Calculus Books

Book SynopsisAn engaging, accessible introduction into how numbers work and why we shouldn't be afraid of them, frommaths expertRachel Riley.Do you know your fractions from your percentages? Your adjacent to your hypotenuse? And who really knows how to do long division, anyway?Puzzled already? Don't blame youBut fret not! You won't be At Sixes and Sevens for long. In this brilliant, well-rounded guide, Countdown''s Rachel Riley will take you back to the very basics, allow you to revisit what you learnt at school (and may have promptly forgotten, *ahem*), build your understanding of maths from the get-go and provide you with the essential toolkit to gain confidence in your numerical abilities.Discover how to divide and conquer, make your decimal debut, become a pythagoras professional and so much more with these easy-to-learn tips and tricks. Packed full of working examples, fool-proof methods, quirky trivia and brainteasers to try from puzzle-pro Dr Gareth Moore, this book is an absolute must-read for anyone and everyone who ever thought maths was above' them. Because the truth is: you can do it. What's more, it can be pretty fun too!

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Book Synopsis“In Gonick’s work, clever design and illustration make complicated ideas or insights strikingly clear.”—New York Times Book ReviewLarry Gonick, master cartoonist, former Harvard instructor, and creator of the New York Times bestselling, Harvey Award-winning Cartoon Guide series now does for calculus what he previously did for science and history: making a complex subject comprehensible, fascinating, and fun through witty text and light-hearted graphics. Gonick’s The Cartoon Guide to Calculus is a refreshingly humorous, remarkably thorough guide to general calculus that, like his earlier Cartoon Guide to Physics and Cartoon History of the Modern World, will prove a boon to students, educators, and eager learners everywhere.Trade Review"How do you humanize calculus and bring its equations and concepts to life? Larry Gonick's clever and delightful answer is to have characters talking, commenting, and joking-all while rigorously teaching equations and concepts and indicating calculus's utility. It's a remarkable accomplishment-and a lot of fun." -- Lisa Randall, Professor of Physics, Harvard University, and author of Knocking on Heaven's Door Gonick is to graphical expositions of advanced materials as Newton or Leibniz is to calculus. The difference is that Gonick has no rival. -- Xiao-Li Meng, Whipple V. N. Jones Professor of Statistics and Department Chair, Harvard University Larry Gonick's sparkling and inventive drawings make a vivid picture out of every one of the hundreds of formulas that underlie Calculus. Even the jokers in the back row will ace the course with this book. -- David Mumford, Professor emeritus of Applied Mathematics at Brown University and recipient of the National Medal of Science I always thought that there are no magic tricks that use calculus. Larry Gonick proves me wrong. His book is correct, clear and interesting. It is filled with magical insights into this most beautiful subject. -- Persi Diaconis, Professor of Mathematics, Stanford It has no mean derivative results about the only derivatives that matter... A spunky tool-toting heroine called Delta Wye seems the perfect role model for our next generation. -- Susan Holmes, Professor of Statistics, Stanford A creative take on an old, and for many, tough subject...Gonick's cartoons and intelligent humor make it a fun read. -- Amy Langville, Recipient of the Distinguished Researcher Award at College of Charleston and South Carolina Faculty of the Year

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Book SynopsisA ''difficult'' subject so simply taught - brilliant book'' - Amazon 5 star review ⭐⭐⭐⭐⭐''This is a great refresher book! Lots of worked out examples, great explanations [and] hundreds of practice problems and solutions'' - Amazon 5 star review ⭐⭐⭐⭐⭐''This book has been very helpful for my calculus class, I recommend it to anyone that needs extra help, or just feel like learning something new.'' - Amazon 5 star review ⭐⭐⭐⭐⭐Calculus: A Complete Introduction is the most comprehensive yet easy-to-use introduction to using calculus. Written by a leading expert, this book will help you if you are studying for an important exam or essay, or if you simply want to improve your knowledge. The book covers all areas of calculus, including functions, gradients, rates of change, differentiation, exponential and logarithm

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Book SynopsisCalculus For Dummies, 2nd Edition (9781119293491) was previously published as Calculus For Dummies, 2nd Edition (9781118791295). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product.Table of ContentsIntroduction 1 Part 1: An Overview of Calculus 5 Chapter 1: What Is Calculus? 7 Chapter 2: The Two Big Ideas of Calculus: Differentiation and Integration — plus Infinite Series 13 Chapter 3: Why Calculus Works 21 Part 2: Warming Up with Calculus Prerequisites 27 Chapter 4: Pre-Algebra and Algebra Review 29 Chapter 5: Funky Functions and Their Groovy Graphs 43 Chapter 6: The Trig Tango 61 Part 3: Limits 73 Chapter 7: Limits and Continuity 75 Chapter 8: Evaluating Limits 89 Part 4: Differentiation 105 Chapter 9: Differentiation Orientation 107 Chapter 10: Differentiation Rules — Yeah, Man, It Rules 127 Chapter 11: Differentiation and the Shape of Curves 147 Chapter 12: Your Problems Are Solved: Differentiation to the Rescue! 171 Chapter 13: More Differentiation Problems: Going Off on a Tangent 193 Part 5: Integration and Infinite Series 207 Chapter 14: Intro to Integration and Approximating Area 209 Chapter 15: Integration: It’s Backwards Differentiation 233 Chapter 16: Integration Techniques for Experts 263 Chapter 17: Forget Dr Phil: Use the Integral to Solve Problems 285 Chapter 18: Taming the Infinite with Improper Integrals 303 Chapter 19: Infinite Series 315 Part 6: The Part of Tens 339 Chapter 20: Ten Things to Remember 341 Chapter 21: Ten Things to Forget 345 Chapter 22: Ten Things You Can’t Get Away With 349 Index 353

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Book SynopsisOn August 10, 1632, five leading Jesuits convened in a sombre Roman palazzo to pass judgment on a simple idea: that a continuous line is composed of distinct and limitlessly tiny parts. The doctrine would become the foundation of calculus, but on that fateful day the judges ruled that it was forbidden. With the stroke of a pen they set off a war for the soul of the modern world. Amir Alexander takes us from the bloody religious strife of the sixteenth century to the battlefields of the English civil war and the fierce confrontations between leading thinkers like Galileo and Hobbes. The legitimacy of popes and kings, as well as our modern beliefs in human liberty and progressive science, hung in the balance; the answer hinged on the infinitesimal. Pulsing with drama and excitement, Infinitesimal will forever change the way you look at a simple line.Trade Review'A well-spun yarn, a cracking read… engaging…unique’ -- History Today‘A gripping and thorough history of the ultimate triumph of the mathematical tool… Infinitesimal will inspire you to dig deeper into the implications of the philosophy of mathematics and knowledge’ * New Scientist *‘A complex story told with skill and verve… Alexander does an excellent job of presenting both sides of the debate.’ * THES Book of the Week *‘Amir Alexander’s enthralling book presents a controversial mathematical breakthrough, vividly describing the players and showing exactly what was at stake.’ * Tony Mann, Director of the Maths Centre, University of Greenwich and Former President of the British *“Bertrand Russell once wrote that mathematics had a ‘beauty cold and austere’… Amir Alexander shows that mathematics can also become entangled in ugliness hot and messy… [a] fascinating narrative.” * New York Times *“[Told with] high drama and thrilling tension.” * Kirkus Reviews (starred review) *‘A gripping tale of mathematical, philosophical, and theological controversies in the run-up to calculus.' * Ian Stewart, author of Professor Stewart's Cabinet of Mathematical Curiosities *‘Clever and enthralling.' -- Simon Schaffer, Professor of the History of Science, University of Cambridge‘A real-world Da Vinci Code’ * Publishers Weekly *‘Fascinating.. Amir Alexander vividly recreates a wonderfully strange chapter of scientific history... You will never look at calculus the same way again.’ -- Jordan Ellenberg, Professor of Mathematics, University of Wisconsin-Madison‘Gripping… Amir Alexander writes with elegance and verve... A page-turner full of fascinating stories about the struggles of remarkable individuals and ideas, Infinitesimal will help you understand the world at a deeper level.’ -- Edward Frenkel, Professor, University of California at Berkeley, and author of Love and Math‘We thought we knew the whole story: Copernicus, Galileo, the sun in the centre, the Church rushing to condemn. Now this remarkable book puts the deeply subversive doctrine of atomism and its accompanying mathematics at the heart of modern science.’ -- Margaret C. Jacob, Distinguished Professor of History, University of California, Los Angeles‘A seamless synthesis of cultural history and storytelling... The history of mathematics has rarely been so readable.’ -- Michael Harris, Professor of Mathematics, Columbia University and Université Paris Diderot‘You may find it hard to believe that illustrious mathematicians, philosophers, and religious thinkers would engage in a bitter dispute over infinitely small quantities. Yet this is precisely what happened in the seventeenth century. In Infinitesimal, Amir Alexander puts this fascinating battle in historical and intellectual context.’ -- Mario Livio, Astrophysicist, Space Telescope Science Institute, and author of Brilliant Blunders: Fr

