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First six chapters include theory of fields and sufficient conditions for weak and strong extrema. Chapter 7 considers application of variation methods to systems with infinite degrees of freedom, and Chapter 8 deals with direct methods in the calculus of variations. Problems follow each chapter and the two appendices. Fresh, lively text is ideal for advanced undergraduate and graduate students in math and physics.
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Customer Reviews (15)

Ok if you are a mathematician
Good if you know the subject or are a mathematician. Arfken's Mathematical Methods for Physicists book has a section that is better than this book. The problem is that it is not extensive.

Very useful
This book is really easy to get into. The first chapter on its own is worth the purchase price. After thumbing through that, I was in a position to start solving interesting problems. I was up and running with the basics within a day or so. Other books on the Calculus of Variations are a lot harder to get into and much more difficult to readily apply to problem solving.

Other books that present variational principles generally do it formally in terms of Frechet and Gateux derivatives on Banach spaces, whereas here, the approach is a little bit more ad hoc. The former approach can be heavy going for people who aren't comfortable with functional analysis and even then, actually solving specific problems can be headache. Fomin and Gelfand on the other hand demand very few prerequisites from their readers.

Encompassing concise and comprehensible
My acquintance with the book starts with a course in
calculus of variations in Israel Institute of Technology - Technion.
I recommend this book as a prepration for the advanced
methods in chapter 8 of Tensors Differential Forms and Variational Principles - David Lovelock and Hanno Rund. Anyone who is about to study a course in General Relativity should take a year or leisure two years and read both books. I also recommend this book to anyone who deals with
shape matching, signal processing and image processing.

Quite clear, straight-forward explanations
I like this book, certainly what I have read of it. I'm now digging around near page 75, and it struck me that for the first time I really "get it" - In particular, this was about the use of the Legendre transform. The authors start with a very gentle introduction of it, and then, while things become more abstract, the text never jumps too far and never leaves the reader too much in puzzling. It's ought to be studied though, and it's not "easy".

Calculus of Variations
This book is a "must have" for those wanting to study topics in functional analysis. ... Read more

Product Description
Book basically divided into two parts. Chapters 1-4 include background material, basic theorems and isoperimetric problems. Chapters 5-12 are devoted to applications, geometrical optics, particle dynamics, the theory of elasticity, electrostatics, quantum mechanics and other topics. Exercises in each chapter. 1952 edition.
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Customer Reviews (5)

Calculus of Variations
The book is very good

Will never collect dust
The importance of variational calculus cannot be overstated, as it now is being applied to myriads of different areas, these going way beyond the routine applications in physics and engineering. Indeed, it has been found to be useful in economics, network engineering, financial modeling, computational radiology, and in the new field of constraint programming. This book is now a classic, and has been the standard reference and textbook for variational calculus since its first date of publication. There have been many excellent books on variational calculus that have appeared since this one first did, but they do not provide the needed intuition in learning the subject. The clarity of the author's presentation in this book is outstanding, and, most importantly, he assigns challenging exercises to test the readers understanding of the subject. When the book was first published, computers were not being used as they are today, and the current strategy in solving variational problems is usually to use some sort of numerical software packages. In addition, the use of symbolic programming languages, such as Mathematica and Maple, have made the derivations in variational calculus almost trivial, and this is a promising trend since it allows the user to concentrate more on the concepts involved in applying it. Also, very efficient numerical routines exist in Fortran, C, and C++ that are designed and optimized for variational problems.

All of the beginning material on the calculus of variations is covered in the book, and its application to Lagrangian and Hamiltonian mechanics, elasticity, quantum mechanics, and electrostatics. Isoperimetric problems are treated, the vibrating string and membrane, and the Sturm-Liouville system and its origin as an eigenvalue problem in variational calculus. In the latter, the reader can see the origing of some of special function solutions, such as the Bessel functions and Laguerre polynomials. The reading of this book will amply prepare the reader for applying it to problems in quantum field theory, economics, radiology, financial engineering, logistics, optimization theory, computational geometry, control theory, and the theory of evolutionary strategies. In addition, the book will allow the reader to move on to more advanced areas of mathematics, such as the theory of minimal surfaces, Morse theory, and geometric measure theory.

