Calculus Books
Springer International Publishing AG Computational Calculus
Book SynopsisThis book offers readers the methods that are necessary to apply the power of calculus to analyze real problems.
£33.24
Princeton University Press Calculus of Variations and Optimal Control Theory
Book SynopsisOffers an introduction to calculus of variations and optimal control theory, and is a self-contained resource for graduate students in engineering, applied mathematics, and related subjects. This book traces the historical development of the subject and features numerous exercises, notes and references at the end of each chapter.Trade Review"Each chapter ends with a rich and useful section of notes and references. The exercises are merely problems or even theorems. The author of the book presents a large list of references and a detailed index of notions, names, and symbols. The graphical presentation of the book is pleasant... [T]his book is well written, it fully deserves all its goals mentioned at the beginning of the review, and is a pleasure to read it."--Marian Muresan, Mathematica "This is an extremely well-crafted textbook. If you plan to teach a first course to advanced students on the calculus of variations and optimal control and you like the selection of topics that the author has chosen to present (and I do), it is the text you need. What impresses me most is the careful balance between the formal derivations and the explanations that precede or accompany the statements and proofs... All in all, it is a first-rate, enjoyable text."--Zvi Artstein, Mathematical Reviews Clippings
£68.00
Spark Calculus II Sparkcharts
Book Synopsis
£999.99
World Scientific Publishing Co Pte Ltd Casual Calculus: A Friendly Student Companion (In
Book SynopsisYes, this is another Calculus book. However, it fits in a niche between the two predominant types of such texts. It could be used as a textbook, albeit a streamlined one — it contains exposition on each topic, with an introduction, rationale, train of thought, and solved examples with accompanying suggested exercises. It could be used as a solution guide — because it contains full written solutions to each of the hundreds of exercises posed inside. But its best position is right in between these two extremes. It is best used as a companion to a traditional text or as a refresher — with its conversational tone, its 'get right to it' content structure, and its inclusion of complete solutions to many problems, it is a friendly partner for students who are learning Calculus, either in class or via self-study.Exercises are structured in three sets to force multiple encounters with each topic. Solved examples in the text are accompanied by 'You Try It' problems, which are similar to the solved examples; the students use these to see if they're ready to move forward. Then at the end of the section, there are 'Practice Problems': more problems similar to the 'You Try It' problems, but given all at once. Finally, each section has Challenge Problems — these lean to being equally or a bit more difficult than the others, and they allow students to check on what they've mastered.The goal is to keep the students engaged with the text, and so the writing style is very informal, with attempts at humor along the way. The target audience is STEM students including those in engineering and meteorology programs.
£166.25
Pearson Education (US) Calculus for the Life Sciences
Book SynopsisMarvin Bittinger For over thirty years Professor Marvin L. Bittinger has been teaching math at the university level. Since 1968 he has been employed as a professor of mathematics education at Indiana University - Purdue University at Indianapolis. Professor Bittinger has authored 159 publications on topics ranging from Basic Mathematics to Algebra and Trigonometry to Brief Calculus. He received his BA in Mathematics from Manchester College in 1963 and his PhD in Mathematics Education from Purdue University in 1968. Special honors include being Distinguished Visiting Professor at the United States Air Force Academy and being elected to the Manchester College Board of Trustees from 1992 to 1999. His hobbies include hiking, baseball, golf, and bowling and he enjoys membership in the Professional Bowler's Association and the Society for the Advancement of Baseball Research. Professor Bittinger has also had the privilege of speaking at aTable of ContentsChapter 1 Functions and Graphs 1.1 Slope and Linear Functions 1.2 Polynomial Functions 1.3 Rational and Radical Functions 1.4 Trigonometric Functions 1.5 Trigonometric Functions and the Unit Circle Summary and Review Test EXTENDED LIFE SCIENCE CONNECTION: Carbon Dioxide Concentrations Chapter 2 Differentiation 2.1 Limits and Continuity: Numerically and Graphically 2.2 Limits: Algebraically 2.3 Average Rates of Change 2.4 Differentiation Using Limits of Difference Quotients 2.5 Differentiation Techniques: Introduction 2.6 Instantaneous Rates of Change 2.7 Differentiation Techniques: The Product and Quotient Rules 2.8 The Chain Rule 2.9 Higher-Order Derivatives Summary and Review Test EXTENDED LIFE SCIENCE CONNECTION: Rate of Epidemic Spread: SARS Chapter 3 Applications of Differentiation 3.1 Using the First Derivatives to Find Maximum and Minimum Values and Sketch Graphs 3.2 Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs 3.3 Graph Sketching: Asymptotes and Rational Functions 3.4 Using Derivatives to Find Absolute Maximum and Minimum Values 3.5 Maximum-Minimum Problems 3.6 Approximation Techniques 3.7 Implicit Differentiation of Related Rates Summary and Review Test EXTENDED LIFE SCIENCE CONNECTION: Polymorphism Chapter 4 Exponential and Logarithmic Functions 4.1 Exponential Functions 4.2 Logarithmic Functions 4.3 Applications: The Uninhibited Growth Model, dP/dt = kP 4.4 Applications: Decay 4.5 The Derivatives of ax and logax Summary and Review Test EXTENDED LIFE SCIENCE CONNECTION: Maximum Sustainable Harvest Chapter 5 Integration 5.1 Integration 5.2 Riemann Sums and Definite Integrals 5.3 Fundamental Theorem of Calculus 5.4 Properties of Definite Integrals 5.5 Integration Techniques: Substitution 5.6 Integration Techniques: Integration by Parts 5.7 Integration Techniques: Tables and Technology 5.8 Volume 5.9 Improper Integrals Summary and Review Test EXTENDED LIFE SCIENCE CONNECTION: Estimating Dinosaur Mass Chapter 6 Matrices 6.1 Matrix Operations 6.2 Solving Linear Systems of Equations 6.3 Finding a Matrix Inverse and Determinant 6.4 Computing Eigenvalues and Eigenvectors 6.5 Solving Difference Equations Summary and Review Test EXTENDED LIFE SCIENCE CONNECTION: Population Growth Chapter 7 Functions of Several Variables 7.1 Functions of Several Variables 7.2 Partial Derivatives 7.3 Maximum-Minimum Problems 7.4 An Application: The Method of Least Squares 7.5 Multiple Integration Summary and Review Test EXTENDED LIFE SCIENCE CONNECTION: Stocking Fish Chapter 8 First-Order Differential Equations 8.1 Differential Equations and Initial-Value Problems 8.2 Linear First-Order Differential Equations 8.3 Stability of Autonomous Differential Equations 8.4 Separable Differential Equations 8.5 Numerical Solutions of Differential Equations Summary and Review Test EXTENDED LIFE SCIENCE CONNECTION: Larvae and Forest Defoliation Chapter 9 Higher-Order and Systems of Differential Equations 9.1 Higher-Order Homogeneous Differential Equations 9.2 Higher-Order Nonhomogeneous Differential Equations 9.3 Systems of Linear Differential Equations 9.4 Matrices and Trajectories 9.5 Models of Population Biology 9.6 Numerical Methods Summary and Review Test EXTENDED LIFE SCIENCE CONNECTION: Epidemics Chapter 10 Probability 10.1 Probability 10.2 Multiplication Trees and Bayes’ Rule 10.3 The Binomial Distribution 10.4 Expected Value and Standard Deviation for Discrete Random Variables 10.5 Continuous Random Variables 10.6 Poisson Process 10.7 The Normal Distribution Summary and Review Test EXTENDED LIFE SCIENCE CONNECTION: Axenic Cultures Appendix A: Review of Basic Algebra Appendix B: Functions Tables Integration Formulas Areas for a Standard Normal Distribution Answers Index
£197.03
John Wiley & Sons Inc Calculus One Variable 10e Chapters 1 12 Student
Book SynopsisPractice calculus with this solutions manual For students using Calculus: One and Several Variables for classroom instruction, this complete solutions manual for chapters 1-12 provides the answer key to the one-variable problems presented in the text. Now in its tenth edition, Calculus: One and Several Variables has become known for its easy-to-understand writing style and balance of theory and application. With this solutions manual, students can apply their knowledge using the problems presented in the first 12 chapters and check their work as they go.Table of ContentsCHAPTER 1 1 CHAPTER 2 20 CHAPTER 3 37 CHAPTER 4 63 CHAPTER 5 125 CHAPTER 6 157 CHAPTER 7 186 CHAPTER 8 220 CHAPTER 9 263 CHAPTER 10 288 CHAPTER 11 322 CHAPTER 12 346
£52.20
Princeton University Press Mathematical Analysis of Deterministic and