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Book SynopsisFinance provides a dramatic example of the successful application of advanced mathematical techniques to the practical problem of pricing financial derivatives. This self-contained 2002 text is designed for first courses in financial calculus aimed at students with a good background in mathematics. Key concepts such as martingales and change of measure are introduced in the discrete time framework, allowing an accessible account of Brownian motion and stochastic calculus: proofs in the continuous-time world follow naturally. The Black-Scholes pricing formula is first derived in the simplest financial context. The second half of the book is then devoted to increasing the financial sophistication of the models and instruments. The final chapter introduces more advanced topics including stock price models with jumps, and stochastic volatility. A valuable feature is the large number of exercises and examples, designed to test technique and illustrate how the methods and concepts can be appTrade Review' … being relatively short and a paperback must make it appealing to students and those who need a quick introduction to the material. … nicely produced and elegantly laid out. I would consider adopting it as a text for a course in this topic. Publication of the International Statistical Institute'This is a well written textbook which should be suitable for final year undergraduate and first year graduate students having some background in probability theory.' Klaus Schrüger, Zentralblatt MATH' … this is a very well-organized text that makes it easy to learn.' Journal of the Royal Statistical Society'… it was necessary to supply the framework of the book with some theory of stochastic analysis and to provide a mathematical explanation of the notions used.' EMS NewsletterTable of ContentsPreface; 1. Single period models; 2. Binomial trees and discrete parameter martingales; 3. Brownian motion; 4. Stochastic calculus; 5. The Black-Scholes model; 6. Different payoffs; 7. Bigger models; Bibliography and further reading; Notation; Index.

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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.

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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

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Book SynopsisVolume Two of an award-winning professor’s introduction to essential concepts of calculus and mathematical modeling for students in the biosciencesTrade Review“Clear, enthusiastic, and communicating a love of maths, this is a useful, engaging and well-written text.”—Becca Asquith, Professor of Mathematical Immunology, Imperial College London"This is a wonderful book, wise and witty. It would have taught me most of the math I needed for my career in research – if I did all the problems."—Stephen Stearns, author of The Evolution of Life Histories and Evolutionary Medicine“This well-written book covers multivariate calculus and dynamical systems within the context of the biological sciences, providing well-chosen, up-to-date biomedical examples. The Markov chain, along with its many interesting applications, is also introduced.”—Hongyu He, Professor of Mathematics, Louisiana State University

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Book SynopsisRobert Adams is an Emeritus Professor in the Mathematics Department at the University of British Columbia. He first joined UBC in 1966 after completing a Ph.D. in Mathematics at the University of Toronto. With a keen interest in computers, mathematical typesetting, and illustration, Professor Adams became the first Canadian author in 1984 to typeset his own textbooks using TeX on a personal computer. Christopher Essex is a Professor in the Department of Applied Mathematics at the University of Western Ontario, an award-winning teacher and author. Dr. Essex did pioneering work on the thermodynamics of photon and neutrino radiation.Table of ContentsChapter P Preliminaries Limits and Continuity Differentiation Transcendental Functions More Applications of Differentiation Integration Techniques of Integration Applications of Integration Conics, Parametric Curves, and Polar Curves Sequence, Series, and Power Series Vectors and Coordinate Geometry in 3-Space Arc length, Metric Spaces, and Applications Vector Functions and Curves Partial Differentiation Applications of Partial Derivatives Multiple Integration Vector Fields Vector Calculus Differential Forms and Exterior Calculus Ordinary Differential Equations More Topics in Differential Equations Appendix 1 Complex Numbers Appendix 2 Complex Functions Appendix 3 Continuous Functions Appendix 4 The Riemann Integral Appendix 5 Doing Calculus with Maple Appendix 6 Doing Calculus with Python

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Book SynopsisThis new fourth edition of the acclaimed and bestselling Div, Grad, Curl, and All That has been carefully revised and now includes updated notations and seven new example exercises.

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Book SynopsisBy spinning 28 engaging mathematical tales, Orlin shows us that calculus is simply another language to express the very things we humans grapple with every day - love, risk, time and, most importantly, change. Divided into two parts, Moments and Eternities, and drawing on everyone from Sherlock Holmes to Mark Twain to David Foster Wallace, Change is the Only Constant unearths connections between calculus, art, literature and a beloved dog named Elvis. This is not just maths for maths'' sake; it''s maths for the sake of becoming a wiser and more thoughtful human.

