
Differential Equations
Theory and Applications
By: David Betounes
Hardcover | 16 November 2009 | Edition Number 2
At a Glance
642 Pages
Revised
24.2 x 17.0 x 3.9
Hardcover
$139.00
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Industry Reviews
From the reviews:
"This book is a comprehensive reader-friendly introduction to differential equations. ... The theory is illustrated by a great number of nice examples, applications, figures. ... I can warmly recommend the book for graduate mathematics, physic students and also students in applied sciences. Finally, I would like to encourage students, their professors and researchers to do computer experiments ... for improving their study, teaching and scientific work." (J¡nos Karsai, Acta Scientiarum Mathematica, Vol. 70, 2004)
"The book under review is intended for a one or two semester graduate course in ordinary differential equations. A novel feature of the book is the incorporation of Maple into the presentation ... . This well written book offers an application-minded instructor great flexibility in designing a course. All the necessary theory is included, as well as wide range of examples from physics and engineering ... ." (J. E. Paullet, Zeitschrift f¼r Angewandte Mathematik und Mechanik, Vol. 84 (6), 2004)
"This book is one of the few graduate differential equations texts that use the computer to enhance the concepts and theory normally taught to first- and second-year graduate students in mathematics. The author gives a comprehensive introduction to the theory of ordinary differential equations with a focus on mechanics and dynamical systems ... ." (Wei Nian Zhang, Mathematical Reviews, 2002 b)
"This book is intended to serve as a comprehensive introduction to the theory of ordinary differential equations ... . the material is organized so that it can be also used in a wider setting within today's modern university and society. ... The book makes every attempt to blend together the traditional theoretical material on differential equations and the new techniques afforded by computer algebra systems ... ." (International Aerospace Abstracts, Vol. 42 (5), 2002)
"This book is intended for first- and second-year graduate students in mathematics and also organized to be used for interdisciplinary courses in applied mathematics, physics, and engineering. ... The book is well written and provides many interesting examples. The author gives a comprehensive introduction to the theory on ordinary differential equations with a focus on mechanics and dynamical systems. The exposition is clear and easily understood." (Yuan Rong, Zentralblatt MATH, Vol. 993 (18), 2002)
From the reviews of the second edition:
"This textbook is intended as a comprehensive introduction to ordinary differential equations for graduate students. ... This is a very readable text that is enhanced with good supporting figures. Especially striking are hand-drawn figures reproduced to look like blackboard sketches. The proofs throughout the book are particularly ... detailed. There are also well-designed exercises for every section in the text. This is one graduate-level graduate differential equations text that really would support self-study." (William J. Satzer, The Mathematical Association of America, February, 2010)
"The book is an introduction to the theory of ordinary differential equations and intended for first- or second-year graduate students. ... The main feature of this book is its comprehensive structure, many examples and illustrations, and complementary electronic material. The electronic material is now provided on the Springer' website and consists of about 40 Maple-worksheets." (Sergiy Yanchuk, Zentralblatt MATH, Vol. 1192, 2010)
"This is the updated edition of a comprehensive introduction to ordinary differential equations from the view point of dynamical systems. ... The book is written in a concise style suitable for advanced undergraduate and beginning graduate students. ... all chapters including the online materials have been revised and enhanced. Moreover, many new examples and exercises have been added." (G. Teschl, Monatshefte f¼r Mathematik, Vol. 162 (3), March, 20
| Introduction | p. 1 |
| Examples of Dynamical Systems | p. 1 |
| Vector Fields and Dynamical Systems | p. 16 |
