Customer Reviews

Differential Forms with Applications to the Physical Sciences (Dover Books on Mathematics)

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This book introduces differential forms to physicists. It is suitable for those with no advanced training in math other than multivarible calculus and linear algebra. No prior knowledge of topology is needed, although it doesn't hurt. The book starts with a good intro explaining the motivation for forms and when they might be useful over tensors and vectors. The exterior algebra is then presented in chapter 2, followed by a chapter on the exterior derivative. The rest of the book prestents a good deal of applications of forms including the statement and proof of the Newton-Leibniz-Ostrogradksy-Gauss-Green-Stokes Theorem, usually just called Stoke's Theorem. I'd recommend this book to any grad student in Theoretical physics that plans on doing research on physics in the 21st century.

I was searching for a good source discussing the differential forms "from the ground up",while passing a course on GR. My motivation in studying this book was Weinberg's mentioning af it as "An extremely readable book" on the topic,in his book on GR.I belive that this is a good book if you have enough time and motivation to study differential forms from basic and without much hurry to use them operationally , but (at least in my opinion) it lacks that degree of clarity that one requires from a book on mathematical physics.To truly understand some parts (even at the early definitions)you may need to spend much more time that you could imagine at the first sight.Some basic ideas are expressed too concise.If you want to learn about differential forms in physics, this book would be some good, but not(I think) during a semester on something else (like GR), beacuse the way of presenting the material is not so stright , nor is operational enough.You may find the books by Lovelock & Rund or by Goldberg & Bishop more useful.

As a survey of the applications of differential forms to various aspects of the physical sciences, this book works quite well. However, for the non-mathematician, there are more intuitive texts available, although it is unlikely that they will have the same scope. The exposition is extremely concise, far more so than, for example, Lovelock and Rund, and at times this makes the text somewhat heavy going. Having said that though, the reader with a reasonable level of mathematical maturity will find the book very rewarding. A reader who is mainly focussed on differential geometry though would probably be better off with Lovelock and Rund.

This book covers the basic math behind the "geometric approach" to tensor calculus. The math required is not heavy, but it requires some considerable mathematical maturity. If talk of "the boundary of a boundary is zero" or "exterior derivative" confuses you, this is a good book. An intuitive approach, not a sea of indices. If you want really heavy stuff then Bishop & Goldberg is good.

Most Dover math books are first rate. This one is.

The good news is there's a lot to like about this book. The bad news is there are at least a few things to really not like about this book.

If you are a physics graduate student, I would recommend you put this book on your reading list. But I would also recommend you postpone this book until you have developed a reasonable degree of comfort with the typical tensor calculus based approach to differential geometry along the lines of Differential Geometry.

To get a better feeling for this book, let's take a look at the contents, starting with the back of the book. The back of the book says "Requiring familiarity with several variable calculus and some knowledge of linear algebra and set theory." So to me that sounds like a background of a third or maybe even second year undergrad. But I feel completely safe in saying that anyone with precisely this level of background will be COMPLETELY blown away by this book.

The back of the book ends with a nice quote from the American Mathematical Monthly telling us that this book is "very readable - indeed, enjoyable." And this is the first problem with this book: it is in fact quite a challenging book masquerading as an easy one.

Inside, we will skip directly to the "Preface to the First Edition". Weighing in at less than two pages, this preface actually provides a valuable high level overview of the book. We are told we are faced with ten chapters. The first chapter in an introduction; chapters 2, 3, and 5 provide the theoretical development; chapters 4, 6, 8, and 9 are applications to differential geometry; chapter 7 contains applications to differential equations; and chapter 10 applications to physics.

In the second paragraph we find the statement "Probably on first reading..." This statement is a powerful omen. You should absolutely expect to read this book at least twice. The particular piece of advice here encourages us to merely try to develop "some intuition for the subject" on our first read rather than working through things in detail. That's good advice.

So, on to chapter 1. Chapter 1 is only four pages. The first two pages motivate differential forms via the change of variables theorem for multiple integrals. That's nice. It's also probably the most concrete part of the book. The last two pages launch into a discussion of the differences between differential forms and tensor methods. That's funny because this book repeatedly emphasizes that the reader doesn't need to know anything about tensors to read this book, but then it can't seem to stop talking about them. Again, make no mistake: if you aren't already pretty comfortable with tensors, you will probably not get much out of this book.

Also, in the midst of this two page discussion we are greeted by a host of undefined terms and concepts. Trust me, this discussion will make much more sense to you on your second trip through the book.

Chapter 2 is a fourteen page whirlwind on exterior algebra. The first thing that struck me about this chapter was the level of abstraction: surprisingly high considering its intended audience of scientists and engineers. This is a very compressed treatment, and nowhere is the level of compression more evident than in the discussion of the Hodge star operator. Yes, everything you need to know to compute the star operator in an arbitrary finite dimensional inner product space is actually in this two page treatment, but wringing it out is pretty challenging.