Book SynopsisElectromagnetic complex media are artificial materials that affect the propagation of electromagnetic waves in surprising ways not usually seen in nature. This book introduces the electromagnetics of complex media through a systematic account of their mathematical theory.Trade Review"This monograph is of a very high standard, allowing the reader to learn many facets of the rapidly growing field of complex media and to get up-to-date information on a number of open research problems."--Vilmos Komornik, Mathematical ReviewsTable of ContentsPreface xi PART 1. MODELLING AND MATHEMATICAL PRELIMINARIES 1 Chapter 1. Complex Media 3 Chapter 2. The Maxwell Equations and Constitutive Relations 9 2.1 Introduction 9 2.2 Fundamentals 9 2.3 Constitutive relations 13 2.4 The Maxwell equations in complex media: A variety of problems 23 Chapter 3. Spaces and Operators 38 3.1 Introduction 38 3.2 Function spaces 38 3.3 Standard difierential and trace operators 45 3.4 Function spaces for electromagnetics 48 3.5 Traces 51 3.6 Various decompositions 52 3.7 Compact embeddings 53 3.8 The operators of vector analysis revisited 54 3.9 The Maxwell operator 56 PART 2. TIME-HARMONIC DETERMINISTIC PROBLEMS 59 Chapter 4. Well Posedness 61 4.1 Introduction 61 4.2 Solvability of the interior problem 62 4.3 The eigenvalue problem 68 4.4 Low chirality behaviour 70 4.5 Comments on exterior domain problems 74 4.6 Towards numerics 77 Chapter 5. Scattering Problems: Beltrami Fields and Solvability 83 5.1 Introduction 83 5.2 Elliptic, circular and linear polarisation of waves 84 5.3 Beltrami fields - The Bohren decomposition 86 5.4 Scattering problems: Formulation 88 5.5 An introduction to BIEs 91 5.6 Properties of Beltrami fields 96 5.7 Solvability 99 5.8 Generalised Muller's BIEs 106 5.9 Low chirality approximations 108 5.10 Miscellanea 109 Chapter 6. Scattering Problems: A Variety of Topics 112 6.1 Introduction 112 6.2 Important concepts of scattering theory 113 6.3 Back to chiral media: Scattering relations and the far-field operator 118 6.4 Using dyadics 124 6.5 Herglotz wave functions 129 6.6 Domain derivative 136 6.7 Miscellanea 140 PART 3. TIME-DEPENDENT DETERMINISTIC PROBLEMS 149 Chapter 7. Well Posedness 151 7.1 Introduction 151 7.2 The Maxwell equations in the time domain 151 7.3 Functional framework and assumptions 152 7.4 Solvability 153 7.5 Other possible approaches to solvability 158 7.6 Miscellanea 162 Chapter 8. Controllability 163 8.1 Introduction 163 8.2 Formulation 163 8.3 Controllability of achiral media: The Hilbert Uniqueness method 165 8.4 The forward and backward problems 167 8.5 Controllability: Complex media 174 8.6 Miscellanea 176 Chapter 9. Homogenisation 180 9.1 Introduction 180 9.2 Formulation 181 9.3 A formal two-scale expansion 184 9.4 The optical response region 188 9.5 General bianisotropic media 199 9.6 Miscellanea 207 Chapter 10. Towards a Scattering Theory 212 10.1 Introduction 212 10.2 Formulation 213 10.3 Some basic strategies 214 10.4 On the construction of solutions 217 10.5 Wave operators and their construction 220 10.6 Complex media electromagnetics 225 10.7 Miscellanea 229 Chapter 11. Nonlinear Problems 231 11.1 Introduction 231 11.2 Formulation 231 11.3 Well posedness of the model 232 11.4 Miscellanea 241 PART 4. STOCHASTIC PROBLEMS 245 Chapter 12. Well Posedness 247 12.1 Introduction 247 12.2 Maxwell equations for random media 248 12.3 Functional setting 249 12.4 Well posedness 250 12.5 Other possible approaches to solvability 255 12.6 Miscellanea 261 Chapter 13. Controllability 263 13.1 Introduction 263 13.2 Formulation 263 13.3 Subtleties of stochastic controllability 264 13.4 Approximate controllability I: Random PDEs 266 13.5 Approximate controllability II: BSPDEs 269 13.6 Miscellanea 272 Chapter 14. Homogenisation 275 14.1 Introduction 275 14.2 Ergodic media 276 14.3 Formulation 279 14.4 A formal two-scale expansion 282 14.5 Homogenisation of the Maxwell system 284 14.6 Miscellanea 288 PART 5. APPENDICES 291 Appendix A. Some Facts from Functional Analysis 293 A.1 Duality 293 A.2 Strong, weak and weak-* convergence 295 A.3 Calculus in Banach spaces 297 A.4 Basic elements of spectral theory 300 A.5 Compactness criteria 303 A.6 Compact operators 304 A.7 The Banach-Steinhaus theorem 308 A.8 Semigroups and the Cauchy problem 308 A.9 Some fixed point theorems 312 A.10 The Lax-Milgram lemma 313 A.11 Gronwall's inequality 314 A.12 Nonlinear operators 315 Appendix B. Some Facts from Stochastic Analysis 316 B.1 Probability in Hilbert spaces 316 B.2 Stochastic processes and random fields 318 B.3 Gaussian measures 319 B.4 The Q- and the cylindrical Wiener process 320 B.5 The Ito integral 321 B.6 Ito formula 324 B.7 Stochastic convolution 325 B.8 SDEs in Hilbert spaces 325 B.9 Martingale representation theorem 326 Appendix C. Some Facts from Elliptic Homogenisation Theory 327 C.1 Spaces of periodic functions 327 C.2 Compensated compactness 329 C.3 Homogenisation of elliptic equations 329 C.4 Random elliptic homogenisation theory 332 Appendix D. Some Facts from Dyadic Analysis (by George Dassios) 334 Appendix E. Notation and abbreviations 341 Bibliography 343 Index 377
£999.99
Princeton University Press Partial Differential Equations An Introduction
Book SynopsisThis textbook provides beginning graduate students and advanced undergraduates with an accessible introduction to the rich subject of partial differential equations (PDEs). It presents a rigorous and clear explanation of the more elementary theoretical aspects of PDEs, while also drawing connections to deeper analysis and applications. The book serTrade Review"This book is unique in that it provides a very comprehensive introduction to the theory of PDEs embedded in specific relevant applications in the field."--Choice "The authors provide not only a clear and rigorous explanation of the more elementary theoretical aspects of partial differential equations, but they are also concerned with tools of applied mathematics in the setting of partial differential equations... This reviewer warmly recommends this volume to mathematical university libraries."--Vicentiu D. Radulescu, Zentralblatt MATH
£68.00
Princeton University Press Topics in Quaternion Linear Algebra
Book SynopsisQuaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quateTrade ReviewOne of Choice's Outstanding Academic Titles for 2015 "Rodman fills a void in the monographic literature with this work."--Choice "The book is self-contained and well organized... Full and detailed proofs are supplied. Another exciting point is the presence of many open problems throughout the book."--Gisele C. Ducati, MatchSciNetTable of Contents*FrontMatter, pg. i*Contents, pg. vii*Preface, pg. xi*Chapter One. Introduction, pg. 1*Chapter Two. The algebra of quaternions, pg. 9*Chapter Three. Vector spaces and matrices: Basic theory, pg. 28*Chapter Four. Symmetric matrices and congruence, pg. 64*Chapter Five. Invariant subspaces and Jordan form, pg. 83*Chapter Six. Invariant neutral and semidefinite subspaces, pg. 131*Chapter Seven. Smith form and Kronecker canonical form, pg. 153*Chapter Eight. Pencils of hermitian matrices, pg. 172*Chapter Nine. Skewhermitian and mixed pencils, pg. 194*Chapter Ten. Indefinite inner products: Conjugation, pg. 228*Chapter Eleven. Matrix pencils with symmetries: Nonstandard involution, pg. 261*Chapter Twelve. Mixed matrix pencils: Nonstandard involutions, pg. 279*Chapter Thirteen. Indefinite inner products: Nonstandard involution, pg. 300*Chapter Fourteen. Matrix equations, pg. 328*Chapter Fifteen. Appendix: Real and complex canonical forms, pg. 339*Bibliography, pg. 353*Index, pg. 361
£73.80
Princeton University Press Zombies and Calculus
Book SynopsisHow can calculus help you survive the zombie apocalypse? Colin Adams, humor columnist for the Mathematical Intelligencer and one of today's most outlandish and entertaining popular math writers, demonstrates how in this zombie adventure novel. Zombies and Calculus is the account of Craig Williams, a math professor at a small liberal arts college iTrade Review"Romance! Danger! Calculus! Zombies! Adams (How to Ace Calculus), professor of mathematics at Williams College and humor columnist for The Mathematical Intelligencer, takes readers on an apocalyptic and educational adventure that's also a Spielbergian sci-fi thriller... Adams keeps the in-story math both appropriate and accessible, saving in-depth discussion for appendices at the end. Calculus fans looking for 'real-world' applications woven into a nail-biter of a story, chock-full of narrow escapes, betrayals, some plucky kids, the family dog, and even a romantic subplot, will delight in this fun, and funny pop-math book."--Publishers Weekly starred review "The author didn't intend to write a classic horror novel, he intended to write one that explained the ideas of calculus and differential equations (particularly the latter) in a humorous and unusual setting. He has succeeded admirably. This is a book that can be enjoyed by both students and by faculty members seeking a way to spice up their lectures."--Mark Hunacek, MAA Reviews "You'll laugh! You'll cry! You'll use calculus! So much for kids who say, 'When am I ever going to use calculus?' When you're trying to survive the zombie apocalypse, that's when."--Nancy Szokan, Washington Post "A fun idea, quite well realized."--M.A.Orthofer, Complete Review "The book well deserves attention from not squeamish math instructors and a wider audience of intelligent readers, curious of a new literary genre that mixes storytelling with gentle mathematical instruction."--Alexander Bogomolny, CTK Insights "Whether you agree or disagree with the idea of mixing zombies with calculus, Adams is a craftsman of the first order. Zombies and Calculus has everything: calculus, calculation of force, statistics, normal deviation, word play, heroics, romance, slapstick, and zombies. Two bloody thumbs up!"--Robert Schaefer, New York Journal of Books "The unusual and clearly explained mathematical set-pieces are appealing... I will be recommending this book to colleagues. Let's hope they're not squeamish."--Noel-Ann Bradshaw, Times Higher Education "Adams combines mathematics and zombies in an exciting, humorous way... Zombies and Calculus would be a helpful supplemental text for a student currently studying calculus as it applies the concepts in a situation--however unrealistic--that makes it easier to understand."--Tara Creel, Deseret News "I have never met a novel in which the hero is actually teaching calculus to the ones he just rescued in between blowing out the little bit of brains from zombie heads, and smashing cars with a snow plough. Quite a reading experience."--Adhemar Bultheel, European Mathematical Society "Adams is clever to employ a zombie apocalypse scenario to demonstrate the usefulness of calculus and mathematics more broadly."--Carrie Bengston, Science Book A Day "Highly recommended for those just being introduced to calculus, or those who need a user-friendly recap of the basics. It may just help them survive--their education, that is, rather than a zombie attack!"--Rob Ashmore, Mathematics Today "Adams tells his tale with gore and humor. If you are a fan of zombies and enjoy offbeat applications of mathematics, this book is for you."--Joseph Bettina, Mathematics Teacher "I think that the book is very successful at doing what it was intended to do, namely to demonstrate the usefulness and relevance of what may seem at the first glance to be very abstract maths while teaching it in an engaging way. I would struggle to name another book that manages to couch the process of solving ODEs in quite so entertaining and humorous a manner."--Andrew Simmons, Mathematical GazetteTable of ContentsIntroduction 1 CHAPTER 1 Hour 6 3 CHAPTER 2 Hour 7 19 CHAPTER 3 Hour 7 1/2 32 CHAPTER 4 Hour 7 3/4 48 CHAPTER 5 Hour 8 63 CHAPTER 6 Hour 9 80 CHAPTER 7 Hour 10 95 CHAPTER 8 Hour 18 111 CHAPTER 9 Hour 24 137 Epilogue 152 APPENDIX A Continuing the Conversations 155 APPENDIX B A Brief Review of Calculus as Explained to Connor by Ellie 191 Acknowledgments 223 Bibliography 225 Index 227
