In addition to expanded explanations, the 11th edition includes new problems, updated figures and examples to help motivate students. The program is primarily intended for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for engaging with the program is a working knowledge of calculus, gained from a normal two?] or three?] semester course sequence or its equivalent. Some familiarity with matrices will also be helpful in the chapters on systems of differential equations.
The book begins with a short review of calculus and ordinary differential equations, then moves on to explore integral curves and surfaces of vector fields, quasi-linear and linear equations of first order, series solutions and the Cauchy Kovalevsky theorem. It then delves into linear partial differential equations, examines the Laplace, wave and heat equations, and concludes with a brief treatment of hyperbolic systems of equations.
Among the most important features of the text are the challenging problems at the end of each section which require a wide variety of responses from students, from providing details of the derivation of an item presented to solving specific problems associated with partial differential equations. Requiring only a modest mathematical background, the text will be indispensable to those who need to use partial differential equations in solving physical problems. It will provide as well the mathematical fundamentals for those who intend to pursue the study of more advanced topics, including modern theory.
This highly useful text shows the reader how to formulate a partial differential equation from the physical problem (constructing the mathematical model) and how to solve the equation (along with initial and boundary conditions). Written for advanced undergraduate and graduate students, as well as professionals working in the applied sciences, this clearly written book offers realistic, practical coverage of diffusion-type problems, hyperbolic-type problems, elliptic-type problems, and numerical and approximate methods. Each chapter contains a selection of relevant problems (answers are provided) and suggestions for further reading.
Nigel Kalton (1946-2010) was an extraordinary mathematician who made major contributions to an amazingly diverse range of fields over the course of his career.
Those familiar with mathematics texts will note the fine illustrations throughout and large number of problems offered at the chapter ends. An answer section is provided. Students weary of plodding mathematical prose will find Professor Flanigan's style as refreshing and stimulating as his approach.
Nigel Kalton (1946-2010) was an extraordinary mathematician who made major contributions to an amazingly diverse range of fields over the course of his career.
Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.
Table of Contents:
Chapter 1. Fourier series
1.1 Basic concepts
1.2 Fourier series and Fourier coefficients
1.3 A minimizing property of the Fourier coefficients. The Riemann-Lebesgue theorem
1.4 Convergence of Fourier series
1.5 The Parseval formula
1.6 Determination of the sum of certain trigonometric series
Chapter 2. Orthogonal systems
2.1 Integration of complex-valued functions of a real variable
2.2 Orthogonal systems
2.3 Complete orthogonal systems
2.4 Integration of Fourier series
2.5 The Gram-Schmidt orthogonalization process
2.6 Sturm-Liouville problems
Chapter 3. Orthogonal polynomials
3.1 The Legendre polynomials
3.2 Legendre series
3.3 The Legendre differential equation. The generating function of the Legendre polynomials
3.4 The Tchebycheff polynomials
3.5 Tchebycheff series
3.6 The Hermite polynomials. The Laguerre polynomials
Chapter 4. Fourier transforms
4.1 Infinite interval of integration
4.2 The Fourier integral formula: a heuristic introduction
4.3 Auxiliary theorems
4.4 Proof of the Fourier integral formula. Fourier transforms
4.5 The convention theorem. The Parseval formula
Chapter 5. Laplace transforms
5.1 Definition of the Laplace transform. Domain. Analyticity
5.2 Inversion formula
5.3 Further properties of Laplace transforms. The convolution theorem
5.4 Applications to ordinary differential equations
Chapter 6. Bessel functions
6.1 The gamma function
6.2 The Bessel differential equation. Bessel functions
6.3 Some particular Bessel functions
6.4 Recursion formulas for the Bessel functions
6.5 Estimation of Bessel functions for large values of x. The zeros of the Bessel functions
6.6 Bessel series
6.7 The generating function of the Bessel functions of integral order
6.8 Neumann functions
Chapter 7. Partial differential equations of first order
7.2 The differential equation of a family of surfaces
7.3 Homogeneous differential equations
7.4 Linear and quasilinear differential equations
Chapter 8. Partial differential equations of second order
8.1 Problems in physics leading to partial differential equations
8.3 The wave equation
8.4 The heat equation
8.5 The Laplace equation
Answers to exercises; Bibliography; Conventions; Symbols; Index
Written on an advanced level, the book is aimed at advanced undergraduates and graduate students with a background in calculus, linear algebra, ordinary differential equations, and complex analysis. Over 260 carefully chosen exercises, with answers, encompass both routing and more challenging problems to help students test their grasp of the material.