With algebra out of the way, it's time for some calculus in chapter 3. Section 3.1 defines differential forms on Euclidean space, and section 3.2 introduces the exterior derivative. Section 3.3 discusses mappings and introduces what is usually called the "pullback" of a differential form. It goes on show how pullbacks relate to the previously introduced notions of exterior multiplication and exterior differentiation: quite nicely, thank you. Section 3.4 and 3.5 are very short sections on changes of variables and a theorem from theoretical mechanics that arises in the transition from Lagrangian mechanics to Hamiltonian mechanics. Section 3.6 is a comparatively long section (2 1/2 pages) that proves the converse of the Poincare lemma. The proof is good, but unenlightening. Section 3.7 provides an example for the converse of the Poincare lemma, and section 3.8 is a brief comment on partial differential equations, and how differential forms often expose them to be simpler than they at first appear. Section 3.9 is a set of problems.

This entire chapter spans only 13 pages!

So with all that out of the way by page 32, we're now ready for some applications in chapter 4. Section 4.1 introduces the important idea of a "moving frame". For anyone with the strongly recommended background in classical differential geometry, this may well be a new idea, and its importance will be one of your key takeaways from this book. Section 4.2 discusses the relationship between orthogonal and skew-symmetric matrices. Section 4.3 introduces the six dimensional frame space. Section 4.4 discusses the Laplacian in curvilinear coordinates via differential forms. Section 4.5 digs into the differential geometry of two dimensional surfaces in three space from the perspective of differential forms. Although this treatment might initially seem somewhat cryptic, the elegance of differential geometry using differential forms starts to become apparent quickly. Section 4.6 is a brief section on the differential form formulation of Maxwell's equations, and 4.7 is a problem set.

Chapter 5 is our last chapter on theory, and it centers around Stokes' Theorem. So far this book has restricted its treatment of differential forms to Euclidean space. The first section in this book is a very short blurb to motivate the move to manifolds which section 5.2 introduces properly. The author provides a stripped down approach to manifolds common in applied math books, and I thought it was well done. Section 5.3 and 5.4 introduce tangent spaces and differential forms on manifolds respectively. Section 5.5 introduces Euclidean simplices as we move toward integration. I found this treatment to be particularly clear. Section 5.6 introduces us to chains and boundaries on manifolds, and section 5.7 defines the integral of a differential p-form over a p-chain. And it is in this section that we encounter one of the truly infamous passages in this book. "As an exercise, one could check that each of the standard tricks used to evaluate surface integrals, etc. fits into the above scheme of things. It hardly seems worth our time here."

Hardly seems worth "our" time? This is the only book I have ever read where the author just directly came out and told me that he didn't think explaining something to me was worth his time. Nice.

Section 5.8 covers Stokes' theorem. And I found this treatment unusually clear. Section 5.9 discusses periods of forms and De Rahm's theorems which are stated and discussed but not proven. Section 5.10 gives some examples of the applications of De Rahm's theorems to surfaces. Section 5.11 is about mappings of chains and the behavior of integrals with respect to pullback. Section 5.12 is a problem set.

Chapter 6 is titled "Applications to Euclidean Space" and is a short grab bag of topics. Section 6.1 deals with volumes and areas of spheres in n dimensions, and the differential form treatment of such. Section 6.2 is on winding numbers and degrees of mapping. Section 6.3 is on the Hopf Invariant for mappings between certain high dimensional spheres. It's less than a page long, so obviously there are not a lot of details, but what is given is quite clear. Section 6.4 is titled "Linking Numbers, The Gauss Integral, and Ampere's Law", and again is very short. There are no exercises for this chapter.

Chapter 7 is "Application to Differential Equations". Weighing in at 30 pages, this is one of the beefiest chapters in the book. Section 7.1 is devoted to the study of harmonic functions in E^n. All of the results in this section are the standard, elementary results in harmonic function theory, but all of these results are now proven relying heavily on differential forms. Results proven include Green's Formula, Green's Symmetric Formula, Gauss' Mean Value Theorem, Maximum/Minimum Principle, and the Uniqueness Principle for the Boundary Value Problem. The idea of the Green's function for an arbitrary domain is introduced, and the Poisson Integral Formula for E^n is determined. The section concludes with a proof of Liouville's Theorem. All of these results will be immediately familiar to anyone with an introductory course in complex analysis, but again, these results are demonstrated for E^n rather than just E^2. And again, the elegance of the differential form approach is readily apparent.

Section 7.2 is a short section on the Heat Equation. Essentially all that is demonstrated here is uniqueness of solutions for the boundary value problem.

Section 7.3 is on the Frobenius Integration Theorem. This is a central result about the existence of integrating factors for linearly independent differential one forms. This section is ten pages, and provides a detailed proof of the Frobenius Theorem. It begins with some motivation, and then proves the result for a single form before stating and proving the general result. The reader will need to be comfortable with the theory of ordinary differential equations to follow this section. Section 7.4 follows this up with three applications of the Frobenius Integration Theorem, thus demonstrating its importance.

