoryofourfriendandcolleague
VictorPetrovichHAVIN.
Preface to the English Translation
ThisisanEnglishtranslationofthebookSmoothFunctionsandMapspublishedby
MCCMEinJune2020.Itreflectstheauthors’longexperienceinteachingattheDepart-
mentofMathematicsandMechanicsoftheSt.PetersburgStateUniversity.Thebook
gives a systematic and rather comprehensive presentation of the theory of smooth
functionsandmapsinthepartdealingwithdifferentialcalculus(problemsrelatedto
integration are not addressed here). It contains numerous examples illustrating the
results,aswellasmorethan150exercises.
Having mastered the material of the book, the reader will be able to read con-
sciously any literature based on the differential calculus of functions and maps of
severalvariables.
WearegratefultotheSpringerpublishinghouseforpublishingthetranslationof
ourbook,whichhopefullywillsignificantlyexpanditsreadership.
WhenpreparingtheEnglishtranslationofthebook,webenefitedfromtheassis-
tanceofmanycolleagues,mostnotablyAndreiLodkinandFedorNazarov.Weare
grateful to the translator Natalia Tsilevich not only for her hard work, but also for
herhelpfulcommentsonourtext.
OurspecialthanksgotoAyaMakarovaforherpatientandhighlycompetenthelp
withourcorrespondencewiththepublisher.
St.Petersburg BorisM.Makarov
AnatoliiN.Podkorytov
vii
Preface
Thisbookisaddressedprimarilytostudentswhowishtobecomeacquaintedwith
the theory of smooth functions and maps to an extent which roughly corresponds
to a university course for mathematics majors, as well as to professors teaching
mathematicalanalysis.
It is based on the lecture courses taught by the authors for many years at the
Department of Mathematics and Mechanics of the St. Petersburg State University.
The material of the first three chapters corresponds to a slightly extended syllabus
ofthefirsthalfofthethirdsemester.Theprerequisitehere,apartfromafamiliarity
withthebasicsofdifferentialcalculus,isaknowledgeoftheprincipalresultsofthe
theoryofcontinuousmapsinEuclideanspaces;tomakethetextcomplete,theyare
presented(withsomeproofs)intheintroduction.Wealsoassumethatthereaderis
familiarwiththebasicsoflinearalgebra.Forconvenience,therelatedsimpleresults
arestatedintheconcludingsectionsoftheintroduction.
Thefirstchapter(exceptthelastsection)isdevotedtothetraditionalsubjectsof
thetheoryofsmoothfunctions:differentiability,Taylor’sformula,extrema,implicit
functions.
In presenting the theory of smooth maps (mostly in Chapter II), we tried to
emphasizetheroleoftheopenmappingtheorem,choosingtouseitwheneverseveral
alternativesareavailable.Althoughthiscouldbedonemuchearlier,wedeliberately
postponed the discussion of relative extrema until the end of the third chapter, in
order to use the notion of manifold and provide the reader with an opportunity
togetageometricunderstandingandappreciatetheroleoftheassumptionthatthe
functionalmatrixcorrespondingtotheconstraintequationsmusthavemaximalrank.
Inanefforttofillinsomegapsintheexistingtextbooks,weincludedinthebook
some nontraditional subjects. Among those are, most notably, problems related to
Whitney’sextensiontheorem,includingStein’stheoremondomainswithminimally
smoothboundary(SectionI.11).Theseandsomeothersections(theglobalinvert-
ibilitytheoremgoingbacktoHadamard(SectionII.5),etc.)aremarkedbyastar(∗)
andcanbeomittedinafirstreading.
Chapters I–III contain more than 150 examples and exercises, which constitute
animportantsupplementtothemaintext.
ix
x Preface
Thefourthchapterisverydifferentincharacterfromthepreviousones.Herewe
consideranimportant,yetmorespecific,problemconcerningthemassivenessofthe
setofcriticalvaluesofasmoothmap.Thematerialofthischapter(Morse’s,Sard’s,
Federer’stheorems)is,ofcourse,beyondthescopeofacompulsorycourse,butcan
form the basis of elective courses and seminars. It requires more prerequisites, in
particular,thereaderisassumedtobefamiliarwiththebasicsofmeasuretheory(in
thischapterweusethenotionofHausdorffmeasureandHausdorffdimension;the
necessarybackgroundmaterialispresentedinSections3–5ofChapterV).
