Customer Reviews

Advanced Calculus: A Differential Forms Approach

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I can't believe that nonone has ever bothered to review this book. This is an outstanding book, filled with valuable insights. The author introduces differential forms right from the beginning. He provides associations with the utility of differential forms in Mathematical Physics through many examples. There is a lot of material in this book that cannot be covered easily, so the prospective reader is advised to be patient and initially skip sections when necessary. This is a book about advanced calculus via differential forms written with great care by someone who has thought things through very thoroughly. The book has all the attributes of a classic:

1. Excellent explanations and plenty of examples. 2. Conceptual clarity of key ideas, a rare feature these days. 3. Solutions of all the exercises in the book (truly a lot). 4. Rigorous but not terse mathematics.

Having read this book the reader can easily proceed to address more advanced topics without hesitation. I personally would have liked to have seen more applications in mathematical physics but this is by no means a criticism. The author wrote a book that is about the concept of differential forms in advanced calculus and in that he has succeeded admirably. Apparently this book was first published in 1969 and has gone in and out of print over the last three decades. So hurry up, go out and buy this book.

This book is very well written, and being a reprint of a second edition, has very few misprints. Forms and their calculus are introduced in the 3D Euclidean space familiar to students of Calclulus 3 or University Physics. Later in the book the fundamentals like Stokes' Theorem are generalized to higher dimensions and manifolds. This book is suitable for anyone who has had Calculus 3 and is interested in a better way to do multiple-variable calculus. (Just to whet your appetite: after reading this book, you won't have to remember Green's Theorem, Stokes' Theorem, or the Divergence Theorem separately, they are all one compact and simple theorem {the one on the cover} in the language of Differential Forms.)

I, personally, had a lot of trouble with Flanders' and Darling's books on differential forms until I read some outside material on the subject. This book, however, starts off on a more basic level, and does not demand half the mathematical maturity that Flanders or Darling do. For more advanced topics, Flanders and Darling are fine books to go to, after getting the basics right here.

Reading this book reminds me of "Feynman Lectures in Physics" : An extremely refreshing view of analysis.

The author's point of view is that the theory of functions of multiple variables is very naturally understood if approached from the differential forms angle. And that the best public for that is the undergrad student. Well, he makes his case.

The book is not written in the usual math style (theoreme,lemma,proof,...) and always exhibit the beauty behind the idea. In the first chapter, forms are introduced very naturally with example taken from work, flows and so on. Chapters 2 and 3 are devoted to integrals, integration and differentiation and that's where he unleashes all the power of forms before you notice it. From the fundamental theorem of calculus ($\int_a^b f(x)dx=F(b)-F(a)$) he deduces the general stokes theorem on integration on manifolds and show why the exterior derivative is defined as it is. Chapter 4 talks about linear algebra, again demystifying the implicit function theorem when exetended to differential maps (chap5). Chapter 6 is where everything get prooved rigourously. Chapter 8 is a real gem, showing various application of forms. There are classical applications such as the integrability conditions, Maxwell Equations and special relativity. And very original ones such as revisiting harmonic functions and functions of complex variables.

I wonder why this book is not taught as a classic textbook everywhere.

I'm just gonna come out and say it: this book is the best treatment of multivariable calculus that I've seen.

Unlike the usual multivariable textbook, this book gives lucid, clear, and elegant explanations and proofs for nearly all principles introduced, i.e. the method of Lagrange Multipliers. The author never keeps you guessing; he starts low and builds up quickly and brilliantly.

Chapters 1-3 serve as a good intro to differential forms and the fundamental theorem, almost worthy of their own book. Chapters 4-6 supply the theory that 1-3 motivate, and while they can get a little tough, if the reader has a pencil and paper at the ready, the results will be reaped. And then there's chapters 7-9. Each one of these chapters could be taken on its own, out of context with the rest of the book, and still serve its purpose. Chapter 8 is a true gem; only a knowledge of the opening chapters is necessary for an understanding of the interesting topics that are presented, and these serve as a great segue into more complicated materials. Chapter 9 is a standard treatment of analysis; again, it is complete enough to deserve its own publication.

What makes this book truly great is its versatility. A reader could start at the beginning, the end, or somewhere in between. There are many, many exercises that range from tests of basic understanding to proofs of more advanced results. Best of all, solutions, or at least hints are available for EVERY problem, which is very underrated. In short, this book is phenomenal for any student with a rudimentary knowledge of basic calculus who is interested in developing a full, broad knowledge multivariable calculus, getting a great intro to real analysis, and learning to connect all these things to more advanced fields.

P.S. This book is worth buying just for an explanation of the theorem on the cover; it is probably the most useful theorem in analysis, and it makes memorizing Green's theorem, Stoke's theorem, etc. seem barbaric.