Section 7.5 is a short section on systems of ordinary differential equations, and Section 7.6 is on the Third Lie Theorem, which is fundamental in the study of Lie groups. This result is proven in detail. There are no exercises for this chapter.

Chapter 8 is on applications to differential geometry. At almost 40 pages, it is the longest chapter in the book. In section 8.1 it picks up where 4.5 left off with surface theory. Section 8.2 extends this treatment to hypersurfaces in E^n in preparation for the abandonment of the embedding space. Several crucial constructs appear in this section including curvature forms and the Riemann curvature tensor.

Section 8.3 takes the plunge into Riemannian geometry, and introduces more key elements of differential geometry including the Christoffel symbols. The section ends with four pages discussing the relationships between the differential form treatment of differential geometry and the more usual tensor treatment. Again, this is very helpful if you are familiar with the tensor treatment. If not...

Section 8.4 is a six page survey of Hodge theory. It states the fundamental result, but only proves uniqueness of the forms asserted to exist. I thought this section was excellent. Also excellent, is section 8.5 on Affine Connections. In particular, the definition of an Affine Connection in the differential form treatment will be quite eye opening to anyone with the traditional tensor background. It was enough to single-handedly sell me on the utility of the differential form treatment of differential geometry. Unfortunately, this section is also home to one of the most painful moments in the book. On the very first page the author mentions the dual basis of one-forms. This will be a readily familiar concept to anyone with a background in differential geometry, which this book claims you don't need. The only problem is, he had never defined this concept. So I checked the index to see if I missed it, but it wasn't even in the index.

We finally encounter the necessary definition in the problem set at the end of chapter 10. Better late than never? I didn't really think so, and this book lost serious points for this egregious organizational error. If I hadn't already known what he was talking about, I would have been baffled.

Section 8.6 is a problem set.

Chapter 9 is a short chapter on applications to group theory. Section 9.1 contains the pertinent definitions of Lie group, structure constants, and so on. 9.2 provides some examples of Lie groups, while 9.3 and 9.4 cover matrix groups. 9.5 discusses bi-invariant forms, and section 9.6 is a problem set.

Chapter 10 is Applications to Physics. The bulk of this fairly lengthy chapter is dedicated to classical mechanics, and is alone worth the price of buying and the trouble of reading this book. After seeing a wide variety of ham-fisted treatments of differential forms by classical mechanics books, I found this chapter extremely clarifying. However, the reader must already know Hamiltonian mechanics to get much of anything out of this chapter. Section 10.1 is a short introduction to phase and state spaces. Section 10.2 is on Hamiltonian systems. Section 10.3 covers integral invariants. And it covers them very well. Section 10.4 is dedicated to brackets: Poisson, Lagrange, and Lie. Section 10.5 is on Contact Transformations. Then we get a short section on fluid mechanics in 10.5, and a problem set in 10.6 to wind things up.

Overall impressions: While the exposition is generally good, there are times when the author makes non-trivial assertions either in derivations or in proofs. I found this a bit exasperating, as just a little bit more text could have provided considerably more clarity in these situations. The notation is overly sparse. In particular, sigma's for sums are NEVER indexed. In all cases it is left to the reader to determine exactly what is being summed over. And this frequently changes even from one step to the next during derivations. The book is filled with numerous, helpful figures, and while they say a picture is worth a thousand words, a book that is this short, covers this much territory, and has a bunch of picture in it, well you can pretty much figure that the exposition is going to have to be pretty terse. And it is. Sometime overly so.

In addition, the index is almost, but not quite, completely worthless. And the bibliography has far too high a proportion of books in languages other than English.

To wrap up this far too long review, I had my doubts about this book. In fact, I had them for quite a while. It was the last chapter, and the second reading that really sold me on it. It definitely has its flaws, but it also really shines in certain spots. Recommended, but make sure you bring the background it requires or you will be left in the dust.

Why is it that any book that contains the word "physics" or "physical" in its title feels free to be sloppy? Flander's book is sloppy. There are no shortcuts to understanding - if you want to know differential geometry you have to learn it right, not from Flander's book. For example: he devotes 3 pages to antisymmetrical tensors. The way he manages to do this is by avoiding precise definitions, or giving elaborate proofs. He does not even mention the word tensor! This is unacceptable for anyone who seeks to truly understand differential geometry. After all, if you intend to invest effort and time learning a subject, shouldn't you do it the right way?
I'm not saying Flander's book is without merit - especially the low price - which might make it worthwhile purchasing it is an additional source of information, but as a primary source it is, in my opinion, a very bad one.
There are many alternatives to Flander's book I suggest you check out before trying your luck with this one. The standard reference is Bishop & Goldberg's "Tensor analysis on manifolds". Another good book is "differential forms and connections" by Darling. A comprehensive book about diff. geometry is "Geometry and Physics" by Frankel. I'm sure others will have their personal favorites, but these are a good place to start.