Thelast,fifth,chapterisanaddendum.Itcontainssomenecessaryresults(mostly
withproofs)whosesubjectsgobeyondthescopeofthebook.Sections1–2areused
inSectionI.11,andtheremainingones,inChapterIV.
Thewholetimewhileworkingonthisbookweenjoyedcontinuedsupportfrom
our colleagues at the Chair of Mathematical Analysis, to whom we are sincerely
grateful.ThemeticulousworkofourscientificeditorO.L.Vinogradovwashighly
usefulandallowedustocorrectnumerousfaults.A.B.Alexandrovreadoneofthe
firstversionsofthemanuscript,andhisenthusiasticreactionwasveryencouraging.
Inadditiontodiverseextra-professionalhelp,whichhadabeneficialeffectonthefate
ofthebook,K.P.Kokhaspreparedelectronicversionsofallfigures,realizingour
(notalwayswellarticulated)wishes;duetohissuggestions,someillustrationsturned
outtobemoreexpressivethanwecouldhope.Thematerialofthebook,especially
that of the last two chapters, has been repeatedly discussed with A. A. Florinsky.
Wehighlyappreciatethecontributionofallthesecolleagues.Specialthanksgoto
ourfriendandcolleagueF.L.Nazarovforhisadviceandcommentswhichallowed
ustomakesomesignificantimprovements.Inparticular,hesuggestedtheproofof
Lemma2inSectionV.5.3,whichisreproducedherewithhisconsent.Lastbutnot
least,thanksareduetoN.V.Tsilevichwhoseencouragementandadvicehavebeen
veryhelpful.
The chapters of the book (except the introduction) are numbered with Roman
numerals. They are divided into sections, which, in turn, consist of subsections
numberedwithtwoArabicnumerals.Thefirstofthemindicatesthenumberofthe
section,andthesecondone,thenumberofthesubsectionwithinit.Allstatements
inagivensubsectionarereferredtointhesamewayasthesubsectionitself.Refer-
encesfromwithinthesamechapterincludeonlythenumberofthesubsection(for
example,“byTheorem2.1”referstoTheorem2.1inthegivenchapter).References
to statements from other chapters are preceded by the number of the chapter (for
example, “see Lemma III.2.3” refers to the lemma in Section 2.3 of Chapter III).
Equationsarenumberedcontinuouslythroughouteachsection.Theendofaproof
ismarkedbythesymbol(cid:3).
St.Petersburg BorisM.Makarov
April2019 AnatoliiN.Podkorytov
Contents
Introduction.Preliminaries.......................................... 1
0.1 TheSpaceR< ....................................... 1
0.2 OpenandClosedSets ................................ 3
0.3 TheLimitofaSequenceinR<......................... 6
0.4 Lindelöf’sLemma ................................... 7
0.5 CompactSets ....................................... 7
0.6 TheLimitofaMap .................................. 9
0.7 Continuity .......................................... 11
0.8 ContinuityonaCompactSet .......................... 13
0.9 UniformContinuity .................................. 15
0.10 TheContinuityoftheInverseMap...................... 16
0.11 Connectedness ...................................... 17
0.12 ConvexSetsinR< ................................... 18
0.13 LinearMapsBetweenEuclideanSpaces ................. 20
Exercises ................................................. 23
ChapterI.DifferentiableFunctions................................... 25
1 PartialDerivativesandIncrements ............................ 25
1.1 PartialDerivatives ................................... 25
1.2 Lagrange’sMeanValueTheorem....................... 26
1.3 TheCriterionofConstancy............................ 28
1.4 TheLipschitzCondition .............................. 28
Exercises ................................................. 29
2 TheDefinitionofaDifferentiableFunction ..................... 29
2.1 Differentiability ..................................... 29
2.2 Differentiability:aNecessaryCondition ................. 32
2.3 Differentiability:aSufficientCondition.................. 32
2.4 Gradient............................................ 34
Exercises ................................................. 34
3 DirectionalDerivatives...................................... 35
3.1 TheDerivativeAlongaVector ......................... 35
xi