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Book SynopsisTable of ContentsIntroduction 1 About This Book 1 Foolish Assumptions 2 Icons Used in This Book 3 Beyond the Book 3 Where to Go from Here 3 Part 1: Getting Started with Pre-Calculus 5 Chapter 1: Pre-Pre-Calculus 7 Pre-Calculus: An Overview 8 All the Number Basics (No, Not How to Count Them!) 9 The multitude of number types: Terms to know 9 The fundamental operations you can perform on numbers 11 The properties of numbers: Truths to remember 11 Visual Statements: When Math Follows Form with Function 12 Basic terms and concepts 13 Graphing linear equalities and inequalities 14 Gathering information from graphs 15 Get Yourself a Graphing Calculator 16 Chapter 2: Playing with Real Numbers 19 Solving Inequalities 19 Recapping inequality how-tos 20 Solving equations and inequalities when absolute value is involved 20 Expressing solutions for inequalities with interval notation 22 Variations on Dividing and Multiplying: Working with Radicals and Exponents 24 Defining and relating radicals and exponents 24 Rewriting radicals as exponents (or, creating rational exponents) 25 Getting a radical out of a denominator: Rationalizing 26 Chapter 3: The Building Blocks of Pre-Calculus Functions 31 Qualities of Special Function Types and Their Graphs 32 Even and odd functions 32 One-to-one functions 32 Dealing with Parent Functions and Their Graphs 33 Linear functions 33 Quadratic functions 33 Square-root functions 34 Absolute-value functions 34 Cubic functions 35 Cube-root functions 36 Graphing Functions That Have More Than One Rule: Piece-Wise Functions 37 Setting the Stage for Rational Functions 38 Step 1: Search for vertical asymptotes 39 Step 2: Look for horizontal asymptotes 40 Step 3: Seek out oblique asymptotes 41 Step 4: Locate the x- and y-intercepts 42 Putting the Results to Work: Graphing Rational Functions 42 Chapter 4: Operating on Functions 49 Transforming the Parent Graphs 50 Stretching and flattening 50 Translations 52 Reflections 54 Combining various transformations (a transformation in itself!) 55 Transforming functions point by point 57 Sharpen Your Scalpel: Operating on Functions 58 Adding and subtracting 59 Multiplying and dividing 60 Breaking down a composition of functions 60 Adjusting the domain and range of combined functions (if applicable) 61 Turning Inside Out with Inverse Functions 63 Graphing an inverse 64 Inverting a function to find its inverse 66 Verifying an inverse 66 Chapter 5: Digging Out and Using Roots to Graph Polynomial Functions 69 Understanding Degrees and Roots 70 Factoring a Polynomial Expression 71 Always the first step: Looking for a GCF 72 Unwrapping the box containing a trinomial 73 Recognizing and factoring special polynomials 74 Grouping to factor four or more terms 77 Finding the Roots of a Factored Equation 78 Cracking a Quadratic Equation When It Won’t Factor 79 Using the quadratic formula 79 Completing the square 80 Solving Unfactorable Polynomials with a Degree Higher Than Two 81 Counting a polynomial’s total roots 82 Tallying the real roots: Descartes’s rule of signs 82 Accounting for imaginary roots: The fundamental theorem of algebra 83 Guessing and checking the real roots 84 Put It in Reverse: Using Solutions to Find Factors 90 Graphing Polynomials 91 When all the roots are real numbers 91 When roots are imaginary numbers: Combining all techniques 95 Chapter 6: Exponential and Logarithmic Functions 97 Exploring Exponential Functions 98 Searching the ins and outs of exponential functions 98 Graphing and transforming exponential functions 100 Logarithms: The Inverse of Exponential Functions 102 Getting a better handle on logarithms 102 Managing the properties and identities of logs 103 Changing a log’s base 105 Calculating a number when you know its log: Inverse logs 105 Graphing logs 106 Base Jumping to Simplify and Solve Equations 109 Stepping through the process of exponential equation solving 109 Solving logarithmic equations 112 Growing Exponentially: Word Problems in the Kitchen 113 Part 2: The Essentials of Trigonometry 117 Chapter 7: Circling in on Angles 119 Introducing Radians: Circles Weren’t Always Measured in Degrees 120 Trig Ratios: Taking Right Triangles a Step Further 121 Making a sine 121 Looking for a cosine 122 Going on a tangent 124 Discovering the flip side: Reciprocal trig functions 125 Working in reverse: Inverse trig functions 126 Understanding How Trig Ratios Work on the Coordinate Plane 127 Building the Unit Circle by Dissecting the Right Way 129 Familiarizing yourself with the most common angles 129 Drawing uncommon angles 131 Digesting Special Triangle Ratios 132 The 45er: 45 -45 -90 triangle 132 The old 30-60: 30 -60 -90 triangle 133 Triangles and the Unit Circle: Working Together for the Common Good 135 Placing the major angles correctly, sans protractor 135 Retrieving trig-function values on the unit circle 138 Finding the reference angle to solve for angles on the unit circle 142 Measuring Arcs: When the Circle Is Put in Motion 146 Chapter 8: Simplifying the Graphing and Transformation of Trig Functions 149 Drafting the Sine and Cosine Parent Graphs 150 Sketching sine 150 Looking at cosine 152 Graphing Tangent and Cotangent 154 Tackling tangent 154 Clarifying cotangent 157 Putting Secant and Cosecant in Pictures 159 Graphing secant 159 Checking out cosecant 161 Transforming Trig Graphs 162 Messing with sine and cosine graphs 163 Tweaking tangent and cotangent graphs 173 Transforming the graphs of secant and cosecant 176 Chapter 9: Identifying with Trig Identities: The Basics 181 Keeping the End in Mind: A Quick Primer on Identities 182 Lining Up the Means to the End: Basic Trig Identities 182 Reciprocal and ratio identities 183 Pythagorean identities 185 Even/odd identities 188 Co-function identities 190 Periodicity identities 192 Tackling Difficult Trig Proofs: Some Techniques to Know 194 Dealing with demanding denominators 195 Going solo on each side 199 Chapter 10: Advanced Identities: Your Keys to Success 201 Finding Trig Functions of Sums and Differences 202 Searching out the sine of a b 202 Calculating the cosine of a b 206 Taming the tangent of a b 209 Doubling an Angle and Finding Its Trig Value 211 Finding the sine of a doubled angle 212 Calculating cosines for two 213 Squaring your cares away 215 Having twice the fun with tangents 216 Taking Trig Functions of Common Angles Divided in Two 217 A Glimpse of Calculus: Traveling from Products to Sums and Back 219 Expressing products as sums (or differences) 219 Transporting from sums (or differences) to products 220 Eliminating Exponents with Power-Reducing Formulas 221 Chapter 11: Taking Charge of Oblique Triangles with the Laws of Sines and Cosines 223 Solving a Triangle with the Law of Sines 224 When you know two angle measures 225 When you know two consecutive side lengths 228 Conquering a Triangle with the Law of Cosines 235 SSS: Finding angles using only sides 236 SAS: Tagging the angle in the middle (and the two sides) 238 Filling in the Triangle by Calculating Area 240 Finding area with two sides and an included angle (for SAS scenarios) 241 Using Heron’s Formula (for SSS scenarios) 241 Part 3: Analytic Geometry and System Solving 243 Chapter 12: Plane Thinking: Complex Numbers and Polar Coordinates 245 Understanding Real versus Imaginary 246 Combining Real and Imaginary: The Complex Number System 247 Grasping the usefulness of complex numbers 247 Performing operations with complex numbers 248 Graphing Complex Numbers 250 Plotting Around a Pole: Polar Coordinates 251 Wrapping your brain around the polar coordinate plane 252 Graphing polar coordinates with negative values 254 Changing to and from polar coordinates 256 Picturing polar equations 259 Chapter 13: Creating Conics by Slicing Cones 263 Cone to Cone: Identifying the Four Conic Sections 264 In picture (graph form) 264 In print (equation form) 266 Going Round and Round: Graphing Circles 267 Graphing circles at the origin 267 Graphing circles away from the origin 268 Writing in center–radius form 269 Riding the Ups and Downs with Parabolas 270 Labeling the parts 270 Understanding the characteristics of a standard parabola 271 Plotting the variations: Parabolas all over the plane 272 The vertex, axis of symmetry, focus, and directrix 273 Identifying the min and max of vertical parabolas 276 The Fat and the Skinny on the Ellipse 278 Labeling ellipses and expressing them with algebra 279 Identifying the parts from the equation 281 Pair Two Curves and What Do You Get? Hyperbolas 284 Visualizing the two types of hyperbolas and their bits and pieces 284 Graphing a hyperbola from an equation 287 Finding the equations of asymptotes 287 Expressing Conics Outside the Realm of Cartesian Coordinates 289 Graphing conic sections in parametric form 290 The equations of conic sections on the polar coordinate plane 292 Chapter 14: Streamlining Systems, Managing Variables 295 A Primer on Your System-Solving Options 296 Algebraic Solutions of Two-Equation Systems 297 Solving linear systems 297 Working nonlinear systems 300 Solving Systems with More than Two Equations 304 Decomposing Partial Fractions 306 Surveying Systems of Inequalities 307 Introducing Matrices: The Basics 309 Applying basic operations to matrices 310 Multiplying matrices by each other 311 Simplifying Matrices to Ease the Solving Process 312 Writing a system in matrix form 313 Reduced row-echelon form 313 Augmented form 314 Making Matrices Work for You 315 Using Gaussian elimination to solve systems 316 Multiplying a matrix by its inverse 320 Using determinants: Cramer’s Rule 323 Chapter 15: Sequences, Series, and Expanding Binomials for the Real World 327 Speaking Sequentially: Grasping the General Method 328 Determining a sequence’s terms 328 Working in reverse: Forming an expression from terms 329 Recursive sequences: One type of general sequence 330 Difference between Terms: Arithmetic Sequences 331 Using consecutive terms to find another 332 Using any two terms 332 Ratios and Consecutive Paired Terms: Geometric Sequences 334 Identifying a particular term when given consecutive terms 334 Going out of order: Dealing with nonconsecutive terms 335 Creating a Series: Summing Terms of a Sequence 337 Reviewing general summation notation 337 Summing an arithmetic sequence 338 Seeing how a geometric sequence adds up 339 Expanding with the Binomial Theorem 342 Breaking down the binomial theorem 344 Expanding by using the binomial theorem 345 Chapter 16: Onward to Calculus 351 Scoping Out the Differences between Pre-Calculus and Calculus 352 Understanding Your Limits 353 Finding the Limit of a Function 355 Graphically 355 Analytically 356 Algebraically 357 Operating on Limits: The Limit Laws 361 Calculating the Average Rate of Change 362 Exploring Continuity in Functions 363 Determining whether a function is continuous 364 Discontinuity in rational functions 365 Part 4: The Part of Tens 367 Chapter 17: Ten Polar Graphs 369 Spiraling Outward 369 Falling in Love with a Cardioid 370 Cardioids and Lima Beans 370 Leaning Lemniscates 371 Lacing through Lemniscates 372 Roses with Even Petals 372 A rose Is a Rose Is a Rose 373 Limaçon or Escargot? 373 Limaçon on the Side 374 Bifolium or Rabbit Ears? 374 Chapter 18: Ten Habits to Adjust before Calculus 375 Figure Out What the Problem Is Asking 375 Draw Pictures (the More the Better) 376 Plan Your Attack — Identify Your Targets 377 Write Down Any Formulas 377 Show Each Step of Your Work 378 Know When to Quit 378 Check Your Answers 379 Practice Plenty of Problems 380 Keep Track of the Order of Operations 380 Use Caution When Dealing with Fractions 381 Index 383

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Book SynopsisThis textbook teaches the fundamentals of calculus, keeping points clear, succinct and focused, with plenty of diagrams and practice but relatively few words. It assumes a very basic knowledge but revises the key prerequisites before moving on. Definitions are highlighted for easy understanding and reference, and worked examples illustrate the explanations. Chapters are interwoven with exercises, whilst each chapter also ends with a comprehensive set of exercises, with answers in the back of the book. Introductory paragraphs describe the real-world application of each topic, and also include briefly where relevant any interesting historical facts about the development of the mathematical subject.This text is intended for undergraduate students in engineering taking a course in calculus. It works for the Foundation and 1st year levels. It has a companion volume Foundation Algebra.Table of Contents1. Prerequisites 2. Derivative – I 3. Derivative – II 4. Applications of Derivatives I. 5. Applications of Derivatives II 6. Integration – I 7. Integration II 8. Definite Integration 9. Numerical Integration 10. Applications of Integration 11. Differential Equations 12. Differential Equation Models. Appendix: Proofs Answers to Exercise Questions.

