Calculus of Variations

Rating:
Summary: A great book by a great mathematician
Review: Gelfand was one of the leaders of the great school of mathematics which, somehow, thrived in Soviet Union. I used uncountable times the copy of our library, as the original English edition, in the excellent translation of R. Silvermann, became very hard to find. I put it in the top of the list of books I wanted to buy. Now Dover put it into their catalogue. Great choice. I already ordered my copy!
This is the best book on the Calculus of Variations. It contains, for instance, a wonderful treatment of Noether's theorem, hardly to be surpassed. The Hamilton-Jacobi equation is also treated with brilliance and clarity. Gelfand (and Fomin!) developed a style in which the precision of the mathematics does not interfere with the general panorama. The applications are very well selected and perfectly illustrate the theory. A great book, a great mathematician who can write, a great translator, by less than 10 bucks!

Rating:
Summary: Another great Silverman translation from the russian
Review: I used this book for a first year grad course in the calculus of variations some years ago. I found the book to be clear and pretty readable. I found the problem sets to be quite workable and covered the material well. I liked the fact that there were answers to many of them. The book is more rigourous than the treatments of this subjects found in math methods books like arfken but it is not highly abstract. (a style found in many math books at this level, and more difficult for me, not a professional mathematician, but a scientist) In fact I have found this quality to be a characteristic of most of the russian applied mathematics books translated by Richard Silverman, many of which are now published by Dover. This practical, clear, and rigourous approach of these books is excellent and I think almost all of these books on Dover have found their way onto my shelves.

Rating:
Summary: The best "introductory" book in Variational Calculus.
Review: Readable books on Calculus of Variations are hard to come by, if not non-existent. This is one of the clearest and most readable and self-contained books in the topic. I used it in a fourth year course at Simon Fraser University, BC, Canada. This is an introductory book meant for undergraduates and it is very well suited for this level. I strongly recommend that all serious students in Physics, Chemistry, engineering, and computing science to take a serious look at the book as a starting point in studying this most underated and relevant topic as a necessary mathematical foundation. Virtually all fundamental laws of Physics, Chemistry, and Engineering have their origins from some form of variational prinicples. If anyone is interested in pursuing this topic, this should definitely be the first book one should read. The student should be equipped with at least one introductory course in Real Ananlysis in order to start studying Variational Calculus. Otherwise, the readable may not find this book readable. Just a practical (cook-book) style applied mathematics course in Calculus and Differential Equations is inadequate as preparation to study this topic. I most sincerely congratulate the author in having done such a superbly well organized job in writing such a highly readable book in this challenging and important topic.

Rating:
Summary: an excellent introduction into the subject
Review: This book is intended primarily for graduate students with math background who wish to expand their applied math skills. Originally a Russian textbook, it offers an excellent overview of the theoretical background for the field without real heavy mathematics. All that is required is knowledge of calculus and a little ODE/PDE. Many classical examples, has an introduction to applications such as analytical mechanics, optimal control theory and approximate solution methods. The last edition I am aware of is pretty old, so this book should be tough to come by. A new edition would be welcome, with more examples and other applications.

Rating:
Summary: Clear but technical
Review: This book, of which I studied the first four chapters for an independent study course (I'm a senior undergrad) are very clear, very full, but beware it is mathematics and it is technical.

To appriciate the material you really should have a year of "advanced calculus" also called "intro. real analysis" at some places. This means the formalities of limits, continuity, derivatives, integration and series. This will prepare you to understand and work through the proofs in the text.

The problems are nice since they are varied (computational, physics, and proofs) and they do come with many answers and some hints, but you might find that having a mechanics book at your side motivates some of the problems.

Work hard, be thorough and there's a lot of important ideas in this text, with chapter 4 being especially relevant to physicists (lots of mechanics and conservation theorems!).

Rating:
Summary: Clear but technical
Review: This book, of which I studied the first four chapters for an independent study course (I'm a senior undergrad) are very clear, very full, but beware it is mathematics and it is technical.

To appriciate the material you really should have a year of "advanced calculus" also called "intro. real analysis" at some places. This means the formalities of limits, continuity, derivatives, integration and series. This will prepare you to understand and work through the proofs in the text.

The problems are nice since they are varied (computational, physics, and proofs) and they do come with many answers and some hints, but you might find that having a mechanics book at your side motivates some of the problems.

Work hard, be thorough and there's a lot of important ideas in this text, with chapter 4 being especially relevant to physicists (lots of mechanics and conservation theorems!).