Items related to Ordinary Differential Equations
Although there is no lack of other books on this subject, even with the same title, the appearance of this new one is fully justified on at least two grounds: its approach makes full use of modern mathematical concepts and terminology of considerable sophistication and abstraction, going well beyond the traditional presentation of the subject; and, at the same time, the resulting enhancement of mathematical abstractness is counterbalanced by a constant appeal to geometrical and physical considerations, presented in the main text and in numerous problems and exercises.In terms of mathematical approach, the text is dominated by two central ideas: the theorem on rectifiability of a vector field (which is equivalent to the usual theorems on existence, uniqueness, and differentiability of solutions) and the theory of one-parameter groups of linear transformations (equivalent to the theory of linear autonomous systems). The book also develops whole congeries of fundamental concepts--like phase space and phase flows, smooth manifolds and tangent bundles, vector fields and one-parameter groups of diffeomorphisms--that remain in the shadows in the traditional coordinate-based approach. All of these concepts are presented in some detail, but without assuming any background on the part of the reader beyond the scope of the standard elementary courses on analysis and linear algebra.In terms of concrete applications, the book introduces the pendulum equation at the very beginning, and the efficacy of various concepts and methods introduced throughout is subsequently tested by applying them to this example. Thus, the section on first integrals leads to the law of conservation of energy; the theorem on differentiation with respect to a parameter finds application in the "method of small parameters"; and the theory of linear equations with periodic coefficients merges naturally with the study of parametric resonance. This geometrical and physical specificity is made still more vivid through the inclusion of 259 line drawings and 260 exercises in which other examples are taken up.
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"Professor Arnold has expanded his classic book to include new material on exponential growth, predator-prey, the pendulum, impulse response, symmetry groups and group actions, perturbation and bifurcation ... . The new edition is highly recommended as a general reference for the essential theory of ordinary differential equations and as a textbook for an introductory course for serious undergraduate or graduate students. ... In the US system, it is an excellent text for an introductory graduate course." (Carmen Chicone, SIAM Review, Vol. 49 (2), 2007)
"Vladimir Arnold’s is a master, not just of the technical realm of differential equations but of pedagogy and exposition as well. ... The writing throughout is crisp and clear. ... Arnold’s says that the book is based on a year-long sequence of lectures for second-year mathematics majors in Moscow. In the U.S., this material is probably most appropriate for advanced undergraduates or first-year graduate students." (William J. Satzer, MathDL, August, 2007)
From the reviews: "... This book is an excellent text for a course whose goal is a mathematical treatment of differential equations and the related physical systems." L'Enseignment Mathematique "... Arnold's book is unique as a sophisticated but accessible introduction to the modern theory, and we should be grateful that it exists in a convenient language." Mathematical Association of America Monthly
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- PublisherMIT Press
- Publication date1973
- ISBN 10 0262010372
- ISBN 13 9780262010375
- BindingHardcover
- Number of pages264
- Rating
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