Customer Reviews

Calculus of Variations with Applications (Mathematics Series)

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I have had this gem of a book for ten years, and I'm still enjoying it. Actually, it turns out that the text itself was written some time ago, but don't let you be put off by that - this book is very scholarly and you really will be enlightened by it.

Now, where do you begin? Usually at University, I guess, this subject is treated in a way that makes most folks think that just about all "Calculus of Variations" problems can be solved with the Euler Lagrange equation, and from there on, you just have to solve the resulting differential equation. And I guess that's sometimes possible. But in the real world, sometimes you end up with that equation simply not working, and your problem is more messy and not expressible using "nice" functions. Now, this is where this book begins. We start by looking at "sufficient" and "necessary" conditions for solutions - and these are not the same things! This at least allows you to work out whether a solution is there, for goodness sake, before you waste time trying to find it.

Ewing does better in later chapters. He showcases a whole slew of problems which deserve special consideration, and this gets at times really exciting, covering all sorts of ideas about what we mean by optimum values for integrals, and how to specify systems of equations when one method doesn't really work too well.

One question which he digs into, which is very entertaining, is the problem of what an integral of a function really means. For example, we all know about the Riemann integral - the limit of a sum - but had you ever heard about the Weierstrass integral - or the Lebesgue integral?

Mr Ewing serves up these exotic and flavorsome new varieties in a most satisfactory fashion, with lots of examples to help.

The text never gets too far away from real problems. How easy that would have been! This book is so amazingly practical and also deeply committed to a really thorough treatment. He also gives excellent commentary on the history of this subject.

Overall I would say that this book is not one which belongs on the shelf with the numerical methods books, or the operational research section (thats what numerical optimisation is called over here in Britain), but I would get this now anyway before it goes out of print (if Dover are so crazy as to do that).

This book has one previous Amazon customer review, with 5 stars and a glowing review. I do not dispute that reviewer's assessment; rather I'll observe that his familiarity with calculus of variations is much deeper than mine. I find this book almost impossible to learn from, although there is much in it that I would like to learn. But I confess that I have difficulty with calculus of variations, and always have had.

For me, calculus of variations breaks neatly into two eras, before Weierstrass and after Weierstrass. Before Weierstrass there was no rigor in the subject, and practitioners from Bernoulli to Lagrange used whatever methods they could devise to solve particular problems, or classes of problems, that attracted their attention and that seemed to be tractable. Weierstrass turned the subject into a rigorous one, and was followed by a number of creative theoreticians, some of whom could also solve practical problems: Bolza, Hilbert, Bliss, McShane (whom I knew) and various others. They created mathematical theory that would have been incomprehensible to Euler, but which eliminates the slop in the subject.

When I look for literature on calculus of variations I seem to find the literature also divided into "before Weierstrass" and "after Weierstrass". I have no trouble with the "before Weierstrass" sort, but I would like to go further. And the trouble I have with this book is that despite the promise of a jacket blurb, it doesn't show how the subject developed from an intuitive art into rigorous mathematics; it determinedly sticks to the "modern" viewpoint. I note with amusement, for example, that Lagrange is mentioned only in one slightly pejorative paragraph on page 109, with no indication of how Lagrange actually solved variational problems.

So this book will sit in my personal collection of math books until I get tired of seeing it around and get rid of it (which is what happened to my copy of Bliss many years ago). Is there some text on calculus of variations that will permit a reader like me, with reasonable knowledge of measure theory, modern algebra and topology, and classical analysis, to learn how the calculus of variations has progressed in the last hundred years? This isn't it.