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Book SynopsisAn authorised reissue of the long out of print classic textbook, Advanced Calculus by the late Dr Lynn Loomis and Dr Shlomo Sternberg both of Harvard University has been a revered but hard to find textbook for the advanced calculus course for decades.This book is based on an honors course in advanced calculus that the authors gave in the 1960's. The foundational material, presented in the unstarred sections of Chapters 1 through 11, was normally covered, but different applications of this basic material were stressed from year to year, and the book therefore contains more material than was covered in any one year. It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis.The prerequisites are a good grounding in the calculus of one variable from a mathematically rigorous point of view, together with some acquaintance with linear algebra. The reader should be familiar with limit and continuity type arguments and have a certain amount of mathematical sophistication. As possible introductory texts, we mention Differential and Integral Calculus by R Courant, Calculus by T Apostol, Calculus by M Spivak, and Pure Mathematics by G Hardy. The reader should also have some experience with partial derivatives.In overall plan the book divides roughly into a first half which develops the calculus (principally the differential calculus) in the setting of normed vector spaces, and a second half which deals with the calculus of differentiable manifolds.Table of ContentsIntroduction; Vector Spaces; Finite-Dimensional Vector Spaces; The Differential Calculus; Compactness and Completeness; Scalar Product Spaces; Differential Equations; Multilinear Functionals; Integration; Differentiable Manifolds; The Integral Calculus on Manifolds; Exterior Calculus; Potential Theory in En; Classical Mechanics.

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Book SynopsisShortlisted for the Royal Society Science Book Prize 2019A magisterial history of calculus (and the people behind it) from one of the world's foremost mathematicians.This is the captivating story of mathematics' greatest ever idea: calculus. Without it, there would be no computers, no microwave ovens, no GPS, and no space travel. But before it gave modern man almost infinite powers, calculus was behind centuries of controversy, competition, and even death. Taking us on a thrilling journey through three millennia, professor Steven Strogatz charts the development of this seminal achievement from the days of Archimedes to today's breakthroughs in chaos theory and artificial intelligence. Filled with idiosyncratic characters from Pythagoras to Fourier, Infinite Powers is a compelling human drama that reveals the legacy of calculus on nearly every aspect of modern civilisation, including science, politics, medicine, philosophy, and much besides.Trade ReviewWarning: this book is dangerous. It will make you love mathematics. Even more, there is a nonzero risk it will turn you into a mathematician. * Nassim Nicholas Taleb, bestselling author of The Black Swan *Fascinating reading. * Scientific American *Eloquent, erudite and charming. A remarkable story. Strogatz is a world class mathematician and a world class science writer. With a light touch and razor-sharp clarity, he tells the remarkable story of a mathematical breakthrough that changed the world - and continues to do so. * Alex Bellos, bestselling author of Alex's Adventures in Numberland *Glorious! A master class in accessible maths writing and a perfect read for anyone who feels like they never quite understood what all the fuss was about. It had me leaping for joy. * Hannah Fry, bestselling author of Hello World and presenter of BBC R4’s The Curious Cases of Rutherford and Fry *Simple, lucid, amusing, informative, and a pleasure to read. If you want to know where calculus came from, how it works, what it's good for, and where it's going next, this is the book for you. * Professor Ian Stewart, author of Significant Figures *A fine, thoughtful attempt to make the greatest stories relating to calculus accessible... After reading Infinite Powers, we should no longer fear calculus. * Literary Review *The most fascinating book I have ever read. If you have even the slightest curiosity about maths and its role in this world, I implore you to read this amazing book. * Jo Boaler, professor of mathematics education, Stanford University *A wide-ranging, humane, thoroughly readable take on one of the greatest ideas our species has ever produced. * Jordan Ellenberg, author of How Not to Be Wrong *Fascinating anecdotes abound in Infinite Powers... [Strogatz] has written a romp through the history of calculus. * Nature *A tour de force. Elegant and ebullient. Strogatz speaks to everyone, reminding us why mathematics matters in a practical sense while all the time highlighting its beauty. * Lisa Randall, Professor of Physics at Harvard University and author of Dark Matter and The Dinosaurs *A highly readable account of calculus and its modern applications - all done with the human touch. * Dr David Acheson, Emeritus Fellow, Oxford University and author of The Calculus Story *An incalculable pleasure. If calculus is the language of the universe, then Steven Strogatz is its Homer. * Daniel Gilbert, author of Stumbling on Happiness *In this engaging book, Steven Strogatz illuminates the importance of calculus and explains its mysteries as only he can. * Sean Carroll, author of The Particle at the End of the Universe *Table of Contents1: Infinity 2: The Man Who Harnessed Infinity 3: Discovering the Laws of Motion 4: The Dawn of Differential Calculus 5: The Crossroads 6: The Vocabulary of Change 7: The Secret Fountain 8: Fictions of the Mind 9: The Logical Universe 10: Making Waves 11: The Future of Calculus

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Book SynopsisTrade Review"...if you use this book frequently it’s definitely worth getting the new edition…" --MAA.org, November 2014 "The integrals are very useful, but this book includes many other features that will be helpful to the reader, especially graduate students. The sections on Hermite and Legendre polynomials are especially helpful for students of Electricity and Magnetism, Quantum Mechanics, and Mathematical physics (they won't have to hunt in several books to find what they need)." --Barry Simon, California Institute of Technology "This book is to the CRC Mathematical Tables as the unabridged Oxford English Dictionary is to Webster's Collegiate. Besides being big, it's easy to find things in, because of the way the integrals are organized into classes...It really helped me through grad school." --Phil Hobbs, Amazon ReviewTable of Contents1. Elementary Functions 2. Indefinite Integrals of Elementary Functions 3. Definite Integrals of Elementary Functions 4. Combinations Involving Trigonometric and Hyperbolic Functions and Power 5. Indefinite Integrals of Special Functions 6. Definite Integrals of Special Functions 7. Associated Legendre Functions 8. Special Functions 9. Hypergeometric Functions 10. Vector Field Theory 11. Algebraic Inequalities 12. Integral Inequalities 13. Matrices and Related Result 14. Determinants 15. Norms 16. Ordinary Differential Equations 17. Fourier, Laplace, and Mellin Transforms 18. The Z-transform

