Calculus of Variations

By Robert Weinstock

Book Overview

This book by Robert Weinstock was written to fill the need for a basic introduction to the calculus of variations. Simply and easily written, with an emphasis on the applications of this calculus, it has long been a standard reference of physicists, engineers, and applied mathematicians. The author begins slowly, introducing the reader to the calculus of variations, and supplying lists of essential formulae and derivations. Later chapters cover isoperimetric problems, geometrical optics, Fermat's principle, dynamics of particles, the Sturm-Liouville eigenvalue-eigenfunction problem, the theory of elasticity, quantum mechanics, and electrostatics. Each chapter ends with a series of exercises which should prove very useful in determining whether the material in that chapter has been thoroughly grasped.
The clarity of exposition makes this book easily accessible to anyone who has mastered first-year calculus with some exposure to ordinary differential equations. Physicists and engineers who find variational methods evasive at times will find this book particularly helpful.
"I regard this as a very useful book which I shall refer to frequently in the future." J. L. Synge, Bulletin of the American Mathematical Society.

• Format: Paperback
• Language: English
• ISBN: 0486630692
• ISBN13: 9780486630694
• Release Date: June 1974
• Publisher: Dover Publications
• Length: 352 Pages
• Weight: 0.87 lbs.
• Dimensions: 0.7" x 5.4" x 8.5"

Customer Reviews

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 andhave the feel of an engineer's book rather than a mathematicians text. Weinstock was a Stanford math professor,so the book isfairly 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 Fominbeing 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 if presentation is rigorous or not. It is much more important that the book can be understood easily, contains a lot of thoroughly selected examples which illustrate general principles. In fact, the book is written in a way allowing one to understand what is the most essential in the subject even if 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 to illustrate its application. I think the most valuable point of this book is the author's rigor in mathematics,which are often neglected by most physicists.Secondly is its master use of deduction. I recommend it heartedly.