ebook img

Multivariable Dynamic Calculus on Time Scales PDF

605 Pages·2017·2.392 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Multivariable Dynamic Calculus on Time Scales

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,

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.