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Book SynopsisRaymond N. Greenwell earned a B.A. in Mathematics and Physics from the University of San Diego, and an M.S. in Statistics, an M.S. in Applied Mathematics, and a Ph.D. in Applied Mathematics from Michigan State University, where he earned the graduate student teaching award in 1979. After teaching at Albion College in Michigan for four years, he moved to Hofstra University in 1983, where he currently is Professor of Mathematics. Raymond has published articles on fluid mechanics, mathematical biology, genetic algorithms, combinatorics, statistics, and undergraduate mathematics education. He is a member of MAA, AMS, SIAM, NCTM, and AMATYC. He has served as governor of the Metropolitan New York Section of the MAA, as well as webmaster and liaison coordinator, and he received a distinguished service award from the Section in 2003. He is an outdoor enthusiast and leads trips in the Sierra Club's Inner City Outings program. Nathan P. Ritchey Table of ContentsR. Algebra Reference R.1 Polynomials R.2 Factoring R.3 Rational Expressions R.4 Equations R.5 Inequalities R.6 Exponents R.7 Radicals 1. Functions 1.1 Lines and Linear Functions 1.2 The Least Squares Line 1.3 Properties of Functions 1.4 Quadratic Functions; Translation and Reflection 1.5 Polynomial and Rational Functions Chapter Review Extended Application: Using Extrapolation to Predict Life Expectancy2. Exponential, Logarithmic, and Trigonometric Functions 2.1 Exponential Functions 2.2 Logarithmic Functions 2.3 Applications: Growth and Decay 2.4 Trigonometric Functions Chapter Review Extended Application: Power Functions 3. The Derivative 3.1 Limits 3.2 Continuity 3.3 Rates of Change 3.4 Definition of the Derivative 3.5 Graphical Differentiation Chapter Review Extended Application: A Model For Drugs Administered Intravenously 4. Calculating the Derivative 4.1 Techniques for Finding Derivatives 4.2 Derivatives of Products and Quotients 4.3 The Chain Rule 4.4 Derivatives of Exponential Functions 4.5 Derivatives of Logarithmic Functions 4.6 Derivatives of Trigonometric Functions Chapter Review Extended Application: Managing Renewable Resources 5. Graphs and the Derivative 5.1 Increasing and Decreasing Functions 5.2 Relative Extrema 5.3 Higher Derivatives, Concavity, and the Second Derivative Test 5.4 Curve Sketching Chapter Review Extended Application: A Drug Concentration Model for Orally Administered Medications 6. Applications of the Derivative 6.1 Absolute Extrema 6.2 Applications of Extrema 6.3 Implicit Differentiation 6.4 Related Rates 6.5 Differentials: Linear Approximation Chapter Review Extended Application: A Total Cost Model for a Training Program 7. Integration 7.1 Antiderivatives 7.2 Substitution 7.3 Area and the Definite Integral 7.4 The Fundamental Theorem of Calculus 7.5 The Area Between Two Curves Chapter Review Extended Application: Estimating Depletion Dates for Minerals 8. Further Techniques and Applications of Integration 8.1 Numerical Integration 8.2 Integration by Parts 8.3 Volume and Average Value 8.4 Improper Integrals Chapter Review Extended Application: Flow Systems 9. Multivariable Calculus 9.1 Functions of Several Variables 9.2 Partial Derivatives 9.3 Maxima and Minima 9.4 Total Differentials and Approximations 9.5 Double Integrals Chapter Review Extended Application: Optimization for a Predator 10. Matrices 10.1 Solution of Linear Systems 10.2 Addition and Subtraction of Matrices 10.3 Multiplication of Matrices 10.4 Matrix Inverses 10.5 Eigenvalues and Eigenvectors Chapter Review Extended Application: Contagion 11. Differential Equations 11.1 Solutions of Elementary and Separable Differential Equations 11.2 Linear First-Order Differential Equations 11.3 Euler's Method 11.4 Linear Systems of Differential Equations 11.5 Non-Linear Systems of Differential Equations 11.6 Applications of Differential Equations Chapter Review Extended Application: Pollution of the Great Lakes 12. Probability

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Book SynopsisThis book contains mathematical preliminaries in which basic definitions of fractional derivatives and spaces are presented. The central part of the book contains various applications in classical mechanics including fields such as: viscoelasticity, heat conduction, wave propagation and variational Hamilton–type principles. Mathematical rigor will be observed in the applications. The authors provide some problems formulated in the classical setting and some in the distributional setting. The solutions to these problems are presented in analytical form and these solutions are then analyzed numerically. Theorems on the existence of solutions will be presented for all examples discussed. In using various constitutive equations the restrictions following from the second law of thermodynamics will be implemented. Finally, the physical implications of obtained solutions will be discussed in detail.Trade Review“The book will be useful to researchers and students looking for applications of fractional calculus in applied mechanics and engineering.” (Zentralblatt MATH, 1 November 2014) Table of ContentsPreface ixPart 1. Mathematical Preliminaries, Definitions and Properties of Fractional Integrals and Derivatives 1 Chapter 1. Mathematical Preliminaries 3Chapter 2. Basic Definitions and Properties of Fractional Integrals and Derivatives 17 Part 2. Mechanical Systems 49 Chapter 3. Restrictions Following from the Thermodynamics for Fractional Derivative Models of a Viscoelastic Body 51 Chapter 4. Vibrations with Fractional Dissipation 83 Chapter 5. Lateral Vibrations and Stability of Viscoelastic Rods 123 Chapter 6. Fractional Diffusion-Wave Equations 185 Chapter 7. Fractional Heat Conduction Equations 257Bibliography 289Index 311

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With emphasis on stochastic aspects of deterministic systems this short book introduces the reader to the basic facts and some special topics of applied ergodic theory. It adresses advanced undergraduate and graduate students from various disciplines, i.e. mathematicians, physicists, electrical and mechanical engineers. Based upon a sound (but non-technical) mathematical introduction, a number of typical examples from applications (mostly from mechanics) are thoroughly discussed. By studying both probabilistic and deterministic features of dynamical systems the reader will develop what might be considered a unified view on chaos and chance as two sides of the same thing.

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Book SynopsisThis book provides a concise and meticulous introduction to functional analysis. Since the topic draws heavily on the interplay between the algebraic structure of a linear space and the distance structure of a metric space, functional analysis is increasingly gaining the attention of not only mathematicians but also scientists and engineers. The purpose of the text is to present the basic aspects of functional analysis to this varied audience, keeping in mind the considerations of applicability. A novelty of this book is the inclusion of a result by Zabreiko, which states that every countably subadditive seminorm on a Banach space is continuous. Several major theorems in functional analysis are easy consequences of this result.The entire book can be used as a textbook for an introductory course in functional analysis without having to make any specific selection from the topics presented here. Basic notions in the setting of a metric space are defined in terms of sequences. These include total boundedness, compactness, continuity and uniform continuity. Offering concise and to-the-point treatment of each topic in the framework of a normed space and of an inner product space, the book represents a valuable resource for advanced undergraduate students in mathematics, and will also appeal to graduate students and faculty in the natural sciences and engineering. The book is accessible to anyone who is familiar with linear algebra and real analysis.Trade Review“The title of this book indicates that it is mainly devoted to linear maps on linear spaces. … All chapters are accompanied by useful exercises of varying levels of difficulty, which help the readers to develop their knowledge on the topics. The solutions of the exercises are given at the end of the book. … This textbook is essentially addressed to people working in engineering and sciences branches.” (Mohammad Sal Moslehian, zbMATH 1352.46001, 2017)Table of ContentsChapter 1. Preliminaries.- Chapter 2. Basic Framework.- Chapter 3. Bounded Linear Maps.- Chapter 4. Dual Spaces, Transposes and Adjoints.- Chapter 5. Spectral Theory.

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Book SynopsisTable of ContentsChapter P Preliminaries Limits and Continuity Differentiation Transcendental Functions More Applications of Differentiation Integration Techniques of Integration Applications of Integration Conics, Parametric Curves, and Polar Curves Sequence, Series, and Power Series Vectors and Coordinate Geometry in 3-Space Arc length, Metric Spaces, and Applications Vector Functions and Curves Partial Differentiation Applications of Partial Derivatives Multiple Integration Vector Fields Vector Calculus Differential Forms and Exterior Calculus Ordinary Differential Equations More Topics in Differential Equations Appendix 1 Complex Numbers Appendix 2 Complex Functions Appendix 3 Continuous Functions Appendix 4 The Riemann Integral Appendix 5 Doing Calculus with Maple Appendix 6 Doing Calculus with Python

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Book SynopsisRobert Adams is an Emeritus Professor in the Mathematics Department at the University of British Columbia. He first joined UBC in 1966 after completing a Ph.D. in Mathematics at the University of Toronto. With a keen interest in computers, mathematical typesetting, and illustration, Professor Adams became the first Canadian author in 1984 to typeset his own textbooks using TeX on a personal computer. Christopher Essex is a Professor in the Department of Applied Mathematics at the University of Western Ontario, an award-winning teacher and author. Dr. Essex did pioneering work on the thermodynamics of photon and neutrino radiation.

