Calculus and Ordinary Differential Equations

Author: David Pearson

Publisher: Elsevier

ISBN: 008092865X

Category: Mathematics

Page: 240

View: 7225

Professor Pearson's book starts with an introduction to the area and an explanation of the most commonly used functions. It then moves on through differentiation, special functions, derivatives, integrals and onto full differential equations. As with other books in the series the emphasis is on using worked examples and tutorial-based problem solving to gain the confidence of students.

Ordinary Differential Equations and Calculus of Variations

Book of Problems

Author: M. V. Makarets,V. Yu Reshetnyak

Publisher: World Scientific

ISBN: 9810221916

Category: Mathematics

Page: 372

View: 6772

This problem book contains exercises for courses in differential equations and calculus of variations at universities and technical institutes. It is designed for non-mathematics students and also for scientists and practicing engineers who feel a need to refresh their knowledge. The book contains more than 260 examples and about 1400 problems to be solved by the students ? much of which have been composed by the authors themselves. Numerous references are given at the end of the book to furnish sources for detailed theoretical approaches, and expanded treatment of applications.

Ordinary Differential Equations

From Calculus to Dynamical Systems

Author: Virginia W. Noonburg

Publisher: The Mathematical Association of America

ISBN: 1939512042

Category: Mathematics

Page: 315

View: 5009

This book presents a modern treatment of material traditionally covered in the sophomore-level course in ordinary differential equations. While this course is usually required for engineering students the material is attractive to students in any field of applied science, including those in the biological sciences. The standard analytic methods for solving first and second-order differential equations are covered in the first three chapters. Numerical and graphical methods are considered, side-by-side with the analytic methods, and are then used throughout the text. An early emphasis on the graphical treatment of autonomous first-order equations leads easily into a discussion of bifurcation of solutions with respect to parameters. The fourth chapter begins the study of linear systems of first-order equations and includes a section containing all of the material on matrix algebra needed in the remainder of the text. Building on the linear analysis, the fifth chapter brings the student to a level where two-dimensional nonlinear systems can be analyzed graphically via the phase plane. The study of bifurcations is extended to systems of equations, using several compelling examples, many of which are drawn from population biology. In this chapter the student is gently introduced to some of the more important results in the theory of dynamical systems. A student project, involving a problem recently appearing in the mathematical literature on dynamical systems, is included at the end of Chapter 5. A full treatment of the Laplace transform is given in Chapter 6, with several of the examples taken from the biological sciences. An appendix contains completely worked-out solutions to all of the odd-numbered exercises. The book is aimed at students with a good calculus background that want to learn more about how calculus is used to solve real problems in today's world. It can be used as a text for the introductory differential equations course, and is readable enough to be used even if the class is being "flipped." The book is also accessible as a self-study text for anyone who has completed two terms of calculus, including highly motivated high school students. Graduate students preparing to take courses in dynamical systems theory will also find this text useful.

Multivariable Calculus, Linear Algebra, and Differential Equations

Author: Stanley I. Grossman

Publisher: Academic Press

ISBN: 1483218031

Category: Mathematics

Page: 992

View: 5634

Multivariable Calculus, Linear Algebra, and Differential Equations, Second Edition contains a comprehensive coverage of the study of advanced calculus, linear algebra, and differential equations for sophomore college students. The text includes a large number of examples, exercises, cases, and applications for students to learn calculus well. Also included is the history and development of calculus. The book is divided into five parts. The first part includes multivariable calculus material. The second part is an introduction to linear algebra. The third part of the book combines techniques from calculus and linear algebra and contains discussions of some of the most elegant results in calculus including Taylor's theorem in "n" variables, the multivariable mean value theorem, and the implicit function theorem. The fourth section contains detailed discussions of first-order and linear second-order equations. Also included are optional discussions of electric circuits and vibratory motion. The final section discusses Taylor's theorem, sequences, and series. The book is intended for sophomore college students of advanced calculus.

Ordinary Differential Equations

An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences

Author: Morris Tenenbaum,Harry Pollard

Publisher: Courier Corporation

ISBN: 0486649407

Category: Mathematics

Page: 808

View: 8630

Skillfully organized introductory text examines origin of differential equations, then defines basic terms and outlines the general solution of a differential equation. Subsequent sections deal with integrating factors; dilution and accretion problems; linearization of first order systems; Laplace Transforms; Newton's Interpolation Formulas, more.

