Multivariable Dynamic Calculus on Time Scales

605 Pages·2017·2.392 MB·English

by  Martin Bohner, Svetlin G. Georgiev

Martin Bohner Svetlin G. Georgiev
(cid:129)
Multivariable Dynamic
Calculus on Time Scales
123
Martin Bohner Svetlin G.Georgiev
Department ofMathematics andStatistics Faculty of Mathematics andInformatics
MissouriUniversity ofScience SofiaUniversity St.Kliment Ohridski
andTechnology Sofia
Rolla, MO Bulgaria
USA
ISBN978-3-319-47619-3 ISBN978-3-319-47620-9 (eBook)
DOI 10.1007/978-3-319-47620-9
LibraryofCongressControlNumber:2016955534
©SpringerInternationalPublishingSwitzerland2016
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
ThetheoryoftimescaleswasintroducedbyStefanHilgerinhisPh.D.thesis[31]in
1988 (supervised by Bernd Aulbach) in order to unify continuous and discrete
analysis and to extend the continuous and discrete theories to cases “in between.”
Since then, research in this area of mathematics has exceeded by far a thousand
publications, and numerous applications toliterallyall branches of sciencesuch as
statistics, biology, economics, finance, engineering, physics, and operations
research have been given. For an introduction to single-variable time scales cal-
culusanditsapplications,wereferthereadertothemonograph[21]byBohnerand
Peterson.
In this book, we offer the reader an overview of recent developments of
multivariable time scales calculus. The book is primarily intended for senior
undergraduatestudentsandbeginninggraduatestudentsofengineeringandscience
courses.Studentsinmathematical andphysicalscienceswillfindmanysectionsof
direct relevance. This book contains nine chapters, and each chapter consists of
results with their proofs, numerous examples, and exercises with solutions. Each
chapter concludes with a section featuring advanced practical problems with
solutions followed by a section on notes and references, explaining its context
within existing literature. Altogether, the book contains 123 definitions, 230
theorems including corollaries, lemmas, and propositions, 275 examples, and
239 exercises including advanced practical problems.
The first three chapters deal with single-variable time scales calculus. Many
of the presented results including their proofs are extracted from [21]. Chapter 1
introduces the most fundamental concepts related to time scales, namely the for-
ward and backward jump operators and the graininess. In addition, the induction
principle on time scales is given. Chapter 2 deals with differential calculus for
single-variablefunctionsontimescales.Thebasicdefinitionofdeltadifferentiation
is due to Hilger. Many examples on differentiation in various time scales are
included, as well as the Leibniz formula for the nth derivative of a product of two
functions.Meanvalueresultsarepresentedthatwillbeusedlateroninthebookin
the multivariable case. Several versions of the chain rule are included. Sufficient
conditions for a local maximum and a minimum are given. Moreover, sufficient
conditions for convexity and concavity of single-variable functions are presented.
A sufficient condition for complete delta differentiability of single-variable func-
tions is given, and the geometric sense of differentiability is discussed and illus-
trated. In Chapter 3, the main concepts for regulated, rd-continuous, and
pre-differentiable functions are introduced. The indefinite integral and the
Riemann delta integral are defined and many of their properties are deduced.
Hilger’s complex plane is introduced. Some elementary functions such as the
exponential functional, hyperbolic functions, and trigonometric functions are
definedandtheirpropertiesaregiven.Moreover,Taylor’sformulaandL’Hôpital’s
rulearepresented.Improperintegralsofthefirstandthesecondkindareintroduced
and studied.
The next two chapters discuss sequences and series of functions as well as
parameter-dependent integrals. The results of these chapters are adopted from
[36, 40]. In Chapter 4, the Dini theorem, the Cauchy criterion for uniform con-
vergence of a function series, the Weierstraß M-test for uniform convergence of a
function series, the Abel test, and the Dirichlet test are presented and numerous
examples are given. Chapter 5 introduces and studies both normal
parameter-dependent integrals and improper parameter-dependent integrals of the
first kind.
The final four chapters deal with multivariable time scales calculus. The pre-
sented results including their proofs are n-dimensional analogues of the
two-dimensional results given in [8, 9, 14, 17]. Chapter 6 is devoted to partial
differentiation on time scales. Definitions for partial derivatives and completely
delta differentiable functions are given. Some sufficient conditions for differentia-
bilityarepresented.Thechainruleandsomeofthepropertiesofimplicitfunctions
aregiven.Thedirectionalderivativeisintroduced.InChapter7,multipleRiemann
integrals over rectangles and over more general sets are introduced. Many of their
propertiesaregiven.Meanvalueresultsarepresented.Chapter8definesthelength
of time scale curves. Line integrals of the first kind and of the second kind are
introduced. Moreover, Green’s formula is derived. Chapter 9 deals with surface
integrals. Many of their properties are given.
The aim of this book is to present a clear and well-organized treatment of the
conceptbehindthedevelopmentofmathematicsaswellassolutiontechniques.The
text material of this book is presented in a readable and mathematically solid
format. Many practical problems are given, displaying the power of multivariable
dynamic calculus on time scales.
