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Ordinary Differential Equations (Universitext) Paperback – June 19, 2006

ISBN-13: 978-3540345633 ISBN-10: 3540345639

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Ordinary Differential Equations (Universitext) + Geometrical Methods in the Theory of Ordinary Differential Equations (Grundlehren der Mathematischen Wissenschaften (Springer Paperback)) + Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics, Vol. 60)
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Product Details

  • Series: Universitext
  • Paperback: 334 pages
  • Publisher: Springer (June 19, 2006)
  • Language: English
  • ISBN-10: 3540345639
  • ISBN-13: 978-3540345633
  • Product Dimensions: 9.2 x 6.2 x 0.8 inches
  • Shipping Weight: 1.2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.2 out of 5 stars  See all reviews (16 customer reviews)
  • Amazon Best Sellers Rank: #422,507 in Books (See Top 100 in Books)

Editorial Reviews

Review

From the reviews:

"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)

Language Notes

Text: English, Russian (translation) --This text refers to an out of print or unavailable edition of this title.

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Customer Reviews

4.2 out of 5 stars
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No doubt the best book on ODE by a master!!
Physicsmind
It is a nice counterexample for those who believe that, to be rigorous, a mathematical book needs to be very hard to read.
Ambrosini Walter
So grab a copy of this book, let it flow, and learn about ODEs.
Raman

Most Helpful Customer Reviews

80 of 85 people found the following review helpful By A Customer on November 19, 2001
Format: Paperback
Be aware there are 2 versions of this book
available in English; this one from MIT press
is (contrary to one of the reviews above) is
translated from the *first* Russian edition;
there is another version from Springer-Verlag
translated from the *third* Russian edition.
They're translated by different people so
some wording etc is different but otherwise
they're similar, though not identical. The
later edition has some reworked passages
and modest amount of new material, but it's
not a hugely different book.
Both are excellent, are are all the other
books & papers I've seen by V.I. Arnol'd.
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65 of 69 people found the following review helpful By Raman on March 11, 2004
Format: Paperback
This has to be one of the most amazing math books I've ever read. Arnol'd seems to do the impossible here - he blends abstract theory with an intuitive exposition while avoiding any tendency to become verbose. By the end of Arnol'd, it's hard not to have a deep understanding of the way that ODEs and their solutions behave.
Arnol'd accomplishes this feat through an intense parsimony of words and topics. Everything in this book builds on the central theme of the relations between vector fields and one-parameter groups of diffeomorphisms, and the topics are illustrated (and often motivated) almost exclusively through problems in classical mechanics, most notably the plane pendulum. Almost no solution techniques are given in this book - expect no mention of integrating factors or Bessel functions. One of the main reasons that the book does so much without bogging down is that the mathematical formalism is minimal and terse - proofs are often one or two lines long, merely mentioning the conceptual justification of a result without detailed, formal constructions.
But the result of this parsimony is that Arnol'd is a very difficult book. To understand every detail and to be able to attempt every problem, I think, basically requires a math degree - lots of linear algebra (for his monumental 116-page chapter on linear systems), a solid background in analysis and topology, and a bit of differential geometry and abstract algebra are prerequsites for a full understanding. (I found the section on the "topological classification of singular points," in particular, nearly incomprehensible with my thin chemistry-major math background.
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45 of 47 people found the following review helpful By mathwonk on December 20, 2005
Format: Paperback
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.
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26 of 29 people found the following review helpful By A Customer on November 9, 1995
Format: Paperback
This is one of the few original books in the area of
differential equations. In his clear style, Arnold
presents the basics of differential equations. He is more
interested in understanding the solutions than in deriving
them by analytical methods. The text is well organized and
there seem to be more figures than proofs (although all
proofs are there, it just that they do not get in the
way). A must, if you are in the area of chaos and dynamical
systems. (RM)
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16 of 19 people found the following review helpful By Ambrosini Walter on August 8, 2000
Format: Paperback
It is hardly needed to add words to the existing positive reviews of the book. In the line of previous comments, I just mention that it is an enjoyable book on a basic subject of great interest also for engineers and physicists. The matter is treated with the evident purpose to make the reader fully aware of the interesting geometrical and dynamic implications of the conclusions reached at each step. It is a nice counterexample for those who believe that, to be rigorous, a mathematical book needs to be very hard to read.
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