A standard work on variational calculus
This is the ultimate applications of variations book. The book develops the variational calculus in a pretty succinct manner and then surveys areas of application such as vibrating membranes, optics, hamiltonian mechanics, elasticity theory, quantum mechanics and electrostatics. The book has a pretty reasonable amount of problems which tend to be fairly soluble and
have the feel of an engineer's book rather than a mathematicians text. Weinstock was a Stanford math professor,so the book is

fairly rigorous but the emphasis is not on proofs. Many books on this subject require an advanced knowledge of analysis including such things as the Lebesgue integral etcetera. This is not required here this is a book that a physicist or engineer will be entirely comfortable with. There are other books that introduce the fundamentals in a shorter space Gelfand and Fomin
being my favorite, but this book covers the basics from a pretty practical perspective and includes more applications than any other calculus of variations book I know of.

The book is good DESPITE its claims for mathematical rigour
Contary to the previous reviewer I generally do not care much ifpresentation is rigorous or not.It is much more important that the book can be understood easily, contains a lot of thoroughly selected exampleswhich illustrate general principles. In fact, the book is written in a wayallowing one to understand what isthe most essential in the subject evenif all proofs were omitted. Such a feature is a fingerprint of a classic.

I strongly recommend it for a standard course on variational calculus.

A brief , but wondeful textbook in Calculus of Variations
When I firstly see this book, I don't take it seriously for its length. But after I read it, I find it's an appropriate textbook for its brevity,rigor in mathematics,and a lot of examples helping you toillustrate its application. I think the most valuable point of this book isthe author's rigor in mathematics,which are often neglected by mostphysicists.Secondly is its master use of deduction. I recommend itheartedly. ... Read more

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Customer Reviews (2)

Geared for classical mechanics
Good books on the calculus of variations are not thick on the ground, and this one is pretty good. It is organised around explicating Hamiltonian mechanics, but with mathematical rigor and clarity.Why not 5 stars then? Some proofs skip a lot of steps. There are not a lot of examples. The typesetting is from 509 years ago, with equations and so on in the text not on a separate line. There are few diagrams.

If your interest is in physics this is a FAR better book to choose than Sagan.

Excellent Book on the Calculus of Variations
Thorough, concise, enjoyable and CHEAP book on the Calculus of variations. One of the few mathematics books that is simultaneously rigorous and entertaining. Illustrated with many examples and applications, drawn fromphysics and engineering. The proofs are brisk, though there is enough thereto guide you on your way to complete understanding. A DEFINITE read for anymathematician or scientist! ... Read more

Product Description
The calculus of variations is one of the oldest subjects in mathematics, yet is very much alive and is still evolving. Besides its mathematical importance and its links to other branches of mathematics, such as geometry or differential equations, it is widely used in physics, engineering, economics and biology.

This book serves both as a guide to the expansive existing literature and as an aid to the non-specialist -- mathematicians, physicists, engineers, students or researchers -- in discovering the subject s most important problems, results and techniques. Despite the aim of addressing non-specialists, mathematical rigor has not been sacrificed; most of the theorems are either fully proved or proved under more stringent conditions.

In this new edition, the chapter on regularity has been significantly expanded and 27 new exercises have been added. The book, containing a total of 103 exercises with detailed solutions, is well designed for a course at both undergraduate and graduate levels.

Contents: Preliminaries; Classical Methods; Direct Methods: Existence; Direct Methods: Regularity; Minimal Surfaces; Isoperimetric Inequality; Solutions to the Exercises. ... Read more

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Customer Reviews (1)

Calculus of Variations by Lev D. Elsgolc
The reader is gently introduced to the problems solved by the Calculus of Variations and each solution is given in sufficient detail. ... Read more

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Excellent text provides basis for thorough understanding of the problems, methods and techniques of the calculus of variations and prepares readers for the study of modern optimal control theory. Treatment limited to extensive coverage of single integral problems in one and more unknown functions. Carefully chosen variational problems and over 400 exercises. "...should find wide acceptance as a text and reference"—American Mathematical Monthly. 1969 edition. Bibliography.
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Customer Reviews (4)

Good but could be better
I really like ("plain") books that are written in the definition, lemma, proposition, theorem, and examples style. Sagan does enunciate propositions, theorems, etc. but he does that after giving proofs for the things he will state latter. I simple don't like this style. Another point that I really dislike in Sagan book is the notation. He doen't change the notation, for instance, when he talks about a function y and the y coordinate. This is a little bit boring. Moreover, the exercises aren't stated clearly (I always have to look at the text, many pages before the exercise, to know what the exercise demands to be done). Besides this the book is very complete. A good but that could be better.