Many of the results presented in this book are based on the work by one of the
authors(MartinBohner)andProfessorGuseinShirinGuseinov,whounexpectedly
passed away on March 20, 2015. Both authors have attended the memorial con-
ferenceforProfessorGuseinovatAtılımUniversityinAnkara,Turkey,July11–13,
2016,andtheyhavedecidedtheretodedicatethisbooktothememoryofProfessor
Guseinov.
Finally, the authors would like to thank Rasheed Al-Salih, Mehdi Nategh, and
Dr. Özkan Öztürk for a careful reading of the manuscript. The authors would also
like to thank Professor Mohtar Kirane (University of La Rochelle, France) for
bringing the subject of multivariable calculus on time scales to the attention of the
secondauthor.Moreover,theauthorsaregratefultotheproductionstaffatSpringer.
Rolla, USA Martin Bohner
Paris, France Svetlin G. Georgiev
December 2016
Contents
1 Time Scales. .... .... .... .... .... ..... .... .... .... .... .... 1
1.1 Forward and Backward Jump, Graininess .. .... .... .... .... 1
1.2 Induction Principle... .... .... ..... .... .... .... .... .... 17
1.3 Advanced Practical Problems... ..... .... .... .... .... .... 19
1.4 Notes and References. .... .... ..... .... .... .... .... .... 22
2 Differential Calculus of Functions of One Variable .. .... .... .... 23
2.1 Differentiable Functions of One Variable... .... .... .... .... 23
2.2 Mean Value Theorems.... .... ..... .... .... .... .... .... 50
2.3 Chain Rules.... .... .... .... ..... .... .... .... .... .... 56
2.4 One-Sided Derivatives.... .... ..... .... .... .... .... .... 69
2.5 Nabla Derivatives.... .... .... ..... .... .... .... .... .... 73
2.6 Extreme Values . .... .... .... ..... .... .... .... .... .... 75
2.7 Convex and Concave Functions. ..... .... .... .... .... .... 82
2.8 Completely Delta Differentiable Functions.. .... .... .... .... 86
2.9 Geometric Sense of Differentiability .. .... .... .... .... .... 87
2.10 Advanced Practical Problems... ..... .... .... .... .... .... 92
2.11 Notes and References. .... .... ..... .... .... .... .... .... 94
3 Integral Calculus of Functions of One Variable. .... .... .... .... 97
3.1 Regulated, rd-Continuous, and Pre-Differentiable Functions .... 97
3.2 Indefinite Integral.... .... .... ..... .... .... .... .... .... 109
3.3 The Riemann Delta Integral.... ..... .... .... .... .... .... 111
3.4 Basic Properties of the Riemann Integral... .... .... .... .... 123
3.5 Some Elementary Functions.... ..... .... .... .... .... .... 149
3.5.1 Hilger’s Complex Plane. ..... .... .... .... .... .... 149
3.5.2 Exponential Function ... ..... .... .... .... .... .... 160
3.5.3 Hyperbolic Functions... ..... .... .... .... .... .... 173
3.5.4 Trigonometric Functions. ..... .... .... .... .... .... 184
3.6 Taylor’s Formula.... .... .... ..... .... .... .... .... .... 190
3.7 L’Hôpital’s Rule. .... .... .... ..... .... .... .... .... .... 199
3.8 Improper Integrals of the First Kind .. .... .... .... .... .... 205
3.9 Improper Integrals of the Second Kind .... .... .... .... .... 226
3.10 Advanced Practical Problems... ..... .... .... .... .... .... 233
3.11 Notes and References. .... .... ..... .... .... .... .... .... 236
4 Sequences and Series of Functions .. ..... .... .... .... .... .... 239
4.1 Uniform Convergence of Sequences of Functions .... .... .... 239
4.2 Uniform Convergence of Series of Functions ... .... .... .... 250
4.3 Advanced Practical Problems... ..... .... .... .... .... .... 259
4.4 Notes and References. .... .... ..... .... .... .... .... .... 260
5 Parameter-Dependent Integrals. .... ..... .... .... .... .... .... 261
5.1 Normal Parameter-Dependent Integrals .... .... .... .... .... 261
5.2 Improper Parameter-Dependent Integrals of the First Kind . .... 271
5.3 Advanced Practical Problems... ..... .... .... .... .... .... 301
5.4 Notes and References. .... .... ..... .... .... .... .... .... 302
6 Partial Differentiation on Time Scales..... .... .... .... .... .... 303
6.1 Basic Definitions .... .... .... ..... .... .... .... .... .... 303
6.2 Partial Derivatives and Differentiability.... .... .... .... .... 342