Though the Japanese abacus may appear mysterious or even primitive, this intriguing tool is capable of amazing speed and accuracy. it is still widely used throughout the shop and markets of Asia and its popularity shows no sign of decline.
This volume is designed for the student desiring a greater understanding of the abacus and its calculative functions. The text provides thorough explanations of the advanced operations involving negative numbers, decimals, different units of measurement, and square roots. Diagrams illustrate bead manipulation, and numerous exercises provide ample practice.
Concise and easy-to-follow, this book will improve your abacus skills and help you perform calculations with greater efficiency and precision.
About the First Edition: I have used this book for both regular PDE and topics courses. It has a wonderful combination of insight and technical detail...Evans' book is evidence of his mastering of the field and the clarity of presentation (Luis Caffarelli, University of Texas)
It is fun to teach from Evans' book. It explains many of the essential ideas and techniques of partial differential equations ...Every graduate student in analysis should read it. (David Jerison, MIT)
I use Partial Differential Equations to prepare my students for their Topic exam, which is a requirement before starting working on their dissertation. The book provides an excellent account of PDE's ...I am very happy with the preparation it provides my students. (Carlos Kenig, University of Chicago)
Evans' book has already attained the status of a classic. It is a clear choice for students just learning the subject, as well as for experts who wish to broaden their knowledge ...An outstanding reference for many aspects of the field. (Rafe Mazzeo, Stanford University.
The "lost notebook" contains considerable material on mock theta functions and so undoubtedly emanates from the last year of Ramanujan's life. It should be emphasized that the material on mock theta functions is perhaps Ramanujan's deepest work. Mathematicians are probably several decades away from a complete understanding of those functions. More than half of the material in the book is on q-series, including mock theta functions; the remaining part deals with theta function identities, modular equations, incomplete elliptic integrals of the first kind and other integrals of theta functions, Eisenstein series, particular values of theta functions, the Rogers-Ramanujan continued fraction, other q-continued fractions, other integrals, and parts of Hecke's theory of modular forms.
Topics include ordinary differential equations in more than two variables, partial differential equations of the first and second orders, Laplace's equation, the wave equation, and the diffusion equation. A helpful Appendix offers information on systems of surfaces, and solutions to the odd-numbered problems appear at the end of the book. Readers pursuing independent study will particularly appreciate the worked examples that appear throughout the text.
Topics include Bessel functions of zero order, modified Bessel functions, definite integrals, asymptotic expansions, and Bessel functions of any real order. More than 200 problems throughout the book enable students to test and extend their understanding of the theory and applications of Bessel functions.
Chapter 1 begins with a discussion of various physical problems and equations that play a central role in applications. The following chapters take up the theory of partial differential equations, including detailed discussions of uniqueness, existence, and continuous dependence questions, as well as techniques for constructing conclusions. Specifically, Chapters 2 through 6 deal with problems in one spatial dimension. Chapter 7 is a detailed introduction to the theory of integral equations; then Chapters 8 through 12 treat problems in more spatial variables. Each chapter begins with a discussion of problems that can be treated by elementary means, such as separation of variables or integral transforms, and which lead to explicit, analytical representations of solutions. The minimal mathematical prerequisites for a good grasp of the material in this book are a course in advanced calculus, or an advanced course in science or engineering, and a basic exposure to matrix methods. Students of mathematics, physics, engineering, and other disciplines will find here an excellent guide to mathematical problem-solving techniques with a broad range of applications. For this edition the authors have provided a new section of Solutions and Hints to selected Problems. Suggestions for further reading complete the text.