Wonderful
This book is neither too naive nor too brief, still I was able to read more than 80 pages in one day, simply because it is written very clearly. An excellent introduction in Calculus of variations, very concise and very clear. Some context is devoted to control theory, which can be considered as a generalization of calculus of variations.
The part on field embedding and Hilbert invariant integral is perfectly wriiten, actually the best that I have ever encountered.

Clear and well written
I used this book as a compliment to a Theoretical Mechanics course. The section on Hamilton's equations is especially well written. Although probably more mathematically rigorous than anything I needed, the style is so silky smooth that anyone interested in mathematical physics will surely enjoy it. And it will be a surprise for many to find that the "proof" of equivalence between the Lagrangian and Hamiltonian formulations presented in most texts is half incomplete.
The book is complemented by good examples, clear notation and quite a number of graphics. Of course, proofs and arguments are absolutely rigorous, but well explained. This is a mathematics text, after all. I strongly recommend it, as well as any other of Sagan's books.

Very nice book on variational calculus
This book is a more thorough treatment of the calculus of variations than some of the other books on the subject that you could buy.It's bit more expensive...I especially reccomend this book if you like Sagan's other book "Boundary and Eigenvalue Problems in Mathematical Physics". If you like his style of writing, and I do, he again shows nice touch ofexplaining the material in writing and then with examples.A good bookwith a nicely written and more thorough treatment of the subject (unlessyou want a full price text that is).. ... Read more

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Suitable for advanced undergraduate and graduate students of mathematics, physics, or engineering, this introduction to the calculus of variations focuses on variational problems involving one independent variable. It also discusses more advanced topics such as the inverse problem, eigenvalue problems, and Noether’s theorem. The text includes numerous examples along with problems to help students consolidate the material.

Customer Reviews (5)

A good introduction
Easy to read but still appropriately rigorous. No real coverage on direct methods or optimal control are perhaps the most glaring omissions. Plenty of worked examples make the book appropriate for self study, but exercise solutions would improve it in this regard.

Excellent book
This is one of the most clear, easily-written book for calculus of variations.Everything is explained, not hiding any trick.
Also, it covers variety of topics, including nonholonomic constraints.

Readers are required to have some knowledge on real analysis. I am an engineer, so I have no background on analysis, but the book is still readable.My only complain is that it would become better if it has engineering examples (most examples are too basic or requirs proof).But still, this is one of the best book on calculus of variations !!

Excellent.
Motivated and an enjoyable read while maintaining rigor and has a lot of examples too. I would recommend it highly for self study.

A very good introductory book to Calculus of Variations
A very good introductory book to Calculus of Variations. Very clear expounded. It goes step by step with many examples like:braquistocrone, isoperimetric problem, Dido's problem and others , used to justify the theory and contents explained. A very easy readable book, amazing to read, one goes deeper and deeper almost without notice it.
Maybe some theoretical questions are not covered here, like in "Gelfand and Fomin"'s book, but I think that this is largely compensated for its clearness.

A delightful read.
Unlike so many math texts which seem to be compiled by computers, this book is very personable, interesting, and easily digested.If you are a senior math undergraduate, or a graduate student in science, this is great bedtime reading; a younger (strong) student who knows calculus would also profit from this book.Books of this quality remind math students how math can be so enjoyable.