6.3 Completely Differentiable Functions .. .... .... .... .... .... 381
6.4 Geometric Sense of Differentiability .. .... .... .... .... .... 397
6.5 Sufficient Conditions for Differentiability... .... .... .... .... 405
6.6 Equality of Mixed Partial Derivatives . .... .... .... .... .... 408
6.7 The Chain Rule. .... .... .... ..... .... .... .... .... .... 417
6.8 The Directional Derivative. .... ..... .... .... .... .... .... 432
6.9 Implicit Functions ... .... .... ..... .... .... .... .... .... 436
6.10 Advanced Practical Problems... ..... .... .... .... .... .... 442
6.11 Notes and References. .... .... ..... .... .... .... .... .... 446
7 Multiple Integration on Time Scales. ..... .... .... .... .... .... 449
7.1 Multiple Riemann Integrals over Rectangles .... .... .... .... 449
7.2 Properties of Multiple Integrals over Rectangles . .... .... .... 472
7.3 Multiple Integration over more General Sets .... .... .... .... 494
7.4 Advanced Practical Problems... ..... .... .... .... .... .... 513
7.5 Notes and References. .... .... ..... .... .... .... .... .... 514
8 Line Integrals ... .... .... .... .... ..... .... .... .... .... .... 517
8.1 Length of Time Scale Curves .. ..... .... .... .... .... .... 517
8.2 Line Integrals of the First Kind. ..... .... .... .... .... .... 532
8.3 Line Integrals of the Second Kind.... .... .... .... .... .... 542
8.4 Green’s Formula .... .... .... ..... .... .... .... .... .... 551
8.5 Advanced Practical Problems... ..... .... .... .... .... .... 559
8.6 Notes and References. .... .... ..... .... .... .... .... .... 561
9 Surface Integrals. .... .... .... .... ..... .... .... .... .... .... 563
9.1 Surface Areas... .... .... .... ..... .... .... .... .... .... 563
9.2 Surface Δ-Integrals .. .... .... ..... .... .... .... .... .... 582
9.3 Advanced Practical Problems... ..... .... .... .... .... .... 596
9.4 Notes and References. .... .... ..... .... .... .... .... .... 597
References..... .... .... .... .... .... ..... .... .... .... .... .... 599
Index.... ..... .... .... .... .... .... ..... .... .... .... .... .... 601
Chapter 1
Time Scales
1.1 ForwardandBackwardJump,Graininess
Definition1.1 Atimescaleisanarbitrarynonemptyclosedsubsetoftherealnum-
bers.
WedenoteatimescalebythesymbolT.WesupposethatatimescaleThasthe
topologythatitinheritsfromtherealnumberswiththestandardtopology.
Example1.2 [1,2],R,andNaretimescales.
Example1.3 [a,b),(a,b],and(a,b)arenottimescalesifa <b.
Example1.4 Theset
{1,2,3}∪[4,5]∪{11,12,13}
isatimescale.
Exercise1.5 Provethatthefollowingsetsaretimescales.
1. hZ={h(cid:2)k :k ∈Z},h ∈R,
2. Pa,b = ∞k=0[k(a+b),k(a+b)+a],a >0,
3. qZ ={qk :k ∈Z}∪{0},q >1,
4. Nn ={kn :k ∈N },n ∈N,
0 0
5. {H :n ∈N },where H aretheso-calledharmonicnumbersdefinedby
n 0 n
(cid:3)n
1
H =0 and H = for n ∈N.
0 n
k
k=1
(cid:4)
6. C = ∞n=0Kn,whereK0 =[0,1],K(cid:5)1iso(cid:6)btainedbyremovingtheopen“middle
third”of K ,i.e.,theopeninterval 1,2 , K isobtainedbyremovingthetwo
0 3 3 2 (cid:5) (cid:6) (cid:5) (cid:6)
openmiddlethirdsof K ,i.e.,theopenintervals 1,2 and 7,8 from K ,and
1 9 9 9 9 1
soon.ThesetC iscalledtheCantorset.
2 1 TimeScales
Definition1.6 Fort ∈T,wedefinetheforwardjumpoperatorσ :T→Tby
σ(t)=inf{s ∈T: s >t}.
Wenotethatσ(t)≥t foranyt ∈T.
Definition1.7 Fort ∈T,wedefinethebackwardjumpoperatorρ :T→Tby
ρ(t)=sup{s ∈T: s <t}.
Wenotethatρ(t)≤t foranyt ∈T.
Definition1.8 Weset
inf∅=supT, sup∅=inf T.
Definition1.9 Fort ∈T,wedefinethefollowing.
1. Ifσ(t)>t,thent iscalledright-scattered.
2. Ift <supTandσ(t)=t,thent iscalledright-dense.
3. Ifρ(t)<t,thent iscalledleft-scattered.
4. Ift >infTandρ(t)=t,thent iscalledleft-dense.
5. If t is left-scattered and right-scattered at the same time, then t is said to be
isolated.
6. Ift isleft-denseandright-denseatthesametime,thent issaidtobedense.
Example1.10 Let √
T={ 2n+1: n ∈N}.
√
Ift = 2n+1forsomen ∈N,then
t2−1
n =
2
and √ √ √ (cid:7)
σ(t)=inf{l ∈N: 2l+1> 2n+1}= 2n+3= t2+2,
√ √ √ (cid:7)
ρ(t)=sup{l ∈N: 2l+1< 2n+1}= 2n−1= t2−2
forn ≥2,n ∈N, √ √
ρ( 3)=sup∅=inf T= 3
forn =1.Because
(cid:7) (cid:7)
t2−2<t < t2+2 for n ≥2,