Since many abstractions and generalizations originate with the real line, the author has made it the unifying theme of the text, constructing the real number system from the point of view of a Cauchy sequence (a step which Dr. Sprecher feels is essential to learn what the real number system is).
The material covered in Elements of Real Analysis should be accessible to those who have completed a course in calculus. To help give students a sound footing, Part One of the text reviews the fundamental concepts of sets and functions and the rational numbers. Part Two explores the real line in terms of the real number system, sequences and series of number and the structure of point sets. Part Three examines the functions of a real variable in terms of continuity, differentiability, spaces of continuous functions, measure and integration, and the Fourier series.
An especially valuable feature of the book is the exercises which follow each section. There are over five hundred, ranging from the simple to the highly difficult, each focusing on a concept previously introduced.
Starting with the theory of elliptic equations and the solution of the Dirichlet problem, the text develops the theory of weak derivatives, proves various inequalities and imbedding problems, and derives smoothness theorems. Part Two concerns evolution equations in Banach space and develops the theory of semigroups. It solves the initial-boundary value problem for parabolic equations and covers backward uniqueness, asymptotic behavior, and lower bounds at infinity. The final section includes independent topics directly related to the methods and results of the previous material, including the analyticity of solutions of elliptic and parabolic equations, asymptotic behavior of solutions of elliptic equations near infinity, and problems in the theory of control in Banach space.
Starting with an introduction to differential equations, the text proceeds to examinations of first- and second-order differential equations, series solutions, the Laplace transform, systems of differential equations, difference equations, nonlinear differential equations and chaos, and partial differential equations. Numerous figures, problems with solutions, and historical notes clarify the text.
Get the confidence and the skills you need to master differential equations!
Need to know how to solve differential equations? This easy-to-follow, hands-on workbook helps you master the basic concepts and work through the types of problems you'll encounter in your coursework. You get valuable exercises, problem-solving shortcuts, plenty of workspace, and step-by-step solutions to every equation. You'll also memorize the most-common types of differential equations, see how to avoid common mistakes, get tips and tricks for advanced problems, improve your exam scores, and much more!
More than 100 Problems!
Detailed, fully worked-out solutions to problems
The inside scoop on first, second, and higher order differential equations
A wealth of advanced techniques, including power series
THE DUMMIES WORKBOOK WAY
Quick, refresher explanations
Hands-on practice exercises
Ample workspace to work out problems
Online Cheat Sheet
A dash of humor and fun
Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
The book is divided into four chapters, with two useful appendices, an excellent bibliography, and an index. A section of exercises enables the student to check his progress. Contents include Volterra Equations, Fredholm Equations, Symmetric Kernels and Orthogonal Systems of Functions, Types of Singular or Nonlinear Integral Equations, and more.
Professor Tricomi has presented the principal results of the theory with sufficient generality and mathematical rigor to facilitate theoretical applications. On the other hand, the treatment is not so abstract as to be inaccessible to physicists and engineers who need integral equations as a basic mathematical tool. In fact, most of the material in this book falls into an analytical framework whose content and methods are already traditional.
Starting with a consideration of the existence and uniqueness theorem, the text advances to the behavior of the characteristics of a first-order equation, boundary problems for second-order linear equations, asymptotic methods, and differential equations in the complex field. The author discusses only ordinary differential equations, excluding coverage of the methods of integration and stressing the importance of reading the properties of the integrals directly from the equations. An extensive bibliography and helpful indexes conclude the text.
The University of Toronto Undergraduate Competition was founded to provide additional competition experience for undergraduates preparing for the Putnam competition, and is particularly useful for the freshman or sophomore undergraduate. Lecturers, instructors, and coaches for mathematics competitions will find this presentation useful. Many of the problems are of intermediate difficulty and relate to the first two years of the undergraduate curriculum. The problems presented may be particularly useful for regular class assignments. Moreover, this text contains problems that lie outside the regular syllabus and may interest students who are eager to learn beyond the classroom.