£18.00
Princeton University Press The Calculus of Happiness
Book SynopsisTrade Review"There are plenty of books about managing your wealth, but The Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health, Wealth, and Love, by Oscar Fernandez of Princeton University Press, sounds intriguing."--Matthew Partridge, Money Week "[E]ngaging... Readers are sure to get a sense of how content from algebra and precalculus can help inform us about important decisions that are almost universally relevant."--Jason M. Graham, MAA Reviews "Brilliant... Where Fernandez's book scores highly is that it goes beyond being a typical self-help manual for the numerate, by presenting example after example of how mathematical topics such as probability, game theory and exponential functions really do make sense of a world that can sometimes seem so subjective. It's also an easy-going analysis of those areas in life that get brushed under the carpet, to be attended to another day. Once you realise it all boils down to maths, you will wake up happier, wealthier and healthier tomorrow morning, and we have Oscar E. Fernandez to thank for that."Nick Smith, Engineering & TechnologyTable of ContentsPreface ix Math Topics Covered by Chapter xiii I: A Healthier You Is Just a Few Equations Away 1 1 How Many Calories Should You Eat Each Day? 3 1.1 The Linear Functions Hidden in Your Diet 4 1.2 The Mathematics of Metabolism 8 1.3 Burn Those Calories! Work Those Quads! 11 1.4 The Calories Required to Digest Food 14 2 Live Longer (and Be Healthier) by Eating the Right Foods 20 2.1 A Game of Macronutrient Musical Chairs 20 2.2 How to Eat More and Be Healthier: Energy Density 29 2.3 Live Long(er) and Prosper with theWaist-Height Ratio 34 II: A Mathematician's Guide to Managing Your Money 41 3 Dissecting Your Monthly Budget 43 3.1 The Return of the King (the Linear Function) 44 3.2 To Expenses, and Beyond! 49 3.3 How Many YearsWill It Take You to Reach Financial Independence? 62 4 How to Beat Wall Street at Its Own Game 69 4.1 How to Make 15% a Year, Guaranteed 70 4.2 The Safest Investments 71 4.3 Quantifying Investment Risk and Return 73 4.4 Stocks, Bonds, and the "All-Weather" Portfolio 77 III: Looking for Love? There May Be an Equation for That 87 5 Finding "The 1" 89 5.1 What the Search for Aliens Can Teach You about FindingYour Soulmate 89 5.2 Why Hiring a Secretary Is Like Dating 92 5.3 The Stable Matching Problem 97 6 Living Happily Ever After with "The 1" 103 6.1 Your Relationship as a Dynamical System 104 6.2 Need Help Making a Joint Decision? There's an Equation for That 108 6.3 How Psychologists Use Math to Predict Divorce 113 Epilogue 118 Acknowledgments 120 Appendix A: Background Content 121 Appendix 1 123 Appendix 2 128 Appendix 3 130 Appendix 4 141 Appendix 5 143 Appendix 6 144 Bibliography 151 Index 157
£22.00
Princeton University Press Zombies and Calculus
Book SynopsisTrade Review"Romance! Danger! Calculus! Zombies! Adams (How to Ace Calculus), professor of mathematics at Williams College and humor columnist for The Mathematical Intelligencer, takes readers on an apocalyptic and educational adventure that's also a Spielbergian sci-fi thriller... Adams keeps the in-story math both appropriate and accessible, saving in-depth discussion for appendices at the end. Calculus fans looking for 'real-world' applications woven into a nail-biter of a story, chock-full of narrow escapes, betrayals, some plucky kids, the family dog, and even a romantic subplot, will delight in this fun, and funny pop-math book."--Publishers Weekly starred review "The author didn't intend to write a classic horror novel, he intended to write one that explained the ideas of calculus and differential equations (particularly the latter) in a humorous and unusual setting. He has succeeded admirably. This is a book that can be enjoyed by both students and by faculty members seeking a way to spice up their lectures."--Mark Hunacek, MAA Reviews "You'll laugh! You'll cry! You'll use calculus! So much for kids who say, 'When am I ever going to use calculus?' When you're trying to survive the zombie apocalypse, that's when."--Nancy Szokan, Washington Post "A fun idea, quite well realized."--M.A.Orthofer, Complete Review "The book well deserves attention from not squeamish math instructors and a wider audience of intelligent readers, curious of a new literary genre that mixes storytelling with gentle mathematical instruction."--Alexander Bogomolny, CTK Insights "Whether you agree or disagree with the idea of mixing zombies with calculus, Adams is a craftsman of the first order. Zombies and Calculus has everything: calculus, calculation of force, statistics, normal deviation, word play, heroics, romance, slapstick, and zombies. Two bloody thumbs up!"--Robert Schaefer, New York Journal of Books "The unusual and clearly explained mathematical set-pieces are appealing... I will be recommending this book to colleagues. Let's hope they're not squeamish."--Noel-Ann Bradshaw, Times Higher Education "Adams combines mathematics and zombies in an exciting, humorous way... Zombies and Calculus would be a helpful supplemental text for a student currently studying calculus as it applies the concepts in a situation--however unrealistic--that makes it easier to understand."--Tara Creel, Deseret News "I have never met a novel in which the hero is actually teaching calculus to the ones he just rescued in between blowing out the little bit of brains from zombie heads, and smashing cars with a snow plough. Quite a reading experience."--Adhemar Bultheel, European Mathematical Society "Adams is clever to employ a zombie apocalypse scenario to demonstrate the usefulness of calculus and mathematics more broadly."--Carrie Bengston, Science Book A Day "Highly recommended for those just being introduced to calculus, or those who need a user-friendly recap of the basics. It may just help them survive--their education, that is, rather than a zombie attack!"--Rob Ashmore, Mathematics Today "Adams tells his tale with gore and humor. If you are a fan of zombies and enjoy offbeat applications of mathematics, this book is for you."--Joseph Bettina, Mathematics Teacher "I think that the book is very successful at doing what it was intended to do, namely to demonstrate the usefulness and relevance of what may seem at the first glance to be very abstract maths while teaching it in an engaging way. I would struggle to name another book that manages to couch the process of solving ODEs in quite so entertaining and humorous a manner."--Andrew Simmons, Mathematical GazetteTable of ContentsIntroduction 1 CHAPTER 1 Hour 6 3 CHAPTER 2 Hour 7 19 CHAPTER 3 Hour 7 1/2 32 CHAPTER 4 Hour 7 3/4 48 CHAPTER 5 Hour 8 63 CHAPTER 6 Hour 9 80 CHAPTER 7 Hour 10 95 CHAPTER 8 Hour 18 111 CHAPTER 9 Hour 24 137 Epilogue 152 APPENDIX A Continuing the Conversations 155 APPENDIX B A Brief Review of Calculus as Explained to Connor by Ellie 191 Acknowledgments 223 Bibliography 225 Index 227
£17.09
Princeton University Press Visual Differential Geometry and Forms A
Book SynopsisTrade Review"Finalist for the PROSE Award in Mathematics, Association of American Publishers""Needham proposes to provide a truly geometric 'visual' explication of differential geometry, and he succeeds brilliantly. I know nothing like it in the literature."---Frank Morgan, EMS Magazine"[The] book offers a truly unique and original take on differential geometry, and it amply deserves inclusion within the pantheon of textbook deities."---Eric Poisson, Notices of the AMS"This is a valuable and beautifully created guide to what can at first seem a confusing area of mathematical physics. There are other contenders that try to teach this subject, but this is the best that I have come across so far and I will continue to enjoy learning from it (and almost certainly teaching from it) over the coming years, I am sure."---Jonathan Shock, Mathemafrica"[Proactively] rethinks the way this important area of mathematics should be considered and taught." * MathSciNet *"The book is a remarkable and highly original approach to the basic stem of differential geometry. And that mathematical trunk has roots and branches in so many other unexpected yet related subjects, each of which can be equally well approached from the same geometrical point of view."---Adhemar Bultheel, MAA Reviews"[Visual Differential Geometry and Forms] its peers. It is fun to read and provides a unique and intuitive approach to differential geometry. The author’s passion for the subject is evident throughout the book. Although Needham’s approach is unorthodox, it is rewarding, and complements the exposition found in standard textbooks."---Sean M. Eli & Krešmir Josić, SIAM Review
