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Ordinary Differential Equations : Ordinary Differential Equations - V. I. Arnold

Ordinary Differential Equations

Ordinary Differential Equations

Paperback Published: 15th July 1978
ISBN: 9780262510189
Number Of Pages: 280
For Ages: 18+ years old

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Few books on Ordinary Differential Equations (ODEs) have the elegant geometric insight of this one, which puts emphasis on the qualitative and geometric properties of ODEs and their solutions, rather than on routine presentation of algorithms.

This college-level textbook treats the subject of ordinary differential equations in an entirely new way. A wealth of topics is presented masterfully, accompanied by many thought-provoking examples, problems, and 259 figures. The author emphasizes the geometrical and intuitive aspects and at the same time familiarizes the student with concepts, such as flows and manifolds and tangent bundles, traditionally not found in textbooks of this level. The exposition is guided by applications taken mainly from mechanics. One can expect this book to bring new life into this old subject. * American Scientist * A fresh modern approach to the geometric qualitative theory of ordinary differential equations...suitable for advanced undergraduates and some graduate students. The notions of vector field, phase space, phase flow, and one parameter groups of transformations dominate the entire presentation. The author is acutely aware of the pitfalls of this abstract approach (e.g., putting the reader to sleep) and does a brilliant job of presenting only the most essential ideas with an easily grasped notation, a minimum formalism, and very careful motivation. * Technometrics *

Prefacep. vii
Frequently Used Notationp. ix
Basic Conceptsp. 1
Phase Spaces and Phase Flowsp. 1
Vector Fields on the Linep. 11
Phase Flows on the Linep. 19
Vector Fields and Phase Flows in the Planep. 24
Nonautonomous Equationsp. 28
The Tangent Spacep. 33
Basic Theoremsp. 48
The Vector Field near a Nonsingular Pointp. 48
Applications to the Nonautonomous Casep. 56
Applications to Equations of Higher Orderp. 59
Phase Curves of Autonomous Systemsp. 68
The Directional Derivative. First Integralsp. 72
Conservative Systems with One Degree of Freedomp. 79
Linear Systemsp. 95
Linear Problemsp. 95
The Exponential of an Operatorp. 97
Properties of the Exponentialp. 104
The Determinant of the Exponentialp. 111
The Case of Distinct Real Eigenvaluesp. 115
Complexification and Decomplexificationp. 119
Linear Equations with a Complex Phase Spacep. 124
Complexification of a Real Linear Equationp. 129
Classification of Singular Points of Linear Systemsp. 139
Topological Classification of Singular Pointsp. 143
Stability of Equilibrium Positionsp. 154
The Case of Purely Imaginary Eigenvaluesp. 160
The Case of Multiple Eigenvaluesp. 167
More on Quasi-Polynomialsp. 176
Nonautonomous Linear Equationsp. 188
Linear Equations with Periodic Coefficientsp. 199
Variation of Constantsp. 208
Proofs of the Basic Theoremsp. 211
Contraction Mappingsp. 211
The Existence, Uniqueness, and Continuity Theoremsp. 213
The Differentiable Manifoldsp. 233
The Tangent Bundle. Vector Fields on a Manifoldp. 243
The Phase Flow Determined by a Vector Fieldp. 250
The Index of a Singular Point of a Vector Fieldp. 254
Sample Examination Problemsp. 269
Bibliographyp. 273
Indexp. 275
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780262510189
ISBN-10: 0262510189
Series: Ordinary Differential Equations
Audience: Professional
For Ages: 18+ years old
For Grades: 12+
Format: Paperback
Language: English
Number Of Pages: 280
Published: 15th July 1978
Publisher: MIT Press Ltd
Country of Publication: US
Dimensions (cm): 22.9 x 14.9  x 2.0
Weight (kg): 0.48