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Ordinary differential equations

Author: V I Arnolʹd
Publisher: [Cambridge], [MIT Press], [1973]
Edition/Format:   Book : EnglishView all editions and formats
Database:WorldCat
Summary:
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  Read more...
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Document Type: Book
All Authors / Contributors: V I Arnolʹd
ISBN: 0262010372 9780262010375 0262510189 9780262510189
OCLC Number: 624013
Notes: Translation of Obyknovennye different︠s︡ialʹnye uravnenii︠a︡.
Description: viii, 280 pages : illustrations ; 23 cm
Contents: Basic concepts --
Basic theorems --
Linear systems --
Proofs of the basic theorems --
Differential equations on manifolds.
Other Titles: Obyknovennye different︠s︡ialʹnye uravnenii︠a︡.
Responsibility: V.I. Arnold ; translated and edited by Richard A. Silverman.

Abstract:

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 the 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.

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