£106.20
John Wiley & Sons Inc Calculus Late Transcendentals International
Book SynopsisThe new edition of Calculus continues to bring together the best of both new and traditional curricula in an effort to meet the needs of even more instructors teaching calculus.Table of ContentsBEFORE CALCULUS 1 0.1 Functions 1 0.2 New Functions from Old 15 0.3 Families of Functions 27 0.4 Inverse Functions 38 1 LIMITS AND CONTINUITY 49 1.1 Limits (An Intuitive Approach) 49 1.2 Computing Limits 62 1.3 Limits at Infinity; End Behavior of a Function 71 1.4 Limits (Discussed More Rigorously) 81 1.5 Continuity 90 1.6 Continuity of Trigonometric Functions 101 2 THE DERIVATIVE 110 2.1 Tangent Lines and Rates of Change 110 2.2 The Derivative Function 122 2.3 Introduction to Techniques of Differentiation 134 2.4 The Product and Quotient Rules 142 2.5 Derivatives of Trigonometric Functions 148 2.6 The Chain Rule 153 2.7 Implicit Differentiation 161 2.8 Related Rates 168 2.9 Local Linear Approximation; Differentials 175 3 THE DERIVATIVE IN GRAPHING AND APPLICATIONS 187 3.1 Analysis of Functions I: Increase, Decrease, and Concavity 187 3.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 197 3.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents 207 3.4 Absolute Maxima and Minima 216 3.5 Applied Maximum and Minimum Problems 224 3.6 Rectilinear Motion 238 3.7 Newton’s Method 246 3.8 Rolle’s Theorem; Mean-Value Theorem 252 4 INTEGRATION 265 4.1 An Overview of the Area Problem 265 4.2 The Indefinite Integral 271 4.3 Integration by Substitution 281 4.4 The Definition of Area as a Limit; Sigma Notation 287 4.5 The Definite Integral 300 4.6 The Fundamental Theorem of Calculus 309 4.7 Rectilinear Motion Revisited Using Integration 322 4.8 Average Value of a Function and its Applications 332 4.9 Evaluating Definite Integrals by Substitution 337 5 APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING 347 5.1 Area Between Two Curves 347 5.2 Volumes by Slicing; Disks and Washers 355 5.3 Volumes by Cylindrical Shells 365 5.4 Length of a Plane Curve 371 5.5 Area of a Surface of Revolution 377 5.6 Work 382 5.7 Moments, Centers of Gravity, and Centroids 391 5.8 Fluid Pressure and Force 400 6 EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS 409 6.1 Exponential and Logarithmic Functions 409 6.2 Derivatives and Integrals Involving Logarithmic Functions 420 6.3 Derivatives of Inverse Functions; Derivatives and Integrals Involving Exponential Functions 427 6.4 Graphs and Applications Involving Logarithmic and Exponential Functions 434 6.5 L’Hôpital’s Rule; Indeterminate Forms 441 6.6 Logarithmic and Other Functions Defined by Integrals 450 6.7 Derivatives and Integrals Involving Inverse Trigonometric Functions 462 6.8 Hyperbolic Functions and Hanging Cables 472 7 PRINCIPLES OF INTEGRAL EVALUATION 488 7.1 An Overview of Integration Methods 488 7.2 Integration by Parts 491 7.3 Integrating Trigonometric Functions 500 7.4 Trigonometric Substitutions 508 7.5 Integrating Rational Functions by Partial Fractions 514 7.6 Using Computer Algebra Systems and Tables of Integrals 523 7.7 Numerical Integration; Simpson’s Rule 533 7.8 Improper Integrals 547 8 MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS 561 8.1 Modeling with Differential Equations 561 8.2 Separation of Variables 568 8.3 Slope Fields; Euler’s Method 579 8.4 First-Order Differential Equations and Applications 586 9 INFINITE SERIES 596 9.1 Sequences 596 9.2 Monotone Sequences 607 9.3 Infinite Series 614 9.4 Convergence Tests 623 9.5 The Comparison, Ratio, and Root Tests 631 9.6 Alternating Series; Absolute and Conditional Convergence 638 9.7 Maclaurin and Taylor Polynomials 648 9.8 Maclaurin and Taylor Series; Power Series 659 9.9 Convergence of Taylor Series 668 9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 678 10 PARAMETRIC AND POLAR CURVES; CONIC SECTIONS 692 10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves 692 10.2 Polar Coordinates 705 10.3 Tangent Lines, Arc Length, and Area for Polar Curves 719 10.4 Conic Sections 730 10.5 Rotation of Axes; Second-Degree Equations 748 10.6 Conic Sections in Polar Coordinates 754 11 THREE-DIMENSIONAL SPACE; VECTORS 767 11.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces 767 11.2 Vectors 773 11.3 Dot Product; Projections 785 11.4 Cross Product 795 11.5 Parametric Equations of Lines 805 11.6 Planes in 3-Space 813 11.7 Quadric Surfaces 821 11.8 Cylindrical and Spherical Coordinates 832 12 VECTOR-VALUED FUNCTIONS 841 12.1 Introduction to Vector-Valued Functions 841 12.2 Calculus of Vector-Valued Functions 848 12.3 Change of Parameter; Arc Length 858 12.4 Unit Tangent, Normal, and Binormal Vectors 868 12.5 Curvature 873 12.6 Motion Along a Curve 882 12.7 Kepler’s Laws of Planetary Motion 895 13 PARTIAL DERIVATIVES 906 13.1 Functions of Two or More Variables 906 13.2 Limits and Continuity 917 13.3 Partial Derivatives 927 13.4 Differentiability, Differentials, and Local Linearity 940 13.5 The Chain Rule 949 13.6 Directional Derivatives and Gradients 960 13.7 Tangent Planes and Normal Vectors 971 13.8 Maxima and Minima of Functions of Two Variables 977 13.9 Lagrange Multipliers 989 14 MULTIPLE INTEGRALS 1000 14.1 Double Integrals 1000 14.2 Double Integrals over Nonrectangular Regions 1009 14.3 Double Integrals in Polar Coordinates 1018 14.4 Surface Area; Parametric Surfaces 1026 14.5 Triple Integrals 1039 14.6 Triple Integrals in Cylindrical and Spherical Coordinates 1048 14.7 Change of Variables in Multiple Integrals; Jacobians 1058 14.8 Centers of Gravity Using Multiple Integrals 1071 15 TOPICS IN VECTOR CALCULUS 1084 15.1 Vector Fields 1084 15.2 Line Integrals 1094 15.3 Independence of Path; Conservative Vector Fields 1111 15.4 Green’s Theorem 1122 15.5 Surface Integrals 1130 15.6 Applications of Surface Integrals; Flux 1138 15.7 The Divergence Theorem 1148 15.8 Stokes’ Theorem 1158 A APPENDICES A GRAPHING FUNCTIONS USING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS A1 B TRIGONOMETRY REVIEW A13 C SOLVING POLYNOMIAL EQUATIONS A27 D SELECTED PROOFS A34 ANSWERS TO ODD-NUMBERED EXERCISES A45 INDEX I-1 WEB APPENDICES (online only) Available for download at www.wiley.com/go/global/anton and in WileyPLUS. E REAL NUMBERS, INTERVALS, AND INEQUALITIES F ABSOLUTE VALUE G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS H DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS I EARLY PARAMETRIC EQUATIONS OPTION J MATHEMATICAL MODELS K THE DISCRIMINANT L SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS WEB PROJECTS: Expanding the Calculus Horizon (online only) Available for download at www.wiley.com/go/global/anton and in WileyPLUS. BLAMMO THE HUMAN CANNONBALL COMET COLLISION HURRICANE MODELING ITERATION AND DYNAMICAL SYSTEMS RAILROAD DESIGN ROBOTICS
£48.59
John Wiley & Sons Inc Mathematical Analysis and Applications
Book SynopsisAn authoritative text that presents the current problems, theories, and applications of mathematical analysis research Mathematical Analysis and Applications: Selected Topics offers the theories, methods, and applications of a variety of targeted topics including: operator theory, approximation theory, fixed point theory, stability theory, minimization problems, many-body wave scattering problems, Basel problem, Corona problem, inequalities, generalized normed spaces, variations of functions and sequences, analytic generalizations of the Catalan, Fuss, and FussCatalan Numbers, asymptotically developable functions, convex functions, Gaussian processes, image analysis, and spectral analysis and spectral synthesis. The authorsa noted team of international researchers in the field highlight the basic developments for each topic presented and explore the most recent advances made in their area of study. The text is presented in such a way that enables the reader to fTable of ContentsPreface xv About the Editors xxi List of Contributors xxiii 1 Spaces of Asymptotically Developable Functions and Applications 1Sergio Alejandro Carrillo Torres and Jorge Mozo Fernández 1.1 Introduction and Some Notations 1 1.2 Strong Asymptotic Expansions 2 1.3 Monomial Asymptotic Expansions 7 1.4 Monomial Summability for Singularly Perturbed Differential Equations 13 1.5 Pfaffian Systems 15 References 19 2 Duality for Gaussian Processes from Random Signed Measures 23Palle E.T. Jorgensen and Feng Tian 2.1 Introduction 23 2.2 Reproducing Kernel Hilbert Spaces (RKHSs) in the Measurable Category 24 2.3 Applications to Gaussian Processes 30 2.4 Choice of Probability Space 34 2.5 A Duality 37 2.A Stochastic Processes 40 2.B Overview of Applications of RKHSs 45 Acknowledgments 50 References 51 3 Many-Body Wave Scattering Problems for Small Scatterers and Creating Materials with a Desired Refraction Coefficient 57Alexander G. Ramm 3.1 Introduction 57 3.2 Derivation of the Formulas for One-Body Wave Scattering Problems 62 3.3 Many-Body Scattering Problem 65 3.3.1 The Case of Acoustically Soft Particles 68 3.3.2 Wave Scattering by Many Impedance Particles 70 3.4 Creating Materials with a Desired Refraction Coefficient 71 3.5 Scattering by Small Particles Embedded in an Inhomogeneous Medium 72 3.6 Conclusions 72 References 73 4 Generalized Convex Functions and their Applications 77Adem Kiliçman and Wedad Saleh 4.1 Brief Introduction 77 4.2 Generalized E-Convex Functions 78 4.3 E𝛼-Epigraph 84 4.4 Generalized s-Convex Functions 85 4.5 Applications to Special Means 96 References 98 5 Some Properties and Generalizations of the Catalan, Fuss, and Fuss–Catalan Numbers 101Feng Qi and Bai-Ni Guo 5.1 The Catalan Numbers 101 5.1.1 A Definition of the Catalan Numbers 101 5.1.2 The History of the Catalan Numbers 101 5.1.3 A Generating Function of the Catalan Numbers 102 5.1.4 Some Expressions of the Catalan Numbers 102 5.1.5 Integral Representations of the Catalan Numbers 103 5.1.6 Asymptotic Expansions of the Catalan Function 104 5.1.7 Complete Monotonicity of the Catalan