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Ordinary Differential Equations (Springer Textbook) |
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Ordinary Differential Equations (Springer Textbook)
List Price: $64.95
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Manufacturer: Springer
Author: Vladimir I. Arnol'd
Binding: Paperback
Publication Date: 1992-05-08
Publisher: Springer
Label: Springer
Number Of Pages: 334
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Editorial Review:
There are dozens of books on ODEs, but none with the elegant geometric insight of Arnol'd's book. Arnol'd puts a clear emphasis on the qualitative and geometric properties of ODEs and their solutions, rather than on theroutine presentation of algorithms for solving special classes of equations.Of course, the reader learns how to solve equations, but with much more understanding of the systems, the solutions and the techniques. Vector fields and one-parameter groups of transformations come right from the startand Arnol'd uses this "language" throughout the book. This fundamental difference from the standard presentation allows him to explain some of the real mathematics of ODEs in a very understandable way and without hidingthe substance. The text is also rich with examples and connections with mechanics. Where possible, Arnol'd proceeds by physical reasoning, using it as a convenient shorthand for much longer formal mathematical reasoning. This technique helps the student get a feel for the subject. Following Arnol'd's guiding geometric and qualitative principles, there are 272 figures in the book, but not a single complicated formula. Also, the text is peppered with historicalremarks, which put the material in context, showing how the ideas have developped since Newton and Leibniz. This book is an excellent text for a course whose goal is a mathematical treatment of differential equations and the related physical systems.
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Customer Reviews
Average Customer Rating: 
changed my life 2008-06-28
Well differential equaitons are all about change, and this book changed my life. I read this more than 30 years ago, and all the mathematics I know, I mean really know, I learned from this book. Along with Aristotle's ethics, it is probably the most important book in my life.
ARNOLD==The MASTER!!! 2006-08-18
No doubt the best book on ODE by a master!! Ecuaciones Diferenciales Ordinarias (Fondos Distribuidos) Kiseliov Krasnov is another great book! Translated in English!! Like Spivak's Calculus on Manifolds, thin but good!!!
MDC 2006-07-13
This is a classic in the field. Excellent presentation and geometric perspective of dynamical systems. Most definitely a book to be kept as reference.
wow! differential equations made appealing 2005-12-20
I had always hated d.e.'s until this book made me see the geometry. And I have only read a few pages.
I never realized before that the existence and uniqueness theorem defines an equivalence relation on the compact manifold, where two points are equivalent iff they lie on the same flow curve. This instantly renders a d.e. visible, and not just some ugly formulas.
He also made me understand for the first time the proof of Reeb's theorem that a compact manifold with a function having only 2 critical points is a sphere. If they are non degenerate at least, the proof is simple. Each critical point has a nbhd looking like a disc. In between, the lack of critical points means there is a one parameter flow from the boundary circle of one disc to the other, i.e. thus the in between stuff is a cylinder.
Hence gluing a disc into each end of a cylinder gives a sphere! It also makes it clear why the sphere may have a non standard differentiable structure, because the diff. structure depends on how you glue in the discs.
What a book. I bought the cheaper older version, thanks to a reviewer here, and I love it. No other book gives me the geometry this forcefully and quickly. Of course I am a mathematician so the vector field and manifold language are familiar to me. But I guess this is a great place for beginners to learn it.
One tiny remark. He does not mind "deceiving you" in the sense of making plausible statements that are actually deep theorems in mathematics to prove. E.g. the fact that in a rectangle it is impossible to join two pairs of opposite corners by continuous curves that do not intersect, is non trivial to prove.
Hence the staement on page 2 that the problem is "solved" merely by introducing the phase plane, is not strictly true, until you prove the intersection statement above. All the phase plane version does for me is render the problem's solution highly plausible, and show the way to solving it. You still have to do it. But it was huge fun thiunking up a fairly elementary winding number argument for this fact.
Good teachers know how to deceive you instructively by making plausible statements that a beginner is willing to accept. I presume a physicist, e.g., would not quarrel with the statement above about curves intersecting.
This is the best differential, equaitons book I know of if you want to understand what they are, as opposed to learn to calculate canned solution fornmulas for special ones. He even makes clear what it is that is special about the special ones, e.g. linear equations are nice not just because the solutions are familiar exponential functions, but because the flow curves exist for all time,...
Amazing 2005-11-18
This is an amazing book. Arnold's style is unique - very intuitive and geometric. This book can be read by non-mathematicians but to really appreciate its beauty, and to understand the proofs that sometimes are just sketched, it takes some mathematical culture. This is the way ordinary differential equations should be taught (but they are not).
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