Ordinary Differential Equations (The MIT Press) 1st Edition by V.I. Arnold (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Review 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—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—

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I was looking for a text on the theory of non-qualitative ODE’s. Which this textbook isn’t. There are many other texts that I would read prior to this one should I ever attempt to learn the qualitative theory of ODEs. Albeit those texts have a different target audience.If you’re interested in the DE connection to linear algebra (all books have this LA connection), topology, and differential geometry then this book IS for you.3/5 because I didn’t read it enough to hate it or love it. Although it is obvious which one I fall into.

⭐It’s taken me a bit to get used to the approach, but it sure is satisfying when things click!

⭐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,…

⭐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.) There are foibles, too, including proofs that satisfy the requirements for some theorem or definition without actually stating what theorem or definition is now applicable. One can detect some mild arrogance in places (after an arduous two-page proof, he mentions “As always in proving obvious theorems, it is easier to carry out the proof of the extension theorem than to read through it.”) Also, a few typos can be found here and there, which sometimes result in confusion.One very curious thing about Arnol’d is that my most brilliant math-major friends find it impenetrable, whereas I know biologists who got through it with no problem. So I guess that, for a mere mortal, reading Arnol’d demands a willingness to have a feel for a big picture without worrying about every epsilon and delta.So grab a copy of this book, let it flow, and learn about ODEs. It’s well worth the effort.

⭐This is one of the most beautiful books written on ordinary differential equation and a most read for any interested person.The ebook format (and all its variants) however does not handle mathematical expressions yet, and formula have to be represented by images. This make the kindle edition unreadable. Add to this the a horrible job done by the publisher in translating it into ebook format.hopefully, the epub,mobi, … formats one day will include the mathml extension. Until then, go for the paper edition.

⭐If you consider buying the kindle edition, then just don’t! Neither the equations, nor the tables are expandable. You will not be able to read them, even if you wear presbyopic eyeglasses! I bought it and had it refunded within five minutes.

⭐This is a great textbook for a theoretical understanding of ordinary differential equations. I suggest attempting this book after one has handled ODEs at an undergraduate level.

⭐Excellent book, expects the reader to have a certain level of mathematical maturity, which is not surprising.

⭐Great book that gives insights for basic differential equations.

⭐Gives you geometric insight of DFEs , different from the monotonous solving methodsa must have for all science students..

⭐Best book for long and strong thesis in mathematics of me

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