Numbers 105 5.1.8 Inequalities of the Catalan Numbers and Function 106 5.1.9 The Bell Polynomials of the Second Kind and the Bessel Polynomials 109 5.2 The Catalan–Qi Function 111 5.2.1 The Fuss Numbers 111 5.2.2 A Definition of the Catalan–Qi Function 111 5.2.3 Some Identities of the Catalan–Qi Function 112 5.2.4 Integral Representations of the Catalan–Qi Function 114 5.2.5 Asymptotic Expansions of the Catalan–Qi Function 115 5.2.6 Complete Monotonicity of the Catalan–Qi Function 116 5.2.7 Schur-Convexity of the Catalan–Qi Function 118 5.2.8 Generating Functions of the Catalan–Qi Numbers 118 5.2.9 A Double Inequality of the Catalan–Qi Function 118 5.2.10 The q-Catalan–Qi Numbers and Properties 119 5.2.11 The Catalan Numbers and the k-Gamma and k-Beta Functions 119 5.2.12 Series Identities Involving the Catalan Numbers 119 5.3 The Fuss–Catalan Numbers 119 5.3.1 A Definition of the Fuss–Catalan Numbers 119 5.3.2 A Product-Ratio Expression of the Fuss–Catalan Numbers 120 5.3.3 Complete Monotonicity of the Fuss–Catalan Numbers 120 5.3.4 A Double Inequality for the Fuss–Catalan Numbers 121 5.4 The Fuss–Catalan–Qi Function 121 5.4.1 A Definition of the Fuss–Catalan–Qi Function 121 5.4.2 A Product-Ratio Expression of the Fuss–Catalan–Qi Function 122 5.4.3 Integral Representations of the Fuss–Catalan–Qi Function 123 5.4.4 Complete Monotonicity of the Fuss–Catalan–Qi Function 124 5.5 Some Properties for Ratios of Two Gamma Functions 124 5.5.1 An Integral Representation and Complete Monotonicity 125 5.5.2 An Exponential Expansion for the Ratio of Two Gamma Functions 125 5.5.3 A Double Inequality for the Ratio of Two Gamma Functions 125 5.6 Some New Results on the Catalan Numbers 126 5.7 Open Problems 126 Acknowledgments 127 References 127 6 Trace Inequalities of Jensen Type for Self-adjoint Operators in Hilbert Spaces: A Survey of Recent Results 135Silvestru Sever Dragomir 6.1 Introduction 135 6.1.1 Jensen’s Inequality 135 6.1.2 Traces for Operators in Hilbert Spaces 138 6.2 Jensen’s Type Trace Inequalities 141 6.2.1 Some Trace Inequalities for Convex Functions 141 6.2.2 Some Functional Properties 145 6.2.3 Some Examples 151 6.2.4 More Inequalities for Convex Functions 154 6.3 Reverses of Jensen’s Trace Inequality 157 6.3.1 A Reverse of Jensen’s Inequality 157 6.3.2 Some Examples 163 6.3.3 Further Reverse Inequalities for Convex Functions 165 6.3.4 Some Examples 169 6.3.5 Reverses of Hölder’s Inequality 174 6.4 Slater’s Type Trace Inequalities 177 6.4.1 Slater’s Type Inequalities 177 6.4.2 Further Reverses 180 References 188 7 Spectral Synthesis and Its Applications 193László Székelyhidi 7.1 Introduction 193 7.2 Basic Concepts and Function Classes 195 7.3 Discrete Spectral Synthesis 203 7.4 Nondiscrete Spectral Synthesis 217 7.5 Spherical Spectral Synthesis 219 7.6 Spectral Synthesis on Hypergroups 238 7.7 Applications 248 Acknowledgments 252 References 252 8 Various Ulam–Hyers Stabilities of Euler–Lagrange–Jensen General (a, b; k = a + b)-Sextic Functional Equations 255John Michael Rassias and Narasimman Pasupathi 8.1 Brief Introduction 255 8.2 General Solution of Euler–Lagrange–Jensen General (a, b; k = a + b)-Sextic Functional Equation 257 8.3 Stability Results in Banach Space 258 8.3.1 Banach Space: Direct Method 258 8.3.2 Banach Space: Fixed Point Method 261 8.4 Stability Results in Felbin’s Type Spaces 267 8.4.1 Felbin’s Type Spaces: Direct Method 268 8.4.2 Felbin’s Type Spaces: Fixed Point Method 269 8.5 Intuitionistic Fuzzy Normed Space: Stability Results 270 8.5.1 IFNS: Direct Method 272 8.5.2 IFNS: Fixed Point Method 279 References 281 9 A Note on the Split Common Fixed Point Problem and its Variant Forms 283Adem Kiliçman and L.B. Mohammed 9.1 Introduction 283 9.2 Basic Concepts and Definitions 284 9.2.1 Introduction 284 9.2.2 Vector Space 284 9.2.3 Hilbert Space and its Properties 286 9.2.4 Bounded Linear Map and its Properties 288 9.2.5 Some Nonlinear Operators 289 9.2.6 Problem Formulation 294 9.2.7 Preliminary Results 294 9.2.8 Strong Convergence for the Split Common Fixed-Point Problems for Total Quasi-Asymptotically Nonexpansive Mappings 296 9.2.9 Strong Convergence for the Split Common Fixed-Point Problems for Demicontractive Mappings 302 9.2.10 Application to Variational Inequality Problems 306 9.2.11 On Synchronal Algorithms for Fixed and Variational Inequality Problems in Hilbert Spaces 307 9.2.12 Preliminaries 307 9.3 A Note on the Split Equality Fixed-Point Problems in Hilbert Spaces 315 9.3.1 Problem Formulation 315 9.3.2 Preliminaries 316 9.3.3 The Split Feasibility and Fixed-Point Equality Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces 316 9.3.4 The Split Common Fixed-Point Equality Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces 320 9.4 Numerical Example 322 9.5 The Split Feasibility and Fixed Point Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces 328 9.5.1 Problem Formulation 328 9.5.2 Preliminary Results 328 9.6 Ishikawa-Type Extra-Gradient Iterative Methods for Quasi-Nonexpansive Mappings in Hilbert Spaces 329 9.6.1 Application to Split Feasibility Problems 334 9.7 Conclusion 336 References 337 10 Stabilities and Instabilities of Rational Functional Equations and Euler–Lagrange–Jensen (a, b)-Sextic Functional Equations 341John Michael Rassias, Krishnan Ravi, and Beri V. Senthil Kumar 10.1 Introduction 341 10.1.1 Growth of Functional Equations 342 10.1.2 Importance of Functional Equations 342 10.1.3 Functional Equations Relevant to Other Fields 343 10.1.4 Definition of Functional Equation with Examples 343 10.2 Ulam Stability Problem for Functional Equation 344 10.2.1 𝜖-Stability of Functional Equation 344 10.2.2 Stability Involving Sum of Powers of Norms 345 10.2.3 Stability Involving Product of Powers of Norms 346 10.2.4 Stability Involving a General Control Function 347 10.2.5 Stability Involving Mixed Product–Sum of Powers of Norms 347 10.2.6 Application of Ulam Stability Theory 348 10.3 Various Forms of Functional Equations 348 10.4 Preliminaries 353 10.5 Rational Functional Equations 355 10.5.1 Reciprocal Type Functional Equation 355 10.5.2 Solution of Reciprocal Type Functional Equation 356 10.5.3 Generalized Hyers–Ulam Stability of Reciprocal Type Functional Equation 357 10.5.4 Counter-Example 360 10.5.5 Geometrical Interpretation of Reciprocal Type Functional Equation 362 10.5.6 An Application of Equation (10.41) to Electric Circuits 364 10.5.7 Reciprocal-Quadratic Functional Equation 364 10.5.8 General Solution of Reciprocal-Quadratic Functional Equation 366 10.5.9 Generalized Hyers–Ulam Stability of Reciprocal-Quadratic Functional Equations 368 10.5.10 Counter-Examples 373 10.5.11 Reciprocal-Cubic and Reciprocal-Quartic Functional Equations 375 10.5.12 Hyers–Ulam Stability of Reciprocal-Cubic and Reciprocal-Quartic Functional Equations 375 10.5.13 Counter-Examples 380 10.6 Euler-Lagrange–Jensen (a, b; k = a + b)-Sextic Functional Equations 384 10.6.1 Generalized Ulam–Hyers Stability of Euler-Lagrange-Jensen Sextic Functional Equation Using Fixed Point Method 384 10.6.2 Counter-Example 387 10.6.3 Generalized Ulam–Hyers Stability of Euler-Lagrange-Jensen Sextic Functional Equation Using Direct Method 389 References 395 11 Attractor of the Generalized Contractive Iterated Function System 401Mujahid Abbas and Talat Nazir 11.1 Iterated Function System 401 11.2 Generalized F-contractive Iterated Function System 407 11.3 Iterated Function System in b-Metric Space 414 11.4 Generalized F-Contractive Iterated Function System in b-Metric Space 420 References 426 12 Regular and Rapid Variations and Some Applications 429Ljubiša D.R. Kočinac, Dragan Djurčić, and Jelena V. Manojlović 12.1 Introduction and Historical Background 429 12.2 Regular Variation 431 12.2.1 The Class Tr(RVs) 432 12.2.2 Classes of Sequences Related to Tr(RVs) 434 12.2.3 The Class ORVs and Seneta Sequences 436 12.3 Rapid Variation 437 12.3.1 Some Properties of Rapidly Varying Functions 438 12.3.2 The Class ARVs 440 12.3.3 The Class KRs,∞ 442 12.3.4 The Class Tr(Rs,∞) 447 12.3.5 Subclasses of Tr(Rs,∞) 448 12.3.6 The Class Γs 451 12.4 Applications to Selection Principles 453 12.4.1 First Results 455 12.4.2 Improvements 455 12.4.3 When ONE has a Winning Strategy? 460 12.5 Applications to Differential Equations 463 12.5.1 The Existence of all Solutions of (A) 464 12.5.2 Superlinear Thomas–Fermi Equation (A) 466 12.5.3 Sublinear Thomas–Fermi Equation (A) 470 12.5.4 A Generalization 480 References 486 13 n-Inner Products, n-Norms, and Angles Between Two Subspaces 493Hendra Gunawan 13.1 Introduction 493 13.2 n-Inner Product Spaces and n-Normed Spaces 495 13.2.1 Topology in n-Normed Spaces 499 13.3 Orthogonality in n-Normed Spaces 500 13.3.1 G-, P-, I-, and BJ- Orthogonality 503 13.3.2 Remarks on the n-Dimensional Case 505 13.4 Angles Between Two Subspaces 505 13.4.1 An Explicit Formula 509 13.4.2 A More General Formula 511 References 513 14 Proximal Fiber Bundles on Nerve Complexes 517James F. Peters 14.1 Brief Introduction 517 14.2 Preliminaries 518 14.2.1 Nerve Complexes and Nerve Spokes 518 14.2.2 Descriptions and Proximities 521 14.2.3 Descriptive Proximities 523 14.3 Sewing Regions Together 527 14.3.1 Sewing Nerves Together with Spokes to Construct a Nervous System Complex 529 14.4 Some Results for Fiber Bundles 530 14.5 Concluding Remarks 534 References 534 15 Approximation by Generalizations of Hybrid Baskakov Type Operators Preserving Exponential Functions 537Vijay Gupta 15.1 Introduction 537 15.2 Baskakov–Szász Operators 539 15.3 Genuine Baskakov–Szász Operators 542 15.4 Preservation of eAx 545 15.5 Conclusion 549 References 550 16 Well-Posed Minimization Problems via the Theory of Measures of Noncompactness 553Józef Banaś and Tomasz Zając 16.1 Introduction 553 16.2 Minimization Problems and Their Well-Posedness in the Classical Sense 554 16.3 Measures of Noncompactness 556 16.4 Well-Posed Minimization Problems with Respect to Measures of Noncompactness 565 16.5 Minimization Problems