Essential PTC® Mathcad Prime® 3.0

A Guide for New and Current Users

Author: Brent Maxfield

Publisher: Academic Press

ISBN: 0124104584

Category: Computers

Page: 584

View: 3087

Learn how to use PTC® Mathcad Prime® 3.0, one of the world’s leading tools for technical computing, in the context of engineering, science, and math applications. Quickly harness the power of PTC Mathcad Prime 3.0 to solve both simple and complex problems. Essential PTC® Mathcad Prime® 3.0 is perfect for college students, first-time users, and experienced Mathcad 15 users who are moving to PTC Mathcad Prime 3.0. Updated from Maxfield’s popular Essential Mathcad, this book introduces the most powerful functions and features of the new PTC Mathcad Prime 3.0 software and teaches how to apply them to create comprehensive calculations for any quantitative subject. Examples from several fields demonstrate the power and utility of PTC Mathcad’s tools while also demonstrating how users can eff ectively incorporate Microsoft® Excel spreadsheets into the software. Learn the basics faster: Chapter 1 introduces many fundamentals of Mathcad, allowing the reader to begin using the program in less time. Learn PTC Mathcad tools in context: Incorporates many applied examples and problems from a wide variety of disciplines. Thorough discussion of many PTC Mathcad tools: Units, arrays, plotting, solving, symbolic calculations, programming, algebra, calculus, differential equations, reading from files, writing to files, and incorporating MS Excel spreadsheets. Includes a link to PTC with instructions on how to purchase the PTC® Mathcad Prime® 3.0 Student Edition (The Student Edition software is intended for educational purposes only.)

A Workbook for Differential Equations

Author: Bernd S. W. Schröder

Publisher: John Wiley & Sons

ISBN: 0470447516

Category: Mathematics

Page: 340

View: 7827

An accessible and hands-on approach to modeling and predicting real-world phenomena using differential equations A Workbook for Differential Equations presents an interactive introduction to fundamental solution methods for ordinary differential equations. The author emphasizes the importance of manually working through computations and models, rather than simply reading or memorizing formulas. Utilizing real-world applications from spring-mass systems and circuits to vibrating strings and an overview of the hydrogen atom, the book connects modern research with the presented topics, including first order equations, constant coefficient equations, Laplace transforms, partial differential equations, series solutions, systems, and numerical methods. The result is a unique guide to understanding the significance of differential equations in mathematics, science, and engineering. The workbook contains modules that involve readers in as many ways as possible, and each module begins with "Prerequisites" and "Learning Objectives" sections that outline both the skills needed to understand the presented material and what new skills will be obtained by the conclusion of the module. Detailed applications are intertwined in the discussion, motivating the investigation of new classes of differential equations and their accompanying techniques. Introductory modeling sections discuss applications and why certain known solution techniques may not be enough to successfully analyze certain situations. Almost every module concludes with a section that contains various projects, ranging from programming tasks to theoretical investigations. The book is specifically designed to promote the development of effective mathematical reading habits such as double-checking results and filling in omitted steps in a computation. Rather than provide lengthy explanations of what readers should do, good habits are demonstrated in short sections, and a wide range of exercises provide the opportunity to test reader comprehension of the concepts and techniques. Rich illustrations, highlighted notes, and boxed comments offer illuminating explanations of the computations. The material is not specific to any one particular software package, and as a result, necessary algorithms can be implemented in various programs, including Mathematica®, Maple, and Mathcad®. The book's related Web site features supplemental slides as well as videos that discuss additional topics such as homogeneous first order equations, the general solution of separable differential equations, and the derivation of the differential equations for a multi-loop circuit. In addition, twenty activities are included at the back of the book, allowing for further practice of discussed topics whether in the classroom or for self-study. With its numerous pedagogical features that consistently engage readers, A Workbook for Differential Equations is an excellent book for introductory courses in differential equations and applied mathematics at the undergraduate level. It is also a suitable reference for professionals in all areas of science, physics, and engineering.

Ordinary Differential Equations and Dynamical Systems

Author: Gerald Teschl

Publisher: American Mathematical Soc.

ISBN: 0821883283

Category: Mathematics

Page: 356

View: 1254

This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students. The first part begins with some simple examples of explicitly solvable equations and a first glance at qualitative methods. Then the fundamental results concerning the initial value problem are proved: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore, linear equations are considered, including the Floquet theorem, and some perturbation results. As somewhat independent topics, the Frobenius method for linear equations in the complex domain is established and Sturm-Liouville boundary value problems, including oscillation theory, are investigated. The second part introduces the concept of a dynamical system. The Poincare-Bendixson theorem is proved, and several examples of planar systems from classical mechanics, ecology, and electrical engineering are investigated. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed. Finally, stability is studied, including the stable manifold and the Hartman-Grobman theorem for both continuous and discrete systems. The third part introduces chaos, beginning with the basics for iterated interval maps and ending with the Smale-Birkhoff theorem and the Melnikov method for homoclinic orbits. The text contains almost three hundred exercises. Additionally, the use of mathematical software systems is incorporated throughout, showing how they can help in the study of differential equations.

Introduction to Partial Differential Equations with Applications

Author: E. C. Zachmanoglou,Dale W. Thoe

Publisher: Courier Corporation

ISBN: 048613217X

Category: Mathematics

Page: 432

View: 9347

This text explores the essentials of partial differential equations as applied to engineering and the physical sciences. Discusses ordinary differential equations, integral curves and surfaces of vector fields, the Cauchy-Kovalevsky theory, more. Problems and answers.