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Book SynopsisTrade Review"There are myriad introductory books on special relativity. This one distinguishes itself by working through the mathematics of relativity in a very detailed yet conversational fashion. . . . Highly recommended." * Choice *“Introducing students in small, careful steps toward an understanding of the notation and physics behind special relativity has not been undertaken at this level since the text by Taylor and Wheeler from thirty years ago. Goldberg’s approach of encouraging the reader to see the simplicity behind the seemingly complex is welcome.” -- Christopher G. Tully, author of "Elementary Particle Physics in a Nutshell"“This engaging book will shape the education of a generation of physicists and astrophysicists. It defines the conceptual and mathematical stage—spacetime—on which physics is performed. From contemporary notation in the early chapters to sophisticated applications in the late chapters, Goldberg's book will not only propel students to more advanced classes, it will ease their entry into research.” -- Daniel Fabrycky, Department of Astronomy and Astrophysics, University of Chicago“Goldberg slings the reader straight in at the deep end . . . but with enough masterly wit to keep you afloat.” * Nature, on "The Universe in the Rearview Mirror" *“Reading this book is like taking a class with the most awesome science professor ever.” -- Annalee Newitz, founding editor of io9, on "The Universe in the Rearview Mirror"“Most physics books can’t really be described as ‘rollicking,’ but most physics books aren't written by Dave Goldberg.” -- Sean Carroll, theoretical physicist at Caltech, author of "The Particle at the End of the Universe," on "The Universe in the Rearview Mirror"

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Book SynopsisTrade Review"There are myriad introductory books on special relativity. This one distinguishes itself by working through the mathematics of relativity in a very detailed yet conversational fashion. . . . Highly recommended." * Choice *“Introducing students in small, careful steps toward an understanding of the notation and physics behind special relativity has not been undertaken at this level since the text by Taylor and Wheeler from thirty years ago. Goldberg’s approach of encouraging the reader to see the simplicity behind the seemingly complex is welcome.” -- Christopher G. Tully, author of "Elementary Particle Physics in a Nutshell"“This engaging book will shape the education of a generation of physicists and astrophysicists. It defines the conceptual and mathematical stage—spacetime—on which physics is performed. From contemporary notation in the early chapters to sophisticated applications in the late chapters, Goldberg's book will not only propel students to more advanced classes, it will ease their entry into research.” -- Daniel Fabrycky, Department of Astronomy and Astrophysics, University of Chicago“Goldberg slings the reader straight in at the deep end . . . but with enough masterly wit to keep you afloat.” * Nature, on "The Universe in the Rearview Mirror" *“Reading this book is like taking a class with the most awesome science professor ever.” -- Annalee Newitz, founding editor of io9, on "The Universe in the Rearview Mirror"“Most physics books can’t really be described as ‘rollicking,’ but most physics books aren't written by Dave Goldberg.” -- Sean Carroll, theoretical physicist at Caltech, author of "The Particle at the End of the Universe," on "The Universe in the Rearview Mirror"

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Book SynopsisThis substantially illustrated manual describes how to use Maple as an investigative tool to explore calculus concepts numerically, graphically, symbolically and verbally.Table of ContentsGetting Started. Functions and Limits. Differentiation. Applications of the Derivative. Integration. Applications of the Definite Integral. Logarithmic and Exponential Functions. Inverse Trigonometric and Hyperbolic Functions. Integration Exact and Approximate. Improper Integrals: Sequences and Infinite Series. Analytic Geometry. Polar Coordinates and Parametric Equations. Vectors and Three Dimensional Space. Vector Functions. Partial Derivatives. Multiple Integrals. Topics in Vector Calculus. Second--Order Differential Equations. References. Index.