for Functionals Defined in Banach Sequence Spaces 568 16.6 Minimization Problems for Functionals Defined in the Classical Space C([a, b])) 576 16.7 Minimization Problems for Functionals Defined in the Space of Functions Continuous and Bounded on the Real Half-Axis 580 References 584 17 Some Recent Developments on Fixed Point Theory in Generalized Metric Spaces 587Poom Kumam and Somayya Komal 17.1 Brief Introduction 587 17.2 Some Basic Notions and Notations 593 17.3 Fixed Points Theorems 596 17.3.1 Fixed Points Theorems for Monotonic and Nonmonotonic Mappings 597 17.3.2 PPF-Dependent Fixed-Point Theorems 600 17.3.3 Fixed Points Results in b-Metric Spaces 602 17.3.4 The generalized Ulam–Hyers Stability in b-Metric Spaces 604 17.3.5 Well-Posedness of a Function with Respect to 𝛼-Admissibility in b-Metric Spaces 605 17.3.6 Fixed Points for F-Contraction 606 17.4 Common Fixed Points Theorems 608 17.4.1 Common Fixed-Point Theorems for Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces 609 17.5 Best Proximity Points 611 17.6 Common Best Proximity Points 614 17.7 Tripled Best Proximity Points 617 17.8 Future Works 624 References 624 18 The Basel Problem with an Extension 631Anthony Sofo 18.1 The Basel Problem 631 18.2 An Euler Type Sum 640 18.3 The Main Theorem 645 18.4 Conclusion 652 References 652 19 Coupled Fixed Points and Coupled Coincidence Points via Fixed Point Theory 661Adrian Petruşel and Gabriela Petruşel 19.1 Introduction and Preliminaries 661 19.2 Fixed Point Results 665 19.2.1 The Single-Valued Case 665 19.2.2 The Multi-Valued Case 673 19.3 Coupled Fixed Point Results 680 19.3.1 The Single-Valued Case 680 19.3.2 The Multi-Valued Case 686 19.4 Coincidence Point Results 689 19.5 Coupled Coincidence Results 699 References 704 20 The Corona Problem, Carleson Measures, and Applications 709Alberto Saracco 20.1 The Corona Problem 709 20.1.1 Banach Algebras: Spectrum 709 20.1.2 Banach Algebras: Maximal Spectrum 710 20.1.3 The Algebra of Bounded Holomorphic Functions and the Corona Problem 710 20.2 Carleson’s Proof and Carleson Measures 711 20.2.1 Wolff’s Proof 712 20.3 The Corona Problem in Higher Henerality 712 20.3.1 The Corona Problem in ℂ 712 20.3.2 The Corona Problem in Riemann Surfaces: A Positive and a Negative Result 713 20.3.3 The Corona Problem in Domains of ℂn 714 20.3.4 The Corona Problem for Quaternionic Slice-Regular Functions 715 20.3.4.1 Slice-Regular Functions f ∶ D → ℍ 715 20.3.4.2 The Corona Theorem in the Quaternions 717 20.4 Results on Carleson Measures 718 20.4.1 Carleson Measures of Hardy Spaces of the Disk 718 20.4.2 Carleson Measures of Bergman Spaces of the Disk 719 20.4.3 Carleson Measures in the Unit Ball of ℂn 720 20.4.4 Carleson Measures in Strongly Pseudoconvex Bounded Domains of ℂn 722 20.4.5 Generalizations of Carleson Measures and Applications to Toeplitz Operators 723 20.4.6 Explicit Examples of Carleson Measures of Bergman Spaces 724 20.4.7 Carleson Measures in the Quaternionic Setting 725 20.4.7.1 Carleson Measures on Hardy Spaces of 𝔹 ⊂ ℍ 725 20.4.7.2 Carleson Measures on Bergman Spaces of 𝔹 ⊂ ℍ 726 References 728 Index 731
£105.26
Macmillan Learning Calculus Early Transcendentals
Book Synopsis
£69.34
Macmillan Learning Calculus Early Transcendentals
Book Synopsis
£69.34
Johns Hopkins University Press Calculus in Context
Book SynopsisCalculus in Context is a compelling exploration-for students and instructors alike-of a discipline that is both rich in conceptual beauty and broad in its applied relevance.Trade ReviewThe depth of detail in each application [offered by Calculus in Context] provides an excellent structure for guiding students through the “why should we care” moments that every calculus class experiences.—Mathematical Association of AmericaRecommended.—ChoiceHahn's book is the perfect choice for college and university teachers who want to teach calculus with reference to its origins and applications.—Zentralblatt MathVery well written in an engaging and enthusiastic style: it is very suitable for first year students, is perhaps not too demanding for students about to enter university, and it is particularly useful to those with more than a passing interest in astronomy. There is plenty to learn for the reader, and the massive text is also a good reference book on calculus. This labour of love from the author more than satisfies the high hopes for a good calculus book... and I highly recommend it.—Peter Shiu, Mathematical GazetteTable of ContentsPrefacePart I1. The Astronomy and Geometry of the Greeks1.1. The Greeks Explain the Universe1.2. Achieving the Impossible?1.3. Greek Geometry1.4. The Pythagorean Theorem1.5. The Radian Measure of an Angle1.6. Greek Trigonometry1.7. Aristarchus Sizes Up the Universe1.8. Problems and Projects2. The Genius of Archimedes2.1. The Conic Sections2.2. The Question of Area2.3. Playing with Squares2.4. The Area of a Parabolic Section2.5. The Method of Archimedes2.6. Problems and Projects3. A New Astronomy3.1. A Fixed Sun at the Center3.2. Copernicus's Model of Earth's Orbit3.3. About the Distances of the Planets from the Sun3.4. Tycho Brahe and Parallax3.5. Kepler's Elliptical Orbits3.6. The Studies of Galileo3.7. The Size of the Solar System3.8. Problems and Projects4. The Coordinate Geometry of Descartes4.1. The Real Numbers4.2. The Coordinate Plane4.3. About the Parabola4.4. About the Ellipse4.5. Quadratic Equations in x and y4.6. Circles and Trigonometry4.7. Problems and Projects5. The Calculus of Leibniz5.1. Straight Lines5.2. Tangent Lines to Curves5.3. The Function Concept5.4. The Derivative of a Function5.5. Fermat, Kepler, and Wine Barrels5.6. The Definite Integral5.7. Cavalieri's Principle5.8. Differentials and the Fundamental Theorem5.9. Volumes of Revolution5.10. Problems and Projects6. The Calculus of Newton6.1. Simple Functions and Areas6.2. The Derivative of a Simple Function6.3. From Simple Functions to Power Series6.4. The Mathematics of a Moving Point6.5. Galileo and Acceleration6.6. Dealing with Forces6.7. The Trajectory of a Projectile6.8. Newton Studies the Motion of the Planets6.9. Connecting Force and Geometry6.10. The Law of Universal Gravitation6.11. Problems and ProjectsPart II7. Differential Calculus7.1. Mathematical Functions7.2. A Study of Limits7.3. Continuous Functions7.4. Differentiable Functions7.5. Computing Derivatives7.6. Some Theoretical Concerns7.7. Derivatives of Trigonometric Functions7.8. Understanding Functions7.9. Graphing Functions7.10. Exponential Functions7.11. Logarithm Functions7.12. Hyperbolic Functions7.13. Final Comments about Graphs7.14. Problems and Projects8. Applications of Differential Calculus8.1. Derivatives as Rates of Change8.1.1. Growth of Organisms8.1.2. Radioactive Decay8.1.3. Cost of Production8.2. The Pulley Problem of L'Hospital8.2.1. The Solution Using Calculus8.2.2. The Solution by Balancing Forces8.3. The Suspension Bridge8.4. An Experiment of Galileo8.4.1. Sliding Ice Cubes and Spinning Wheels8.4.2. Torque and Rotational Inertia8.4.3. The Mathematics behind Galileo's Experiment8.5. From Fermat's Principle to the Reflecting Telescope8.5.1. Fermat's Principle and the Reflection of Light8.5.2. The Refraction of Light8.5.3. About Lenses8.5.4. Refracting and Reflecting Telescopes8.6. Problems and Projects9. The Basics of Integral Calculus9.1. The Definite Integral of a Function9.2. Volume and the Definite Integral9.3. Lengths of Curves and the Definite Integral9.4. Surface Area and the Definite Integral9.5. The Definite Integral and the Fundamental Theorem9.6. Area as Antiderivative9.7. Finding Antiderivatives9.7.1. Integration by Substitution9.7.2. Integration by Parts9.7.3. Some Algebraic Moves9.8. Inverse Functions9.9. Inverse Trigonometric and Hyperbolic Functions9.9.1. Trigonometric Inverses9.9.2. Hyperbolic Inverses9.10. Trigonometric and Hyperbolic Substitutions9.11. Some Integral Formulas9.12. The Trapezoidal and Simpson Rules9.13. One Loop of the Sine Curve9.14. Problems and Projects10. Applications of Integral Calculus10.1. Estimating the Weight of Domes10.1.1. The Hagia Sophia10.1.2. The Roman Pantheon10.2. The Cables of a Suspension Bridge10.3. From Pocket Watch to Pseudosphere10.3.1. Volume and Surface Area of Revolution of the Tractrix10.3.2. The Pseudosphere10.4. Calculating the Motion of a Planet10.4.1. Determining Position in Terms of Time10.4.2. Determining Speed and Direction10.4.3. Earth, Jupiter, and Halley10.5. Integral Calculus and the Action of Forces10.5.1. Work and Energy, Impulse and Momentum10.5.2. Analysis of Springs10.5.3. The Force in a Gun Barrel10.5.4. The Springfield Rifle10.6. Problems and Projects11. Basics of Differential Equations11.1. First-Order Separable Differential Equations11.2. The Method of Integrating Factors11.3. Direction Fields and Euler's Method11.4. The Polar Coordinate System11.5. The Complex Plane11.6. Second-Order Differential Equations11.7. The Basics of Power Series11.8. Taylor and Maclaurin Series11.9. Solving a Second-Order Differential Equation11.10. Free Fall with Air Resistance11.10.1. Going Up11.10.2. Coming Down11.10.3. Bullets and Ping-Pong Balls11.11. Systems with Springs and Damping Elements11.11.1. The Family Sedan and the Stock Car11.12. More about Hanging Cables11.13. Problems and Projects12. Polar Calculus and Newton's Planetary Orbits12.1. Graphing Polar Equations12.2. The Conic Sections in Polar Coordinates12.3. The Derivative of a Polar Function12.4. The Lengths of Polar Curves12.5. Areas in Polar Coordinates12.6. Equiangular Spirals12.7. Centripetal Force in Cartesian Coordinates12.8. Going Polar12.9. From Conic Section to Inverse Square Law and Back Again12.10. Gravity and Geometry12.11. Spiral Galaxies12.12. Problems and ProjectsReferencesImage Credits and NotesIndex
£80.50
W. H. Freeman Single Variable Student Solutions Manual for
Book SynopsisContains worked-out solutions to all odd-numbered exercises in the single variable section of the Taalman/Kohn Calculus text.