Ingeniously relying on elementary algebra and just a smidgen of calculus, Professor Walker demonstrates how the underlying ideas behind wavelet analysis can be applied to solve significant problems in audio and image processing, as well in biology and medicine.
Nearly twice as long as the original, this new edition provides
· 104 worked examples and 222 exercises, constituting a veritable book of review material
· Two sections on biorthogonal wavelets
· A mini-course on image compression, including a tutorial on arithmetic compression
· Extensive material on image denoising, featuring a rarely covered technique for removing isolated, randomly positioned clutter
· Concise yet complete coverage of the fundamentals of time-frequency analysis, showcasing its application to audio denoising, and musical theory and synthesis
· An introduction to the multiresolution principle, a new mathematical concept in musical theory
· Expanded suggestions for research projects
· An enhanced list of references
· FAWAV: software designed by the author, which allows readers to duplicate described applications and experiment with other ideas.
To keep the book current, Professor Walker has created a supplementary website. This online repository includes ready-to-download software, and sound and image files, as well as access to many of the most important papers in the field.
Volume III concentrates on the classical aspects of gauge theory, describing the four fundamental forces by the curvature of appropriate fiber bundles. This must be supplemented by the crucial, but elusive quantization procedure.
The book is arranged in four sections, devoted to realizing the universal principle force equals curvature:
Part I: The Euclidean Manifold as a Paradigm
Part II: Ariadne's Thread in Gauge Theory
Part III: Einstein's Theory of Special Relativity
Part IV: Ariadne's Thread in Cohomology
For students of mathematics the book is designed to demonstrate that detailed knowledge of the physical background helps to reveal interesting interrelationships among diverse mathematical topics. Physics students will be exposed to a fairly advanced mathematics, beyond the level covered in the typical physics curriculum.
Quantum Field Theory builds a bridge between mathematicians and physicists, based on challenging questions about the fundamental forces in the universe (macrocosmos), and in the world of elementary particles (microcosmos).
The book is divided into three parts. The first introduces the framework for the above-mentioned classification, methodically developing a geometric and topological formulation applicable to all physical laws and properties; the second applies this formulation to a detailed study of particle dynamics, electromagnetism, deformable solids, fluid dynamics, heat conduction, and gravitation. The third part further analyses the general structure of the classification diagram for variables and equations of physical theories.
Suitable for a diverse audience of physicists, engineers, and mathematicians, The Mathematical Structure of Classical and Relativistic Physics offers a valuable resource for studying the physical world. Written at a level accessible to graduate and advanced undergraduate students in mathematical physics, the book can be used as a research monograph across various areas of physics, engineering and mathematics, and as a supplemental text for a broad range of upper-level scientific coursework.
The author devotes the first four chapters to proofs of classical theorems on boundary values and boundary integral representations of analytic functions in the unit disc, including generalizations to Dirichlet algebras. The fifth chapter contains the factorization theory of Hp functions, a discussion of some partial extensions of the factorization, and a brief description of the classical approach to the theorems of the first five chapters. The remainder of the book addresses the structure of various Banach spaces and Banach algebras of analytic functions in the unit disc.
Enhanced with 100 challenging exercises, a bibliography, and an index, this text belongs in the libraries of students, professional mathematicians, as well as anyone interested in a rigorous, high-level treatment of this topic.
Incorporating many of the techniques of applied mathematics, the book also contains most of the concepts of rigorous analysis usually found in a course in advanced calculus. These techniques and concepts are presented in a setting where their need is clear and their application immediate. Chapters I through IV cover the one-dimensional wave equation, linear second-order partial differential equations in two variables, some properties of elliptic and parabolic equations and separation of variables, and Fourier series. Chapters V through VIII address nonhomogeneous problems, problems in higher dimensions and multiple Fourier series, Sturm-Liouville theory, and general Fourier expansions and analytic functions of a complex variable.