| Nonautonomous Systems | p. 26 |
| Fixed Points | p. 29 |
| Reduction to 1st-Order, Autonomous | p. 30 |
| Summary | p. 34 |
| Techniques, Concepts and Examples | p. 37 |
| Euler's Numerical Method | p. 38 |
| The Geometric View | p. 38 |
| The Analytical View | p. 40 |
| Gradient Vector Fields | p. 43 |
| Fixed Points and Stability | p. 50 |
| Limit Cycles | p. 55 |
| The Two-Body Problem | p. 60 |
| Jacobi Coordinates | p. 62 |
| The Central Force Problem | p. 63 |
| Summary | p. 77 |
| Existence and Uniqueness: The Flow Map | p. 79 |
| Picard Iteration | p. 82 |
| Existence and Uniqueness Theorems | p. 86 |
| Maximum Interval of Existence | p. 96 |
| The Flow Generated by a Time-Dependent Vector Field | p. 100 |
| The Flow for Autonomous Systems | p. 108 |
| Summary | p. 117 |
| Linear Systems | p. 119 |
| Existence and Uniqueness for Linear Systems | p. 124 |
| The Fundamental Matrix and the Flow | p. 128 |
| Homogeneous, Constant Coefficient Systems | p. 154 |
| The Geometry of the Integral Curves | p. 161 |
| Real Eigenvalues | p. 163 |
| Complex Eigenvalues | p. 174 |
| Canonical Systems | p. 201 |
| Diagonalizable Matrices | p. 204 |
| Complex Diagonalizable Matrices | p. 208 |
| The Nondiagonalizable Case: Jordan Forms | p. 209 |
| Summary | p. 218 |
| Linearization & Transformation | p. 221 |
| Linearization | p. 221 |
| Transforming Systems of DEs | p. 237 |
| The Spherical Coordinate Transformation | p. 243 |
| Some Results on Differentiable Equivalence | p. 248 |
| The Linearization and Flow Box Theorems | p. 258 |
| Stability Theory | p. 267 |
| Stability of Fixed Points | p. 269 |
| Linear Stability of Fixed Points | p. 272 |
| Computation of the Matrix Exponential for Jordan Forms | p. 272 |
| Nonlinear Stability | p. 283 |
| Liapunov Functions | p. 285 |
| Stability of Periodic Solutions | p. 296 |
| Integrable Systems | p. 333 |
| First Integrals (Constants of the Motion) | p. 334 |
| Integrable Systems in the Plane | p. 339 |
| Integrable Systems in 3-D | p. 345 |
| Integrable Systems in Higher Dimensions | p. 359 |
| Newtonian Mechanics | p. 371 |
| The N-Body Problem | p. 372 |
| Fixed Points | p. 375 |
| Initial Conditions | p. 376 |
| Conservation Laws | p. 377 |
| Stability of Conservative Systems | p. 385 |
| Euler's Method and the N-body Problem | p. 394 |
| Discrete Conservation Laws | p. 403 |
| The Central Force Problem Revisited | p. 412 |
| Effective Potentials | p. 415 |
| Qualitative Analysis | p. 416 |
| Linearization and Stability | p. 420 |
| Circular Orbits | p. 421 |
| Analytical Solution | p. 423 |
| Rigid-Body Motions | p. 436 |
| The Rigid-Body Differential Equations | p. 443 |
| Kinetic Energy and Moments of Inertia | p. 450 |
| The Degenerate Case | p. 458 |
| Euler's Equation | p. 459 |
| The General Solution of Euler's Equation | p. 463 |
| Hamiltonian Systems | p. 475 |
| 1-Dimensional Hamiltonian Systems | p. 478 |
| Conservation of Energy | p. 481 |
| Conservation Laws and Poisson Brackets | p. 489 |
| Lie Brackets and Arnold's Theorem | p. 502 |
| Arnold's Theorem | p. 505 |
| Liouville's Theorem | p. 528 |
| Elementary Analysis | p. 535 |
| Multivariate Calculus | p. 535 |
| The Chain Rule | p. 541 |
| The Inverse and Implicit Function Theorems | p. 542 |
| Taylor's Theorem and The Hessian | p. 548 |
| The Change of Variables Formula | p. 552 |
| Lipschitz Maps and Linearization | p. 553 |
| Norms | p. 554 |
| Lipschitz Functions | p. 555 |
| The Contraction Mapping Principle | p. 559 |
| The Linearization Theorem | p. 565 |
| Linear Algebra | p. 579 |
| Vector Spaces and Direct Sums | p. 579 |
| Bilinear Forms | p. 582 |
| Inner Product Spaces | p. 584 |
| The Principal Axes Theorem | p. 588 |
| Generalized Eigenspaces | p. 591 |
| Matrix Analysis | p. 602 |
| Power Series with Matrix Coefficients | p. 609 |
| Electronic Contents | p. 613 |
| Bibliography | p. 615 |
| Index | p. 621 |
| Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9781441911629
ISBN-10: 1441911626
Published: 16th November 2009
Format: Hardcover
Language: English
Number of Pages: 642
Audience: Professional and Scholarly
Publisher: Springer Nature B.V.
Country of Publication: GB
Edition Number: 2
Edition Type: Revised
Dimensions (cm): 24.2 x 17.0 x 3.9
Weight (kg): 1.07
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