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Macmillan Learning Student Solutions Manual for Calculus
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Kaplan Publishing AP Calculus Flashcards, Fourth Edition:
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Fitzhenry & Whiteside Advantage Calculus: Calculus Problems and Answers
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Hansebooks Differential and Integral Calculus on the Basis
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Wiley-VCH Verlag GmbH Analysis kompakt für Dummies
Book SynopsisAn der Analysis kommen Sie nicht vorbei: Sei es nun in der Schule oder wenn Sie Natur-, Ingenieurs- oder Wirtschaftswissenschaften studieren. Dieses Buch hilft Ihnen, wenn Sie sich einen schnellen Überblick über das Thema verschaffen wollen. Mark Ryan erklärt Ihnen leicht verständlich, was Sie über Grenzwerte, Ableitungen und Integrale unbedingt wissen sollten. Übungsaufgaben helfen Ihnen dabei, das Gelernte zu verinnerlichen. So ist dies Ihr perfekter Nachhilfelehrer für die Tasche: freundlich, kompetent, günstig.Table of ContentsEinführung 19 Teil I: Analysis – ein Überblick 25 Kapitel 1: Was ist Analysis? 27 Kapitel 2: Die beiden wichtigen Konzepte der Analysis: Differenziation und Integration 33 Kapitel 3: Warum die Analysis funktioniert 39 Teil II: Die Voraussetzungen für die Analysis 45 Kapitel 4: Überblick über Vor-Algebra und Algebra 47 Kapitel 5: Verrückte Funktionen und ihre wunderbaren Graphen 63 Kapitel 6: Trigonometrie ist Trumpf! 81 Teil III: Grenzwerte 85 Kapitel 7: Grenzwerte und Stetigkeit 87 Kapitel 8: Grenzwerte auswerten 97 Teil IV: Differenziation 107 Kapitel 9: Differenziation – Orientierung 109 Kapitel 10: Regeln für die Differenziation – was sein muss, muss sein! 127 Kapitel 11: Differenziation und die Form von Kurven 137 Kapitel 12: Wunschlos glücklich: Der Differenziation sei Dank! 157 Teil V: Integration 177 Kapitel 13: Integration und Flächenannäherung – ein Einstieg 179 Kapitel 14: Integration: Differenziation rückwärts 195 Kapitel 15: Integrationstechniken für Profis 219 Kapitel 16: Grau ist alle Theorie: Mit Integralen echte Probleme lösen 233 Teil VI: Der Top-Ten-Teil 253 Kapitel 17: Zehn Dinge, die Sie sich merken sollten 255 Kapitel 18: Zehn Dinge, die Sie vergessen können 257 Anhang: Lösungen 259 Abbildungsverzeichnis 279 Stichwortverzeichnis 283
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Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Analysis 1
Book SynopsisAusführlicher Einblick in die Anfänge der Analysis: von der Einführung der reellen Zahlen bis hin zu fortgeschrittenen Themen wie Differentialformen auf Mannigfaltigkeiten, asymptotische Betrachtungen, Fourier-, Laplace- und Legendre-Transformationen, elliptische Funktionen und Distributionen. Ausgerichtet auf naturwissenschaftliche Fragestellungen und in detaillierter Herangehensweise an die Integral- und Differentialrechnung. Mit einer Fülle hilfreicher Beispiele, Aufgaben und Anwendungen. In Band 1: vollständige Übersicht zur Integral- und Differentialrechnung einer Variablen, erweitert um die Differentialrechnung mehrerer Variablen. Trade ReviewAus den Rezensionen der englischen Ausgabe: "Diese profunde Einführung [Math.Analysis I und II] in die Analysis sollte in keiner mathematischen Bibliothek fehlen, selbst bei budgetären Restriktionen, trotz der Überfülle an Einführungsbüchern. Eine genaue, bewußte Lektüre dieses profunden Werks könnte mögliche künftige Autoren mittelmäßiger Analysisbücher vielleicht abschrecken. [...]Meisterhaft wird hier intuitives Verstehen gefördert, vermittelt durch anschauliche geometrische Denkweisen, heuristische Ideen und induktive Vorgangsweisen, ohne Exaktheitsansprüche hintanzustellen oder konkrete Details oder Anwendungen auch nur ansatzweise zu vernachlässigen. Der Aufbau ist in vieler Hinsicht ungewöhnlich, eröffnet frühe Einblicke und Weitblicke und regt zum Denken an [...], ist auch der historischen Entwicklung angemessen und bietet eine wichtige Alternative zu den vielen "eleganten" Zugängen, bei denen die Vermittlung wichtiger nötiger Entwicklungsschritte für ein aktives Verständnis zu kurz kommt. Der umfassende, Nachbardisziplinen laufend berührende Zugang trägt reiche Früchte, ebenso die facettenreiche Fülle an Erklärungen der Wurzeln und Essenz der grundlegenden Konzepte und Resultate, die Beschreibungen von Zusammenhängen und Ausblicke auf weitere Entwicklungen mit vielen in Einführungsbüchern leider eher unüblichen Anwendungen und Querbezügen [...]. Man erwirbt mit diesem Werk zusätzlich ein vollständiges, umfangreiches und wertvolles "Problem-Buch". Bei aller reichhaltiger Fülle stellt sich die Mathematik hier aber immer als eine Einheit dar, in ihrer auf den heutigen Stellenwert Bezug nehmenden historischen und philosophischen Entwicklung, geprägt durch, an passender Stelle kompetent gewürdigte, bedeutende große schöpferische Persönlichkeiten. [...] Dieses vorzügliche Werk atmet den Geist einer bewunderungswürdigen, vielschichtigen Forscher- und Lehrerpersönlichkeit." H.Rindler, Monatshefte für Mathematik 146, Issue 4, 2005 "Die vorliegenden zwei Bände sind die englische Übersetzung eines russischen Werkes, das bereits Anfang der achtziger Jahre erschienen ist und inzwischen bereits zum vierten Mal aufgelegt wurde. Die Bücher beinhalten auf über 1200 Seiten die klassische Analysis in einer zeitgemäßen Darstellung sowie Querverbindungen zu Algebra, Differenzailgleichungen, Differenzialgeometrie, komplexe Analysis und Funktionalanlaysis. Addressaten sind Studenten (und Lehrende), die neben einer strengen mathematischen Theorie auch konkrete Anwendungen suchen... Dieses ausgezeichnete Werk kann Studienanfängern und fortgeschrittenen Studierenden uneingeschränkt empfohlen werden, aber auch Lehrende werden viele Anregungen darin finden." M.Kronfellner (Wien), IMN - Internationale Mathematische Nachrichten 59, Issue 198, 2005, S. 36-37 Aus den Rezensionen: "Der umfangreiche Band enthält den … Stoff einer Analysisvorlesung … Viel Raum wird … der Behandlung der Grundlagen gewidmet. … Im weiteren Verlauf beleben dann immer wieder naturwissenschaftliche und technische Anwendungen die mathematische Theorie. Jeder Abschnitt endet mit Aufgabenstellungen. Bei aller mathematischen Strenge sind die Ausführungen verständlich und vermeiden nicht unbedingt erforderliche abstrakte Ausweitungen … Empfehlenswert als Begleitlektüre zum Studium." (Wolfgang Grölz, in: ekz-Informationsdienst Einkaufszentrale für öffentliche Bibliotheken, 2006, Issue 52)Table of ContentsInhaltsverzeichnis 1 Allgemeine mathematische Begriffe und Schreibweisen . . . . . 1 1.1 Logische Symbole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Bindew¨orter und Klammern . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Hinweise zu Beweisen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Einige besondere Schreibweisen. . . . . . . . . . . . . . . . . . . . . . 3 1.1.4 Abschließende Anmerkungen . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.5 ¨Ubungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Mengen und elementare Mengenoperationen . . . . . . . . . . . . . . . . 5 1.2.1 Der Begriff einer Menge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Teilmengen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Elementare Mengenoperationen . . . . . . . . . . . . . . . . . . . . . 9 1.2.4 ¨Ubungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Funktionen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.1 Der Begriff einer Funktion (Abbildung) . . . . . . . . . . . . . . 12 1.3.2 Elementare Klassifizierung von Abbildungen . . . . . . . . . . 17 1.3.3 Zusammengesetzte Funktionen. Inverse Abbildungen . . . 18 1.3.4 Funktionen als Relationen. Der Graph einer Funktion . . 20 1.3.5 ¨Ubungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4 Erg¨anzungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4.1 Die M¨achtigkeit einer Menge (Kardinalzahlen) . . . . . . . . 27 1.4.2 Axiome der Mengenlehre . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4.3 S¨atze in der Sprache der Mengenlehre . . . . . . . . . . . . . . . . 31 1.4.4 ¨Ubungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 Die reellen Zahlen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1 Axiome und Eigenschaften der reellen Zahlen . . . . . . . . . . . . . . . 38 2.1.1 Definition der Menge der reellen Zahlen . . . . . . . . . . . . . . 38 2.1.2 Algebraische Eigenschaften der reellen Zahlen . . . . . . . . . 42 2.1.3 Das Vollst¨andigkeitsaxiom. Die kleinste obere Schranke 46 2.2 Klassen reeller Zahlen und Berechnungen . . . . . . . . . . . . . . . . . . . 49 2.2.1 Die nat¨urlichen Zahlen. Mathematische Induktion . . . . . 49 XVI Inhaltsverzeichnis 2.2.2 Rationale und irrationale Zahlen . . . . . . . . . . . . . . . . . . . . 52 2.2.3 Das archimedische Prinzip . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.2.4 Geometrische Interpretation. Gesichtspunkte beim Rechnen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.2.5 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.3 Wichtige S¨atze zur Vollst¨andigkeit . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.3.1 Der Satz zur Intervallschachtelung . . . . . . . . . . . . . . . . . . . 74 2.3.2 Der Satz zur endlichen ¨Uberdeckung . . . . . . . . . . . . . . . . . 75 2.3.3 Der Satz vom H¨aufungspunkt . . . . . . . . . . . . . . . . . . . . . . . 76 2.3.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.4 Abz¨ahlbare und ¨uberabz¨ahlbare Mengen . . . . . . . . . . . . . . . . . . . 78 2.4.1 Abz¨ahlbare Mengen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.4.2 Die M¨achtigkeit des Kontinuums . . . . . . . . . . . . . . . . . . . . 80 2.4.3 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3 Grenzwerte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.1 Der Grenzwert einer Folge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.1.1 Definitionen und Beispiele . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.1.2 Eigenschaften des Grenzwertes einer Folge . . . . . . . . . . . . 86 3.1.3 Existenz des Grenzwertes einer Folge . . . . . . . . . . . . . . . . 90 3.1.4 Elementares zu Reihen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.1.5 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.2 Der Grenzwert einer Funktion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.2.1 Definitionen und Beispiele . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.2.2 Eigenschaften des Grenzwertes einer Funktion . . . . . . . . . 116 3.2.3 Grenzwert auf einer Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3.2.4 Existenz des Grenzwertes einer Funktion . . . . . . . . . . . . . 137 3.2.5 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4 Stetige Funktionen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.1 Wichtige Definitionen und Beispiele . . . . . . . . . . . . . . . . . . . . . . . . 157 4.1.1 Stetigkeit einer Funktion in einem Punkt . . . . . . . . . . . . . 157 4.1.2 Unstetigkeitsstellen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.2 Eigenschaften stetiger Funktionen . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.2.1 Lokale Eigenschaften . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.2.2 Globale Eigenschaften stetiger Funktionen . . . . . . . . . . . . 167 4.2.3 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5 Differentialrechnung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.1 Differenzierbare Funktionen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.1.1 Problemstellung und einleitende Betrachtungen . . . . . . . 181 5.1.2 In einem Punkt differenzierbare Funktionen . . . . . . . . . . . 186 5.1.3 Tangenten und geometrische Interpretation der Ableitung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 5.1.4 Die Rolle des Koordinatensystems . . . . . . . . . . . . . . . . . . . 192 Inhaltsverzeichnis XVII 5.1.5 Einige Beispiele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.1.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.2 Wichtige Ableitungsregeln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.2.1 Differentiation und arithmetische Operationen . . . . . . . . 201 5.2.2 Differentiation einer verketteten Funktion (Kettenregel) 205 5.2.3 Differentiation einer inversen Funktion . . . . . . . . . . . . . . . 208 5.2.4 Ableitungstabelle der Elementarfunktionen . . . . . . . . . . . 213 5.2.5 Differentiation einer sehr einfachen impliziten Funktion 213 5.2.6 Ableitungen h¨oherer Ordnung . . . . . . . . . . . . . . . . . . . . . . . 