Ordinary Differential Equations and Stability Theory

An Introduction

Author: David A. Sánchez

Publisher: Courier Corporation

ISBN: 0486638286

Category: Mathematics

Page: 164

View: 5786

Beginning with a general discussion of the linear equation, topics developed include stability theory for autonomous and nonautonomous systems. Two appendices are also provided, and there are problems at the end of each chapter — 55 in all. Unabridged republication of the original (1968) edition. Appendices. Bibliography. Index. 55 problems.

The Inverse Problem of the Calculus of Variations for Ordinary Differential Equations

Author: Ian Anderson,Gerard Thompson

Publisher: American Mathematical Soc.

ISBN: 082182533X

Category: Mathematics

Page: 110

View: 9706

This monograph explores various aspects of the inverse problem of the calculus of variations for systems of ordinary differential equations. The main problem centers on determining the existence and degree of generality of Lagrangians whose system of Euler-Lagrange equations coincides with a given system of ordinary differential equations. The authors rederive the basic necessary and sufficient conditions of Douglas for second order equations and extend them to equations of higher order using methods of the variational bicomplex of Tulcyjew, Vinogradov, and Tsujishita. What emerges is a fundamental dichotomy between second and higher order systems: the most general Lagrangian for any higher order system can depend only upon finitely many constants. The authors present an algorithm, based upon exterior differential systems techniques, for solving the inverse problem for second order equations. A number of new examples illustrate the effectiveness of this approach. The monograph also contains a study of the inverse problem for a pair of geodesic equations arising from a two dimensional symmetric affine connection. The various possible solutions to the inverse problem for these equations are distinguished by geometric properties of the Ricci tensor.

General Linear Methods for Ordinary Differential Equations

Author: Zdzislaw Jackiewicz

Publisher: John Wiley & Sons

ISBN: 9780470522158

Category: Mathematics

Page: 480

View: 6279

Learn to develop numerical methods for ordinary differential equations General Linear Methods for Ordinary Differential Equations fills a gap in the existing literature by presenting a comprehensive and up-to-date collection of recent advances and developments in the field. This book provides modern coverage of the theory, construction, and implementation of both classical and modern general linear methods for solving ordinary differential equations as they apply to a variety of related areas, including mathematics, applied science, and engineering. The author provides the theoretical foundation for understanding basic concepts and presents a short introduction to ordinary differential equations that encompasses the related concepts of existence and uniqueness theory, stability theory, and stiff differential equations and systems. In addition, a thorough presentation of general linear methods explores relevant subtopics such as pre-consistency, consistency, stage-consistency, zero stability, convergence, order- and stage-order conditions, local discretization error, and linear stability theory. Subsequent chapters feature coverage of: Differential equations and systems Introduction to general linear methods (GLMs) Diagonally implicit multistage integration methods (DIMSIMs) Implementation of DIMSIMs Two-step Runge-Kutta (TSRK) methods Implementation of TSRK methods GLMs with inherent Runge-Kutta stability (IRKS) Implementation of GLMs with IRKS General Linear Methods for Ordinary Differential Equations is an excellent book for courses on numerical ordinary differential equations at the upper-undergraduate and graduate levels. It is also a useful reference for academic and research professionals in the fields of computational and applied mathematics, computational physics, civil and chemical engineering, chemistry, and the life sciences.

Advanced Calculus


Author: Lynn Harold Loomis,Shlomo Sternberg

Publisher: World Scientific Publishing Company

ISBN: 9814583952

Category: Mathematics

Page: 596

View: 6934

An 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.

Linear Ordinary Differential Equations

Author: Earl A. Coddington,Robert Carlson

Publisher: SIAM

ISBN: 9781611971439

Category: Differential equations

Page: 334

View: 696

Linear Ordinary Differential Equations, a text for advanced undergraduate or beginning graduate students, presents a thorough development of the main topics in linear differential equations. A rich collection of applications, examples, and exercises illustrates each topic. The authors reinforce students' understanding of calculus, linear algebra, and analysis while introducing the many applications of differential equations in science and engineering. Three recurrent themes run through the book. The methods of linear algebra are applied directly to the analysis of systems with constant or periodic coefficients and serve as a guide in the study of eigenvalues and eigenfunction expansions. The use of power series, beginning with the matrix exponential function leads to the special functions solving classical equations. Techniques from real analysis illuminate the development of series solutions, existence theorems for initial value problems, the asymptotic behavior solutions, and the convergence of eigenfunction expansions.

Differential Equations For Dummies

Author: Steven Holzner

Publisher: John Wiley & Sons

ISBN: 0470178140

Category: Mathematics

Page: 360

View: 6052

The fun and easy way to understand and solve complex equations 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.