Having said that, it is not intended to be an advanced reference.But if you are a researcher, you know that. ... Read more

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Shortly after the invention of differential and integral calculus, the calculus of variations was developed. The new calculus looks for functions that minimize or maximize some quantity, such as the brachistochrone problem, which was solved by Johann Bernoulli, Leibniz, Newton, Jacob Bernoulli and l'Hopital and is sometimes considered as the starting point of the calculus of variations. In Woodhouse's book, first published in 1810, he has interwoven the historical progress with the scientific development of the subject. The reader will have the opportunity to see how calculus, during its first one hundred years, developed by seemingly tiny increments to become the highly polished subject that we know today. Here, Woodhouse's interweaving of history and science gives his special point of view on the mathematics. As he states in his preface: ``Indeed the authors who write near the beginnings of science are, in general, the most instructive; they take the reader more along with them, show him the real difficulties and, which is the main point, teach him the subject, the way they themselves learned it.'' ... Read more

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The updated 2nd edition of this book presents a variety of image analysis applications, reviews their precise mathematics and shows how to discretize them. For the mathematical community, the book shows the contribution of mathematics to this domain, and highlights unsolved theoretical questions. For the computer vision community, it presents a clear, self-contained and global overview of the mathematics involved in image procesing problems. The second edition offers a review of progress in image processing applications covered by the PDE framework, and updates the existing material. The book also provides programming tools for creating simulations with minimal effort.

Customer Reviews (1)

It is my bible!
This book not only includes the state-of-art image processig techniques using math methods, but also provides very good numerical schemes for these methods. This book is written by mathematicians, but engineers still can easily implemente many commonly used image processing algorithm, just follow the detailed numerical difference schemes provided in the appendix. ... Read more

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This textbook on the calculus of variations leads the reader from the basics to modern aspects of the theory. One-dimensional problems and the classical issues such as Euler-Lagrange equations are treated, as are Noether's theorem, Hamilton-Jacobi theory, and in particular geodesic lines, thereby developing some important geometric and topological aspects. The basic ideas of optimal control theory are also given. The second part of the book deals with multiple integrals. After a review of Lebesgue integration, Banach and Hilbert space theory and Sobolev spaces (with complete and detailed proofs), there is a treatment of the direct methods and the fundamental lower semicontinuity theorems. Subsequent chapters introduce the basic concepts of the modern calculus of variations, namely relaxation, Gamma convergence, bifurcation theory and minimax methods based on the Palais-Smale condition. The prerequisites are knowledge of the basic results from calculus of one and several variables. After having studied this book, the reader will be well equipped to read research papers in the calculus of variations. ... Read more

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The subject of calculus of variations is to find optimal solutions to engineering problems where the optimum may be a certain quantity, a shape, or a function. Applied Calculus of Variations for Engineers addresses this very important mathematical area applicable to many engineering disciplines. Its unique, application-oriented approach sets it apart from the theoretical treatises of most texts. It is aimed at enhancing the engineer’s understanding of the topic as well as aiding in the application of the concepts in a variety of engineering disciplines.

The first part of the book presents the fundamental variational problem and its solution via the Euler–Lagrange equation. It also discusses variational problems subject to constraints, the inverse problem of variational calculus, and the direct solution techniques of variational problems, such as the Ritz, Galerkin, and Kantorovich methods. With an emphasis on applications, the second part details the geodesic concept of differential geometry and its extensions to higher order spaces. It covers the variational origin of natural splines and the variational formulation of B-splines under various constraints. This section also focuses on analytic and computational mechanics, explaining classical mechanical problems and Lagrange’s equations of motion.

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This long-awaited book by two of the foremost researchers and writers in the field is the first part of a treatise that will cover the subject in breadth and depth, paying special attention to the historical origins, partly in applications, e.g. from geometrical optics, of parts of the theory. A variety of aids to the reader are provided: the detailed table of contents, an introduction to each chapter, section and subsection, an overview of the relevant literature (in Vol. 2) plus the references in the Scholia to each chapter, in the (historical) footnotes, and in the bibliography, and finally an index of the examples used throughout the book. Later volumes will deal with direct methods and regularity theory. Both individually and collectively these volumes will undoubtedly become standard references. ... Read more

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Real Analysis and Applications starts with a streamlined, but complete, approach to real analysis. It finishes with a wide variety of applications in Fourier series and the calculus of variations, including minimal surfaces, physics, economics, Riemannian geometry, and general relativity. The basic theory includes all the standard topics: limits of sequences, topology, compactness, the Cantor set and fractals, calculus with the Riemann integral, a chapter on the Lebesgue theory, sequences of functions, infinite series, and the exponential and Gamma functions. The applications conclude with a computation of the relativistic precession of Mercury's orbit, which Einstein called "convincing proof of the correctness of the theory [of General Relativity]." The text not only provides clear, logical proofs, but also shows the student how to derive them. The excellent exercises come with select solutions in the back. This is a text that makes it possible to do the full theory and significant applications in one semester. Frank Morgan is the author of six books and over one hundred articles on mathematics. He is an inaugural recipient of the Mathematical Association of America's national Haimo award for excellence in teaching. With this applied version of his Real Analysis text, Morgan brings his famous direct style to the growing numbers of potential mathematics majors who want to see applications along with the theory. The book is suitable for undergraduates interested in real analysis. ... Read more