The first three chapters of this book are worthy of separate publication. They could be read by any bright undergraduate with full comprehension, and they introduce in a marvelously clear way the unifying power of forms defined on an ambient Euclidean space using basic examples from physics (work and flow).

Throughout the author clearly demonstrates the need for mathematical rigor. Whenever he uses an informal example or argument, he will always conclude the section by analyzing why a rigorous argument is needed and often outlining how such an argument could be achieved. Later on in the book (in the sixth chapter) he will finally develop all the arguments rigorously in full depth.

After this third chapter, however, the book starts becoming less elegant and more tedious. Linear algebra is discussed--but without any of the modern notation! Vectors are a rare character here, and matrices are scantly used other than to define ideas. Instead, you will be bombarded with a hoard of individual variable names. Keeping track of exactly what's going on with all the variable names and summations becomes a task of mental endurance, not ingenuity or understanding. Some modern terminology is actually discussed, such as vector spaces and linear transformations, but not until the end of the chapter on linear algebra, effectively defeating the point. It is as if the material were tacked on just to make the book conform more to the standard content coverage.

Note that here you will not find a k-form defined as a member of the k-th exterior power of the cotangent bundle of a manifold. Rather than such an abstract definition, this book is far more down-to-earth and hence will allow readers who do not have serious mathematical training to grasp the power and beauty of forms. Depending on your previous familiarity with forms and on your mathematical background, this is a plus or a minus. For me, it was certainly a plus because until recently the abstract definition I provided above was meaningless garbage to me.

Overall, this is a book that would be best thumbed through at a book store so you can decide if it's worth your time and if the author's style meets your taste. It's a very well-written book with plenty of fresh insights and a novel approach. Mistakes are nearly impossible to find. The author has a powerful and humbling command of mathematics. Unfortunately, the notation was often too outdated for my taste and hindered not only my enjoyment of the book but also my ability to fully understand concepts that appear difficult here because of the onslaught of symbols but which are really rather straightforward in modern notation. But I suppose some people may prefer the different notation.

Once in a long while you get to find a gem hidden between pages of a text. Edwards offers a powerful and stunning presentation of differential forms unlike any other. I wish I could say something other than "read it and see", but any review I can offer will not do justice to Edwards' stunning effort. The presentation is VISUAL while honoring the rigor of sound mathematics. It's simply -- stunning.

Think of it this way... Feynman is to physics as Edwards is to differential forms.

Chapter eight is nothing short of artistic expression in mathematics and science. That chapter, alone, should be studied by educators and integrated into modern instruction.

The differential forms approach has considerable intuitive appeal as well as capturing more useful math for the physics or engineering student than the conventional approach. Edwards is a little too much the mathematician. The text misses the mark for the typical physics or engineering student who has taken only the usual calculus sequence and needs a little more intuitive introduction and to be led into the abstraction more gently. A more geometric approach might have been useful. I would have introduced the wedge product explicitly with a geometric explanation in terms of vectors.

My objective in purchasing the book was to fill in my background on the subject the easy way after pretty much figuring out what it is all about. For that the book is fine. But back in 1959 when I took advanced calculus, I think I would have found the books difficult without a good teacher to help me along. The book is probably not what I'll use for a course.

I found this book to be a beautiful introduction to multivariable calculus using the language of differential forms. This is a must-have perspective on multivariable calculus. However it is very long. Over 400 pages! This book also assumes very little background, and thus it uses ugly and painstaking notation, rather than more comprehensible and easier-to-read abstract notation such as matricies, (except for the 4th and 8th chapters), etc. Still I recommend this book to anyone wanting a different perspective on multivariable calculus.

This book tries to be accessible to a very wide audience. In Australia, where I did my undergraduate studies this could have been read by a beginning first year university student.

The author does not even suppose a prior knowledge of linear algebra.

The result is that the reader will confronted with a swarm of u's v's,w's,x's, y's and so on, rather than more abstract notions of functions. In fact this book looks decidedly 19th century in places. This is the opposite to a book by Serge Lang ,Dieudonné or Rudin. To be fair the author has gone to great lengths to motivate the mathematics and for this reason it may well be very popular with engineers and physicists.

However, in my mind it makes it harder to see the mathematical forest for the trees. On the other hand, if you are prepared to slog though it all you will be able to use this book on its own as a self study book.

In the end I found a book more suited to me for the concepts of forms: V. Arnold's Mathematical Methods of Classical Mechanics.

When I first read this book, it was before really learning algebraic topology. If you are like this, and you do not know algebraic topology, then this book will not be suited for you.

This book really makes the relation between homology (in algebraic topology) and integrals (in calculus and analysis) explicit. Or it begins that way.

So after learning algebraic topology, then returning to this book, what a pleasure it was to read! It was so beautiful how this relationship was described, etc.

But if you are, e.g., a physics or engineering student and want something a little more advanced in calculus...don't use this book.