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Book SynopsisDesigned to help motivate the learning of advanced calculus by demonstrating its relevance in the field of statistics, this successful text features detailed coverage of optimization techniques and their applications in statistics while introducing the reader to approximation theory. The Second Edition provides substantial new coverage of the material, including three new chapters and a large appendix that contains solutions to almost all of the exercises in the book. Applications of some of these methods in statistics are discusses.Trade Review"This is an exceptional book, which I would recommend for anyone beginning a career in statistical research." (Journal of the American Statistical Association, September 2004)Table of ContentsPreface xv Preface to the First Edition xvii 1. An Introduction to Set Theory 1 1.1. The Concept of a Set 1 1.2. Set Operations 2 1.3. Relations and Functions 4 1.4. Finite Countable and Uncountable Sets 6 1.5. Bounded Sets 9 1.6. Some Basic Topological Concepts 10 1.7. Examples in Probability and Statistics 13 Further Reading and Annotated Bibliography 15 Exercises 17 2. Basic Concepts in Linear Algebra 21 2.1. Vector Spaces and Subspaces 21 2.2. Linear Transformations 25 2.3. Matrices and Determinants 27 2.3.1. Basic Operations on Matrices 28 2.3.2. The Rank of a Matrix 33 2.3.3. The Inverse of a Matrix 34 2.3.4. Generalized Inverse of a Matrix 36 2.3.5. Eigenvalues and Eigenvectors of a Matrix 36 2.3.6. Some Special Matrices 38 2.3.7. The Diagonalization of a Matrix 38 2.3.8. Quadratic Forms 39 2.3.9. The Simultaneous Diagonalization of Matrices 40 2.3.10. Bounds on Eigenvalues 41 2.4. Applications of Matrices in Statistics 43 2.4.1. The Analysis of the Balanced Mixed Model 43 2.4.2. The Singular-Value Decomposition 45 2.4.3. Extrema of Quadratic Forms 48 2.4.4. The Parameterization of Orthogonal Matrices 49 Further Reading and Annotated Bibliography 50 Exercises 53 3. Limits and Continuity of Functions 57 3.1. Limits of a Function 57 3.2. Some Properties Associated with Limits of Functions 63 3.3. The o O Notation 65 3.4. Continuous Functions 66 3.4.1. Some Properties of Continuous Functions 71 3.4.2. Lipschitz Continuous Functions 75 3.5. Inverse Functions 76 3.6. Convex Functions 79 3.7. Continuous and Convex Functions in Statistics 82 Further Reading and Annotated Bibliography 87 Exercises 88 4. Differentiation 93 4.1. The Derivative of a Function 93 4.2. The Mean Value Theorem 99 4.3. Taylor’s Theorem 108 4.4. Maxima and Minima of a Function 112 4.4.1. A Sufficient Condition for a Local Optimum 114 4.5. Applications in Statistics 115 4.5.1. Functions of Random Variables 116 4.5.2. Approximating Response Functions 121 4.5.3. The Poisson Process 122 4.5.4. Minimizing the Sum of Absolute Deviations 124 Further Reading and Annotated Bibliography 125 Exercises 127 5. Infinite Sequences and Series 132 5.1. Infinite Sequences 132 5.1.1. The Cauchy Criterion 137 5.2. Infinite Series 140 5.2.1. Tests of Convergence for Series of Positive Terms 144 5.2.2. Series of Positive and Negative Terms 158 5.2.3. Rearrangement of Series 159 5.2.4. Multiplication of Series 162 5.3. Sequences and Series of Functions 165 5.3.1. Properties of Uniformly Convergent Sequences and Series 169 5.4. Power Series 174 5.5. Sequences and Series of Matrices 178 5.6. Applications in Statistics 182 5.6.1. Moments of a Discrete Distribution 182 5.6.2. Moment and Probability Generating Functions 186 5.6.3. Some Limit Theorems 191 5.6.3.1. The Weak Law of Large Numbers ŽKhinchine’s Theorem. 192 5.6.3.2. The Strong Law of Large Numbers ŽKolmogorov’s Theorem. 192 5.6.3.3. The Continuity Theorem for Probability Generating Functions 192 5.6.4. Power Series and Logarithmic Series Distributions 193 5.6.5. Poisson Approximation to Power Series Distributions 194 5.6.6. A Ridge Regression Application 195 Further Reading and Annotated Bibliography 197 Exercises 199 6. Integration 205 6.1. Some Basic Definitions 205 6.2. The Existence of the Riemann Integral 206 6.3. Some Classes of Functions That Are Riemann Integrable 210 6.3.1. Functions of Bounded Variation 212 6.4. Properties of the Riemann Integral 215 6.4.1. Change of Variables in Riemann Integration 219 6.5. Improper Riemann Integrals 220 6.5.1. Improper Riemann Integrals of the Second Kind 225 6.6. Convergence of a Sequence of Riemann Integrals 227 6.7. Some Fundamental Inequalities 229 6.7.1. The Cauchy_Schwarz Inequality 229 6.7.2. H¨older’s Inequality 230 6.7.3. Minkowski’s Inequality 232 6.7.4. Jensen’s Inequality 233 6.8. Riemann_Stieltjes Integral 234 6.9. Applications in Statistics 239 6.9.1. The Existence of the First Negative Moment of a Continuous Distribution 242 6.9.2. Transformation of Continuous Random Variables 246 6.9.3. The Riemann_Stieltjes Representation of the Expected Value 249 6.9.4. Chebyshev’s Inequality 251 Further Reading and Annotated Bibliography 252 Exercises 253 7. Multidimensional Calculus 261 7.1. Some Basic Definitions 261 7.2. Limits of a Multivariable Function 262 7.3. Continuity of a Multivariable Function 264 7.4. Derivatives of a Multivariable Function 267 7.4.1. The Total Derivative 270 7.4.2. Directional Derivatives 273 7.4.3. Differentiation of Composite Functions 276 7.5. Taylor’s Theorem for a Multivariable Function 277 7.6. Inverse and Implicit Function Theorems 280 7.7. Optima of a Multivariable Function 283 7.8. The Method of Lagrange Multipliers 288 7.9. The Riemann Integral of a Multivariable Function 293 7.9.1. The Riemann Integral on Cells 294 7.9.2. Iterated Riemann Integrals on Cells 295 7.9.3. Integration over General Sets 297 7.9.4. Change of Variables in n-Tuple Riemann Integrals 299 7.10. Differentiation under the Integral Sign 301 7.11. Applications in Statistics 304 7.11.1. Transformations of Random Vectors 305 7.11.2. Maximum Likelihood Estimation 308 7.11.3. Comparison of Two Unbiased Estimators 310 7.11.4. Best Linear Unbiased Estimation 311 7.11.5. Optimal Choice of Sample Sizes in Stratified Sampling 313 Further Reading and Annotated Bibliography 315 Exercises 316 8. Optimization in Statistics 327 8.1. The Gradient Methods 329 8.1.1. The Method of Steepest Descent 329 8.1.2. The Newton_Raphson Method 331 8.1.3. The Davidon_Fletcher_Powell Method 331 8.2. The Direct Search Methods 332 8.2.1. The Nelder_Mead Simplex Method 332 8.2.2. Price’s Controlled Random Search Procedure 336 8.2.3. The Generalized Simulated Annealing Method 338 8.3. Optimization Techniques in Response Surface Methodology 339 8.3.1. The Method of Steepest Ascent 340 8.3.2. The Method of Ridge Analysis 343 8.3.3. Modified Ridge Analysis 350 8.4. Response Surface Designs 355 8.4.1. First-Order Designs 356 8.4.2. Second-Order Designs 358 8.4.3. Variance and Bias Design Criteria 359 8.5. Alphabetic Optimality of Designs 362 8.6. Designs for Nonlinear Models 367 8.7. Multiresponse Optimization 370 8.8. Maximum Likelihood Estimation and the EM Algorithm 372 8.8.1. The EM Algorithm 375 8.9. Minimum Norm Quadratic Unbiased Estimation of Variance Components 378 8.10. Scheff´e’s Confidence Intervals 382 8.10.1. The Relation of Scheff´e’s Confidence Intervals to the F-Test 385 Further Reading and Annotated Bibliography 391 Exercises 395 9. Approximation of Functions 403 9.1. Weierstrass Approximation 403 9.2. Approximation by Polynomial Interpolation 410 9.2.1. The Accuracy of Lagrange Interpolation 413 9.2.2. A Combination of Interpolation and Approximation 417 9.3. Approximation by Spline Functions 418 9.3.1. Properties of Spline Functions 418 9.3.2. Error Bounds for Spline Approximation 421 9.4. Applications in Statistics 422 9.4.1. Approximate Linearization of Nonlinear Models by Lagrange Interpolation 422 9.4.2. Splines in Statistics 428 9.4.2.1. The Use of Cubic Splines in Regression 428 9.4.2.2. Designs for Fitting Spline Models 430 9.4.2.3. Other Applications of Splines in Statistics 431 Further Reading and Annotated Bibliography 432 Exercises 434 10. Orthogonal Polynomials 437 10.1. Introduction 437 10.2. Legendre Polynomials 440 10.2.1. Expansion of a Function Using Legendre Polynomials 442 10.3. Jacobi Polynomials 443 10.4. Chebyshev Polynomials 444 10.4.1. Chebyshev Polynomials of the First Kind 444 10.4.2. Chebyshev Polynomials of the Second Kind 445 10.5. Hermite Polynomials 447 10.6. Laguerre Polynomials 451 10.7. Least-Squares Approximation with Orthogonal Polynomials 453 10.8. Orthogonal Polynomials Defined on a Finite Set 455 10.9. Applications in Statistics 456 10.9.1. Applications of Hermite Polynomials 456 10.9.1.1. Approximation of Density Functions and Quantiles of Distributions 456 10.9.1.2. Approximation of a Normal Integral 460 10.9.1.3. Estimation of Unknown Densities 461 10.9.2. Applications of Jacobi and Laguerre Polynomials 462 10.9.3. Calculation of Hypergeometric Probabilities Using Discrete Chebyshev Polynomials 462 Further Reading and Annotated Bibliography 464 Exercises 466 11. Fourier Series 471 11.1. Introduction 471 11.2. Convergence of Fourier Series 475 11.3. Differentiation and Integration of Fourier Series 483 11.4. The Fourier Integral 488 11.5. Approximation of Functions by Trigonometric Polynomials 495 11.5.1. Parseval’s Theorem 496 11.6. The Fourier Transform 497 11.6.1. Fourier Transform of a Convolution 499 11.7. Applications in Statistics 500 11.7.1. Applications in Time Series 500 11.7.2. Representation of Probability Distributions 501 11.7.3. Regression Modeling 504 11.7.4. The Characteristic Function 505 11.7.4.1. Some Properties of Characteristic Functions 510 Further Reading and Annotated Bibliography 510 Exercises 512 12. Approximation of Integrals 517 12.1. The Trapezoidal Method 517 12.1.1. Accuracy of the Approximation 518 12.2. Simpson’s Method 521 12.3. Newton_Cotes Methods 523 12.4. Gaussian Quadrature 524 12.5. Approximation over an Infinite Interval 528 12.6. The Method of Laplace 531 12.7. Multiple Integrals 533 12.8. The Monte Carlo Method 535 12.8.1. Variation Reduction 537 12.8.2. Integrals in Higher Dimensions 540 12.9. Applications in Statistics 541 12.9.1. The Gauss_Hermite Quadrature 542 12.9.2. Minimum Mean Squared Error Quadrature 543 12.9.3. Moments of a Ratio of Quadratic Forms 546 12.9.4. Laplace’s Approximation in Bayesian Statistics 548 12.9.5. Other Methods of Approximating Integrals in Statistics 549 Further Reading and Annotated Bibliography 550 Exercises 552 Appendix. Solutions to Selected Exercises 557 General Bibliography 652 Index 665

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Book SynopsisTrade Review"This is a book that should be part of the personal library of any mathematician."---Mark Hunacek, Mathematical Gazette