218 5.2.7 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 5.3 Die zentralen S¨atze der Differentialrechnung . . . . . . . . . . . . . . . . 223 5.3.1 Der Satz von Fermat und der Satz von Rolle . . . . . . . . . . 223 5.3.2 Der Mittelwertsatz und der Satz von Cauchy. . . . . . . . . . 225 5.3.3 Die Taylorschen Formeln . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 5.3.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 5.4 Differentialrechnung zur Untersuchung von Funktionen . . . . . . . 246 5.4.1 Bedingungen f¨ur die Monotonie einer Funktion . . . . . . . . 246 5.4.2 Bedingungen f¨ur ein inneres Extremum einer Funktion . 247 5.4.3 Bedingungen f¨ur die Konvexit¨at einer Funktion . . . . . . . 253 5.4.4 Die Regel von L’Hˆopital . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 5.4.5 Das Konstruieren von Graphen von Funktionen . . . . . . . 263 5.4.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 5.5 Komplexe Zahlen und Elementarfunktionen . . . . . . . . . . . . . . . . . 276 5.5.1 Komplexe Zahlen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 5.5.2 Konvergenz in C und Reihen mit komplexen Gliedern . . 280 5.5.3 Eulersche Formel und Elementarfunktionen . . . . . . . . . . . 285 5.5.4 Analytischer Zugang zur Potenzreihendarstellung . . . . . . 288 5.5.5 Algebraische Abgeschlossenheit des K¨orpers C . . . . . . . . 293 5.5.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 5.6 Beispiele zur Differentialrechnung in den Naturwissenschaften . 301 5.6.1 Bewegung eines K¨orpers mit ver¨anderlicher Masse . . . . . 302 5.6.2 Die barometrische H¨ohenformel . . . . . . . . . . . . . . . . . . . . . 304 5.6.3 Radioaktiver Zerfall und Kernreaktoren . . . . . . . . . . . . . . 306 5.6.4 In der Atmosph¨are fallende K¨orper . . . . . . . . . . . . . . . . . . 308 5.6.5 Die Zahl e und ein erneuter Blick auf exp x . . . . . . . . . . . 310 5.6.6 Schwingungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 5.6.7 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 5.7 Stammfunktionen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 5.7.1 Stammfunktionen und das unbestimmte Integral . . . . . . 321 5.7.2 Allgemeine Methoden zur Bestimmung einer Stammfunktion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 5.7.3 Stammfunktionen rationaler Funktionen . . . . . . . . . . . . . . 329 5.7.4 Stammfunktionen der Form R R(cos x, sin x) dx . . . . . . . . 333 5.7.5 Stammfunktionen der Form R R(x, y(x)) dx . . . . . . . . . . . 335 5.7.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 XVIII Inhaltsverzeichnis 6 Integralrechnung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 6.1 Definition des Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 6.1.1 Problemstellung und einf¨uhrende Betrachtungen . . . . . . 345 6.1.2 Definition des Riemannschen Integrals . . . . . . . . . . . . . . . 347 6.1.3 Die Menge der integrierbaren Funktionen . . . . . . . . . . . . . 349 6.1.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 6.2 Linearit¨at, Additivit¨at und Monotonie des Integrals . . . . . . . . . . 365 6.2.1 Das Integral als lineare Funktion auf dem Raum R[a, b] 365 6.2.2 Das Integral als eine additive Intervallfunktion . . . . . . . . 365 6.2.3 Absch¨atzung, Monotonie und Mittelwertsatz . . . . . . . . . . 368 6.2.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 6.3 Das Integral und die Ableitung . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 6.3.1 Das Integral und die Stammfunktion . . . . . . . . . . . . . . . . . 377 6.3.2 Fundamentalsatz der Integral- und Differentialrechnung 380 6.3.3 Partielle Integration und Taylorsche Formel . . . . . . . . . . . 381 6.3.4 ¨Anderung der Variablen in einem Integral . . . . . . . . . . . . 383 6.3.5 Einige Beispiele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 6.3.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 6.4 Einige Anwendungen der Integralrechnung . . . . . . . . . . . . . . . . . . 393 6.4.1 Additive Intervallfunktionen und das Integral . . . . . . . . . 393 6.4.2 Bogenl¨ange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 6.4.3 Die Fl¨ache eines krummlinigen Trapezes . . . . . . . . . . . . . . 402 6.4.4 Volumen eines Drehk¨orpers . . . . . . . . . . . . . . . . . . . . . . . . . 404 6.4.5 Arbeit und Energie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 6.4.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 6.5 Uneigentliche Integrale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 6.5.1 Definition, Beispiele und wichtige Eigenschaften . . . . . . . 413 6.5.2 Konvergenz eines uneigentlichen Integrals . . . . . . . . . . . . 418 6.5.3 Uneigentliche Integrale mit mehr als einer Singularit¨at . 425 6.5.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 7 Funktionen mehrerer Variabler . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 7.1 Der Raum Rm und seine Unterr¨aume . . . . . . . . . . . . . . . . . . . . . . 432 7.1.1 Die Menge Rm und der Abstand in dieser Menge . . . . . . 432 7.1.2 Offene und abgeschlossene Mengen in Rm . . . . . . . . . . . . 433 7.1.3 Kompakte Mengen in Rm . . . . . . . . . . . . . . . . . . . . . . . . . . 436 7.1.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 7.2 Grenzwerte und Stetigkeit von Funktionen mehrerer Variabler . 438 7.2.1 Der Grenzwert einer Funktion . . . . . . . . . . . . . . . . . . . . . . . 438 7.2.2 Stetigkeit einer Funktion mehrerer Variabler . . . . . . . . . . 444 7.2.3 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 Inhaltsverzeichnis XIX 8 Differentialrechnung mit Funktionen mehrerer Variabler . . . 451 8.1 Die lineare Struktur auf Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 8.1.1 Rm als Vektorraum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 8.1.2 Lineare Transformationen L : Rm ! Rn . . . . . . . . . . . . . . 452 8.1.3 Die Norm in Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 8.1.4 Die euklidische Struktur auf Rm . . . . . . . . . . . . . . . . . . . . . 455 8.2 Das Differential einer Funktion mehrerer Variabler . . . . . . . . . . . 456 8.2.1 Differenzierbarkeit und das Differential in einem Punkt . 456 8.2.2 Partielle Ableitung einer Funktion mit reellen Werten . . 457 8.2.3 Die Jacobimatrix in koordinatenweiser Darstellung . . . . 460 8.2.4 Partielle Ableitungen und Differenzierbarkeit in einem Punkt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 8.3 Die wichtigsten Gesetze der Differentiation . . . . . . . . . . . . . . . . . 462 8.3.1 Linearit¨at der Ableitung . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 8.3.2 Ableitung verketteter Abbildungen (Kettenregel) . . . . . . 465 8.3.3 Ableitung einer inversen Abbildung . . . . . . . . . . . . . . . . . . 470 8.3.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 8.4 Reelle Funktionen mehrerer Variabler . . . . . . . . . . . . . . . . . . . . . . 478 8.4.1 Der Mittelwertsatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 8.4.2 Eine hinreichende Bedingung f¨ur die Differenzierbarkeit 480 8.4.3 Partielle Ableitungen h¨oherer Ordnung . . . . . . . . . . . . . . . 481 8.4.4 Die Taylorsche Formel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 8.4.5 Extrema von Funktionen mehrerer Variabler . . . . . . . . . . 486 8.4.6 Einige geometrische Darstellungen . . . . . . . . . . . . . . . . . . . 493 8.4.7 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 8.5 Der Satz zur impliziten Funktion . . . . . . . . . . . . . . . . . . . . . . . . . . 504 8.5.1 Einleitung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 8.5.2 Ein einfacher Satz zur impliziten Funktion . . . . . . . . . . . . 506 8.5.3 ¨Ubergang zur Gleichung F(x1, . . . , xm, y) = 0 . . . . . . . . . 510 8.5.4 Der Satz zur impliziten Funktion . . . . . . . . . . . . . . . . . . . . 513 8.5.5 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 8.6 Einige Korollare zum Satz zur impliziten Funktion . . . . . . . . . . . 522 8.6.1 Der Satz zur inversen Funktion. . . . . . . . . . . . . . . . . . . . . . 522 8.6.2 Lokale Reduktion in kanonische Form . . . . . . . . . . . . . . . . 527 8.6.3 Funktionale Abh¨angigkeit . . . . . . . . . . . . . . . . . . . . . . . . . . 532 8.6.4 Lokale Zerlegung eines Diffeomorphismus . . . . . . . . . . . . . 534 8.6.5 Das Morse-Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 8.6.6 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 8.7 Fl¨achen in Rn und bedingte Extrema . . . . . . . . . . . . . . . . . . . . . . 542 8.7.1 k-dimensionale Fl¨achen in Rn . . . . . . . . . . . . . . . . . . . . . . . 542 8.7.2 Der Tangentialraum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 8.7.3 Extrema mit Nebenbedingungen . . . . . . . . . . . . . . . . . . . . 552 8.7.4 ¨Ubungen und Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 XX Inhaltsverzeichnis Einige Aufgaben aus den Halbjahrespr¨ufungen . . . . . . . . . . . . . . . . 571 1. Einf¨uhrung der Analysis (Zahlen, Funktionen, Grenzwerte) . . . . . . 571 2. Differentialrechnung in einer Variablen . . . . . . . . . . . . . . . . . . . . . . . 572 3. Integration und Einf¨uhrung mehrerer Variabler . . . . . . . . . . . . . . . . 574 4. Differentialrechnung mehrerer Variabler . . . . . . . . . . . . . . . . . . . . . . 575 Pr¨ufungsgebiete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 1. Erstes Semester . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 1.1. Einleitung und Differentialrechnung in einer Variablen . . . . 579 2. Zweites Semester . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 2.1. Integration. Differentialrechnung mit mehreren Variablen . 581 Literaturverzeichnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 1. Klassische Werke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 1.1 Orginalquellen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 1.2 Wichtige umfassende grundlegende Werke . . . . . . . . . . . . . . . 585 1.3 Klassische Vorlesungen in Analysis aus der ersten H¨alfte des 20. Jahrhunderts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 2. Lehrb¨ucher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 3. Studienunterlagen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 4. Weiterf¨uhrende Literatur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 Namensverzeichnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 Sachverzeichnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
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Springer Fachmedien Wiesbaden Etwas mehr Analysis: Eine Einführung in die
Book SynopsisDieser Band für Studierende der Mathematik ab dem zweiten Semester setzt den ersten Band »Etwas Analysis« fort und führt in die klassische mehrdimensionale Analysis ein. Wieder wird Wert auf eine klare Darstellung mit einer möglichst einfachen Notation gelegt, die Denkweisen der Analysis werden herausgearbeitet. Ausgehend von Kurven werden die mehrdimensionale Differenziation und Analysis entwickelt. Dies führt bis zum Begriff der eingebetteten Mannigfaltigkeit, dem natürlichen Ort der Theorie der Extrema mit Nebenbedingungen. Die gewöhnlichen Differenzialgleichungen nehmen einen breiten Raum ein und bieten gleichzeitig einen Einstieg in die Theorie der dynamischen Systeme. Zu jedem Kapitel gibt es zahlreiche Aufgaben, deren vollständige Lösungen auf der Website des Verlages unter „Zusätzliche Informationen“ bereit gestellt werden. Dieser Band findet seine Fortsetzung im dritten Band "Noch mehr Analysis".Table of ContentsKurven und Wege.- Mehrdimensionale Differenziation.- Mehrdimensionale Analysis.– Wegintegrale.- Lineare Differenzialgleichungen.- Gewöhnliche Differenzialgleichungen.- Integration von Vektorfeldern.- Gleichgewichtspunkte.- Periodische Lösungen.
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Springer Fachmedien Wiesbaden Analysis 2: Differentialrechnung im IRn,
Book SynopsisDer vorliegende Band stellt den zweiten Teil eines Analysis-Kurses für Studierende der Mathematik und Physik im ersten Studienjahr dar und beschäftigt sich mit der mehrdimensionalen Differentialrechnung sowie mit gewöhnlichen Differentialgleichungen. Bei der Darstellung wurde angestrebt, allzu große Abstraktionen zu vermeiden und die Theorie durch viele konkrete Beispiele zu erläutern, insbesondere solche, die für die Physik relevant sind. Für die vorliegende Neuauflage wurde der Text vor allem in den ersten drei Paragraphen überarbeitet und dabei die topologischen Grundlagen ausführlicher dargestellt.Dieses seit vier Jahrzehnten bewährte Standardwerk enthält zahlreiche Übungsaufgaben. Das zugehörige Übungsbuch mit Lösungen unterstützt die Studierenden beim Selbststudium (zum Beispiel bei Prüfungsvorbereitungen).Table of Contents
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