Customer Reviews (1)

Great Intro to Advanced Math
" Real Analysis and Applications" by Frank Morgan is probably one of the best introductory analysis books that I have come across. It gives a basic overview over real analysis without getting too hung up on a specific topic. Though it includes most important proofs, it stays away from too complex and lengthy ones. The book contains a very detailed explanation that goes with every proof. The author keeps complicated matter as simple as possible and approaches it somewhat with humor. It is a great introductory book, especially for people that haven't been exposed to the material before. If you are already an analysis professional you probably might want to stay away from this book, since this book is really primarily for the advanced calculus beginner. ... Read more

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In this second English edition of Carathéodory's famous work (originally published in German), the two volumes of the first edition have been combined into one (with a combination of the two indexes into a single index). There is a deep and fundamental relationship between the differential equations that occur in the calculus of variations and partial differential equations of the first order: in particular, to each such partial differential equation there correspond variational problems. This basic fact forms the rationale for Carathéodory's masterpiece. Includes a Guide to the Literature and an Index.From the Preface: "The book consists of two parts. In the first part, I have made an attempt to simplify the presentation of the theory of partial differential equations to the first order so that its study will require little time and also be accessible to the average student of mathematics ... The second part, which contains the Calculus of Variations, can also be read independently if one refers back to earlier sections in Part I ... I have never lost sight of the fact that the Calculus of Variations, as it is presented in Part II, should above all be a servant of Mechanics. Therefore, I have in particular prepared everything from the very outset for treatment in multidimensional spaces." ... Read more

Customer Reviews (1)

supreme
I have followed the Analytical Mechanics class in my gradute school from this book.
It is supreme and Goldsteins book only is a cheap make up in compared to this book.
Thank you for folowing this book Dr.Yurdahan Guler, at that time
I used to think that you were making our life miserable,but now
I think it was the right thing to do to follow a book written by a true genius. ... Read more

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The calculus of variations is used to find functions that optimize quantities expressed in terms of integrals. Optimal control theory seeks to find functions that minimize cost integrals for systems described by differential equations. This book is an introduction to both the classical theory of the calculus of variations and the more modern developments of optimal control theory from the perspective of an applied mathematician. It focuses on understanding concepts and how to apply them. The range of potential applications is broad: the calculus of variations and optimal control theory have been widely used in numerous ways in biology, criminology, economics, engineering, finance, management science, and physics. Applications described in this book include cancer chemotherapy, navigational control, and renewable resource harvesting. The prerequisites for the book are modest: the standard calculus sequence, a first course on ordinary differential equations, and some facility with the use of mathematical software. It is suitable for an undergraduate or beginning graduate course, or for self study. It provides excellent preparation for more advanced books and courses on the calculus of variations and optimal control theory. ... Read more

Customer Reviews (1)

Very methodical introduction
The material was introduced in a very organized way, building up from simple examples that motivate the ideas of the calculus of variation and eventually moving into the extension of those ideas into the theory of optimal control.If you know only basic calculus, you know enough to work through the material in this book.

Each chapter comes with a few problems that verify one's understanding.The problems are simple enough that, if you really understand the material, you can work them quickly and determine rapidly whether you got the idea or missed something.Nice diagrams help to visualize what is going on, even for some of the problems.

When the material transitions to the theory of optimal control, it gets slightly confusing, I think because the basic ideas of optimal control are not first layed out separately from the description of how to solve them with the calculus of variations.

The book is a "primer".For a more advanced treatment of the calculus of variations (but not optimal control), consider the excellent but inexpensive "Calculus of Variations" by I. M. Gelfand and S. V. Fomin.
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Highly esteemed author

Topics covered are relevant and timely