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on 13 July 2001
For many years I had terrible problems having no proper text of Calculus of Variations with examples of its applications to Geometry and Theoretical Physics. Finally, I found and recommended to the students the old edition of this very book. It worked very well: a slim book, written with characteristic clarity the authors are famous for, which brought numerous useful examples of applications and very well working problems and exercises. Now this book is awailable as a cheap Dover edition. I can recommend it to physicists and mathematicians, students and teachers. This is a classic text. It is a great joy to read it, and it helps to study and to teach.
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on 29 January 2015
Hardly something to love, but I do rate this as a must for anyone wanting to get into this subject. This is a fine example of the rigorous yet clear exposition developed by Soviet mathematicians of the 50s, 60s and beyond. There's only one problem: I am somewhat familiar with this material yet still find it tough going despite a wonderful text and clever writers
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on 24 October 2005
I found this to be a book different from most usual texts that don't talk about applications as much as they should. It was a great treat to see the perfect blend of rigorous theory and intuitive explanation using examples. A great book to get you started...
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on 3 August 2009
This is arguably the finest peice of work on the subject but a word of caution is required. This is a concise and difficult work on the subject and really should only be considered by postgraduates. I don't think it's fair to consider it as an introduction to this very powerful and underused subject.
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on 17 January 2009
The book is an excellent mathematical exposition of the Calculus of Variations. However, I had bought the book in order to a) understand the principles and b) apply it to some optimisation problems in engineering that I was dealing with. If the book had addressed the latter more directly, it would have been ***** review from me. For someone working on engineering problems, the first 2 chapters are excellent if studied in parallel with worked examples from elsewhere.
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VINE VOICEon 2 July 2014
* Introduction

I have chosen this book to further explore the topics around `Calculus of Variations'. Previously, I read and reviewed another book on this mathematical topic named `The Introduction to the Calculus of Variation's' that is also published by Dover. Just to make it clear this following book is not written by the author of the first book. This second volume is very well bound for a paperback, and its texts and graphics are both in black and white. Also of note is that the font sizes are adequate for those of us who may require glasses to read. The contents of this book are based on a Moscow state university series of courses, but with the original author's permissions, these have been built upon and explored more deeply by the author who is also its translator into English. Dover, the publisher, has a tendency to reprint older, perfectly usable Math books at budget prices. This book has been read from May to July 2014.

* A-level, H.N.D, Undergraduate, Graduate?

The book is designed for advanced undergraduate (3rd year level Hons) and graduate level studies.

* What is Calculus of Variations used for?

The basic reasons for the study of this topic is to calculate finite -difference approximations to functions using linear methods with in areas arising in topics such as Analysis, Mechanics, Geometry that must apply technique's using continuously differential functions that are within [a, b]. These can be accommodated several variables members in these approximations. Such calculations, such as to derive the length of a function. Or a where a derivative is zero, so finding a local maximum or minimum for example. These equations are different in that they can possess several variables. Unlike usual Mechanics having physical equivalents of up to the forth derivative, these can possess equations with many finite n - derivations.

* Concisely, what topics are covered?

The general equation that drives these calculations, and the whole book, is a generally applicable `Euler's equation'. These can apply `Taylor series' to generate the whole membership of a custom function series to be solved. This short and rather wonderful equation is reused and built upon several times to explore a whole raft of technique's; some using symbolic notation upon matrices, many others applying symbolic techniques using partial differential equations in the usual level of mathematical short - hand to keep the book shorter than it would otherwise be. It's the method that is being explored and is more important and not the specific equation. To explore the partial differential equations -that can be non-linear- the Euler equation is expanded to Hamilton - Jacobi equation representing canonical equation equivalents representing the characteristic system. This is further explored symbolically to apply to say for an example, 3-D (x, y, z, t) fields by using first derivative equivalents to map a finite number of particles to find where its reaches zero. As before, the classical mechanics equations exploring conservative systems in equilibrium with n - finite number of particles and kinetic and potential energies with observation of Lagrangian rules. Often present with multiple partial integrals. This level of mechanics I felt it became very helpfully explained, but not delving too deeply into another topic when exploring this component of variation topics.

* Summary

I hope my humble summing up is not just a tissue of my butchered misunderstandings! But I have really enjoyed this book and have a better comprehension of these deeper calculus techniques that are written so much within concise mathematical symbolic notation. Its not at a super deep level, but its helpful along the ways to gain new comprehensions.
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on 24 May 2014
This book provides good coverage of the material but its commentary is awful. The second sentence is an example of this.

People buy books such as this because they do not know the material. Obfuscating that material by adhering to opaque verbiage is unhelpful.

It's about time publishers rejected pretentious presentations in favour of lucid, transparent text.

It would not be a mistake to buy this book but be prepared to work much harder that you ought to to understand it!
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on 7 May 2015
A must read for students of optimal control. Rigorous proofs of the main results are given. Many interesting problems are included at the end of each chapter. I found the discussion of Jacobi condition and conjugate points illuminating.
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on 18 May 2016
The book offers a complete mathematical introduction to the calculus of variations, but in an advanced level. Graduate students would benefit from it.
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on 6 February 2015
The book is clear. Suits perfectly for my Calculus of Variations-course :)
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