The last four chapters are devoted to the evaluation of integrals by complex variable methods, solutions based on the Fourier and Laplace transforms, and numerical approximation methods. Numerous exercises are included throughout the text, with solutions at the back.
New to the Third Edition
New chapter on special topics, including discrete Cauchy–Euler equations; gamma, beta, and digamma functions; Lambert W-function; Euler polynomials; functional equations; and exact discretizations of differential equations New chapter on the application of difference equations to complex problems arising in the mathematical modeling of phenomena in engineering and the natural and social sciences Additional problems in all chapters Expanded bibliography to include recently published texts related to the subject of difference equations
Suitable for self-study or as the main text for courses on difference equations, this book helps readers understand the fundamental concepts and procedures of difference equations. It uses an informal presentation style, avoiding the minutia of detailed proofs and formal explanations.
Foundations of Applied Mathematics, Volume 1: Mathematical Analysis includes several key topics not usually treated in courses at this level, such as uniform contraction mappings, the continuous linear extension theorem, Daniell–Lebesgue integration, resolvents, spectral resolution theory, and pseudospectra. Ideas are developed in a mathematically rigorous way and students are provided with powerful tools and beautiful ideas that yield a number of nice proofs, all of which contribute to a deep understanding of advanced analysis and linear algebra. Carefully thought out exercises and examples are built on each other to reinforce and retain concepts and ideas and to achieve greater depth. Associated lab materials are available that expose students to applications and numerical computation and reinforce the theoretical ideas taught in the text. The text and labs combine to make students technically proficient and to answer the age-old question, "When am I going to use this?
"...complete, up-to-date coverage of computational complexity theory...the book promises to become the standard reference on computational complexity." -Zentralblatt MATH
A thorough revision based on advances in the field of computational complexity and readers’ feedback, the Second Edition of Theory of Computational Complexity presents updates to the principles and applications essential to understanding modern computational complexity theory. The new edition continues to serve as a comprehensive resource on the use of software and computational approaches for solving algorithmic problems and the related difficulties that can be encountered.
Maintaining extensive and detailed coverage, Theory of Computational Complexity, Second Edition, examines the theory and methods behind complexity theory, such as computational models, decision tree complexity, circuit complexity, and probabilistic complexity. The Second Edition also features recent developments on areas such as NP-completeness theory, as well as:A new combinatorial proof of the PCP theorem based on the notion of expander graphs, a research area in the field of computer science Additional exercises at varying levels of difficulty to further test comprehension of the presented material End-of-chapter literature reviews that summarize each topic and offer additional sources for further study
Theory of Computational Complexity, Second Edition, is an excellent textbook for courses on computational theory and complexity at the graduate level. The book is also a useful reference for practitioners in the fields of computer science, engineering, and mathematics who utilize state-of-the-art software and computational methods to conduct research.A thorough revision based on advances in the field of computational complexity and readers’ feedback, the Second Edition of Theory of Computational Complexity presents updates to the principles and applications essential to understanding modern computational complexity theory. The new edition continues to serve as a comprehensive resource on the use of software and computational approaches for solving algorithmic problems and the related difficulties that can be encountered. Maintaining extensive and detailed coverage, Theory of Computational Complexity, Second Edition, examines the theory and methods behind complexity theory, such as computational models, decision tree complexity, circuit complexity, and probabilistic complexity. The Second Edition also features recent developments on areas such as NP-completeness theory, as well as: • A new combinatorial proof of the PCP theorem based on the notion of expander graphs, a research area in the field of computer science • Additional exercises at varying levels of difficulty to further test comprehension of the presented material • End-of-chapter literature reviews that summarize each topic and offer additional sources for further study Theory of Computational Complexity, Second Edition, is an excellent textbook for courses on computational theory and complexity at the graduate level. The book is also a useful reference for practitioners in the fields of computer science, engineering, and mathematics who utilize state-of-the-art software and computational methods to conduct research.