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Book SynopsisCalculus Essentials For Dummies(9781119591207) was previously published asCalculus Essentials For Dummies (9780470618356). While this version features a newDummiescover and design, the content is the same as the prior release and should not be considered a new or updated product. Many colleges and universities require students to take at least one math course, and Calculus I is often the chosen option.Calculus Essentials For Dummiesprovides explanations of key concepts for students who may have taken calculus in high school and want to review the most important concepts as they gear up for a faster-paced college course. Free of review and ramp-up material,Calculus Essentials For Dummiessticks to the point with content focused on key topics only. It provides discrete explanations of critical concepts taught in a typical two-semester high school calculus class or a college level Calculus I course, from limits and differentiation to integration and infinite series. This guide is also Table of ContentsIntroduction 1 About This Book 1 Conventions Used in This Book 2 Foolish Assumptions 2 Icons Used in This Book 3 Where to Go from Here 3 Chapter 1: Calculus: No Big Deal 5 So What is Calculus Already? 5 Real-World Examples of Calculus 7 Differentiation 8 Integration 9 Why Calculus Works 11 Limits: Math microscopes 11 What happens when you zoom in 12 Chapter 2: Limits and Continuity 15 Taking it to the Limit 15 Three functions with one limit 15 One-sided limits 17 Limits and vertical asymptotes 18 Limits and horizontal asymptotes 18 Instantaneous speed 19 Limits and Continuity 21 The hole exception 22 Chapter 3: Evaluating Limits 25 Easy Limits 25 Limits to memorize 25 Plug-and-chug limits 26 “Real” Limit Problems 26 Factoring 27 Conjugate multiplication 27 Miscellaneous algebra 28 Limits at Infinity 29 Horizontal asymptotes 30 Solving limits at infinity 31 Chapter 4: Differentiation Orientation 33 The Derivative: It’s Just Slope 34 The slope of a line 35 The derivative of a line 36 The Derivative: It’s Just a Rate 36 Calculus on the playground 36 The rate-slope connection 38 The Derivative of a Curve 39 The Difference Quotient 40 Average and Instantaneous Rate 46 Three Cases Where the Derivative Does Not Exist 47 Chapter 5: Differentiation Rules 49 Basic Differentiation Rules 49 The constant rule 49 The power rule 49 The constant multiple rule 50 The sum and difference rules 51 Differentiating trig functions 52 Exponential and logarithmic functions 52 Derivative Rules for Experts 53 The product and quotient rules 53 The chain rule 54 Differentiating Implicitly 59 Chapter 6: Differentiation and the Shape of Curves 61 A Calculus Road Trip 61 Local Extrema 63 Finding the critical numbers 63 The First Derivative Test 65 The Second Derivative Test 66 Finding Absolute Extrema on a Closed Interval 69 Finding Absolute Extrema over a Function’s Entire Domain 71 Concavity and Inflection Points 73 Graphs of Derivatives 75 The Mean Value Theorem 78 Chapter 7: Differentiation Problems 81 Optimization Problems 81 The maximum area of a corral 81 Position, Velocity, and Acceleration 83 Velocity versus speed 84 Maximum and minimum height 86 Velocity and displacement 87 Speed and distance travelled 88 Acceleration 89 Tying it all together 90 Related Rates 91 A calculus crossroads 91 Filling up a trough 94 Linear Approximation 97 Chapter 8: Introduction to Integration 101 Integration: Just Fancy Addition 101 Finding the Area under a Curve 103 Dealing with negative area 105 Approximating Area 105 Approximating area with left sums 105 Approximating area with right sums 108 Approximating area with midpoint sums 110 Summation Notation 112 Summing up the basics 112 Writing Riemann sums with sigma notation 113 Finding Exact Area with the Definite Integral 116 Chapter 9: Integration: Backwards Differentiation 119 Antidifferentiation: Reverse Differentiation 119 The Annoying Area Function 121 The Fundamental Theorem 124 Fundamental Theorem: Take Two 126 Antiderivatives: Basic Techniques 128 Reverse rules 128 Guess and check 130 Substitution 132 Chapter 10: Integration for Experts 137 Integration by Parts 137 Picking your u 139 Tricky Trig Integrals 141 Sines and cosines 141 Secants and tangents 144 Cosecants and cotangents 147 Trigonometric Substitution 147 Case 1: Tangents 148 Case 2: Sines 150 Case 3: Secants 151 Partial Fractions 152 Case 1: The denominator contains only linear factors 152 Case 2: The denominator contains unfactorable quadratic factors 153 Case 3: The denominator contains repeated factors 155 Equating coefficients 155 Chapter 11: Using the Integral to Solve Problems 157 The Mean Value Theorem for Integrals and Average Value 158 The Area between Two Curves 160 Volumes of Weird Solids 162 The meat-slicer method 162 The disk method 163 The washer method 165 The matryoshka doll method 166 Arc Length 168 Improper Integrals 171 Improper integrals with vertical asymptotes 171 Improper integrals with infinite limits of integration 173 Chapter 12: Eight Things to Remember 175 a2- b2 = (a - b)(a + b) 175 0/5 = 0 But 5/0 is Undefined 175 SohCahToa 175 Trig Values to Know 176 sin2ϴ + cos2ϴ = 1 176 The Product Rule 176 The Quotient Rule 176 Your Sunglasses 176 Index 177

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Book Synopsis

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Book Synopsis

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Book Synopsis

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Book Synopsis

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Book SynopsisThis monograph provides a thorough treatment of parameter-dependent extremal problems with local minimum values that remain unchanged under changes of the parameter. The authors consider the theory as well the practical treatment of those problems, both in finite-dimensional as well as in infinite-dimensional spaces. Various applications are considered, e.g., variational calculus, control theory and bifurcations theory. Thorough treatment of parameter-dependent extremal problems with local minimum values. Includes many applications, e.g., variational calculus, control theory and bifurcations theory. Intended for specialists in the field of nonlinear analysis and its applications as well as for students specializing in these subjects.Trade Review"Overall this is a book which should interest many researchers and students working on either variational problems or topological methods for nonlinear boundary value problems."James R. Ward in: Mathematical Reviews 2009b "The book is carefully written an it can be read by graduate students. Physicists and engineers who use variational methods will also find here a good source of information."In: EMS Newsletter 9/2008

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Book SynopsisThis volume contains survey articles and original research papers, presenting the state of the art on applying the symbolic approach of Gröbner bases and related methods to differential and difference equations. The contributions are based on talks delivered at the Special Semester on Gröbner Bases and Related Methods hosted by the Johann Radon Institute of Computational and Applied Mathematics, Linz, Austria, in May 2006.Trade Review"Many a researchers and many a mathematics graduate will find its pages filled with a contagious enthusiasm packing a treasure of wisdom and delight into eleven chapters."In: Current Engeneering Pratice 2009

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Book SynopsisThis text is a self-contained and unified approach to Bernstein functions and their subclasses, bringing together old and establishing new connections. Applications of Bernstein functions in different fields of mathematics are given, with special attention to interpretations in probability theory. An extensive list of complete Bernstein functions with their representations is provided. A self-contained and unified approach to the topic With applications to various fields of mathematics, such as probability theory, potential theory, operator theory, integral equations, functional calculi and complex analysis With an extensive list of complete Bernstein functions. Additional material and corrections can be found on the authors' website.

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Book SynopsisIn recent years, technological progress created a great need for complex mathematical models. Many practical problems can be formulated using optimization theory and they hope to obtain an optimal solution. In most cases, such optimal solution can not be found. So, non-convex optimization problems (arising, e.g., in variational calculus, optimal control, nonlinear evolutions equations) may not possess a classical minimizer because the minimizing sequences have typically rapid oscillations. This behavior requires a relaxation of notion of solution for such problems; often we can obtain such a relaxation by means of Young measures. This monograph is a self-contained book which gathers all theoretical aspects related to the defining of Young measures (measurability, disintegration, stable convergence, compactness), a book which is also a useful tool for those interested in theoretical foundations of the measure theory. It provides a complete set of classical and recent compactness results in measure and function spaces. The book is organized in three chapters: The first chapter covers background material on measure theory in abstract frame. In the second chapter the measure theory on topological spaces is presented. Compactness results from the first two chapters are used to study Young measures in the third chapter. All results are accompanied by full demonstrations and for many of these results different proofs are given. All statements are fully justified and proved.Table of ContentsPreface 1 Weak Compactness in Measure Spaces 1.1 Measure spaces 1.2 Radon-Nikodym theorem. The dual of L1 1.3 Convergences in L1(l) and ca(A) 1.4 Weak compactness in ca(A) and L1(l) 1.5 The bidual of L1(l) 1.6 Extensions of Dunford-Pettis' theorem 2 Bounded Measures on Topological Spaces 2.1 Regular measures 2.2 Polish spaces. Suslin spaces 2.3 Narrow topology 2.4 Compactness results 2.5 Metrics on the space (Rca+(BT ), T) 2.6 Wiener measure 3 Young Measures 3.1 Preliminaries 3.2 Definitions. Examples 3.3 The stable topology 3.4 The subspace M(S) Y(S) 3.5 Compactness 3.6 Biting lemma 3.7 Product of Young measures 3.8 Jordan finite tight sets 3.9 Strong compactness in Lp(μ,E) References Index

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Book Synopsisthis monographis based on two courses in computational mathematics and operative research, which were given by the author in recent years to doctorate and PhD students. The text focuses on an aspect of the theory of inverse problems, which is usually referred to as identification of parameters (numbers, vectors, matrices, functions) appearing in differential– or integrodifferential– equations. The parameters of such equations are either quite unknown or partially unknown, however knowledge about these is usually essential as they describe the intrinsic properties of the material or substance under consideration.

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Book SynopsisThe book constructs explicitly the fundamental solution of the sub-Laplacian operator for a family of model domains in Cn+1. This type of domain is a good point-wise model for a Cauchy-Rieman (CR) manifold with diagonalizable Levi form. Qualitative results for such operators have been studied extensively, but exact formulas are difficult to derive. Exact formulas are closely related to the underlying geometry and lead to equations of classical types such as hypergeometric equations and Whittaker’s equations.

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Book SynopsisDieses Buch konzentriert das aktuelle Gesamtwissen zum Proximity-Konzept und stellt es dem Leser in gut strukturierter Form dar. Hauptaugenmerk liegt auf den vielfältigen Möglichkeiten, die sich aus dem Proximity-Konzept der räumlichen Nähe und seiner Verallgemeinerung im Nearness-Konzept ergeben.

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