Customer Reviews

Differential Forms in Mathematical Physics

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As a mathematician interested in physical applications of differential geometry, I was eager to peruse this much esteemed textbook. Its scope is fantastic, including basic topology, calculus on R^n, manifolds, Lie groups, fiber bundles, differential forms, Frobenius' theorem, integration, de Rham's theorems, connections, symplectic geometry, relativity, and gauge theories. However, the number of small errors, both in definitions and examples, is inexcusable. An occassional typographical error is to be expected in any publication of this sort, but Westenholz's text contains a conceptual error on nearly every page which I read.

For example, on page 4 we are told that, "A topological space is a non-empty set E together with a family T = (U_i | i element of I) satisfying..." It is by no means necessary that the topology be a family or "indexed set", as it sometimes called (see Noll: Finite Dimensional Spaces). Further down the page, we are given our first example of a topological space: "A toplogy on the real line R can be defined by the class of all open intervals T = {U_i = (a_i, b_i) | a_i, b_i element of R} ... by a straightforward verification T is seen to satisfy the axioms..." Unfortunately, the union of two disjoint open intervals is not an open interval, as would be required by the axioms for T, were T defined correctly. Westenholz has actually given us a base for the usual topology of R, but it remains for the reader to catch this oversight. On page 5, regarding the neighborhood system B(x) of a point x element of E, we read, "The following properties of neighborhoods may be used to define a topology on E: ... "(V1) there exists V element of B(x) and x is an element of V; (V2) for all V_1, V_2 element of B(x) there exists a V_3 element of B(x) such that V_3 is included in V_1 intersect V_2; (V3) if V element of B(x) and y element of V, then there exists U element of B(y) such that U is included in V." If we interpret Westenholz's words as meaning that (V2) and (V3) are properties of neighborhood systems in some preestablished topology, then (V3) is nonsensical, because y could be an endpoint of a closed interval in the usual topology of R. On the other hand, if we interpret Westenholz's words as meaning that (V1), (V2), and (V3) are axioms for a topology in terms of a new primitive concept of neighborhood system, then they are insufficient to ensure that the finite intersection or arbitrary union of such neighborhoods are elements of the topology.

Technicalities such as these continue right on through the core chapters on differientiable manifolds. To the book's credit, there are a great many concepts, like the codifferential, which other works, such as Lichnerowicz's book on magnetohydrodynamics, employ but do not define. Westenholz's book is the only place I have found definitions for such; this attests to the comprehensive scope of the text, but, owing to the large number of errors in familar topics, I would not trust this novel content without first verifying it elsewhere.

As Westenholz says in the preface, "...one of the goals of the present book is to develop an intuition and working knowledge of the subject...without insisting on an extremely high degree of mathematical rigour or precision...". I believe that in a book such as this, it is appropriate to sacrifice details of interest primarily to mathematicians, but what *is* included should be correct. "Analysis, Manifolds and Physics: Part I and II" by Choquet-Bruhat et al share the same goal as Westenholz's book but do not contain a surfeit of errors and pay more attention to the equally important subject of functional analysis. For the classical kernel-index method approach to tensor analysis, of interest in its own right, try "Tensor Analysis for Physicists" by Schouten.

I recently saw this book at a local university library, and it didn't meet the build-up (based on the previous review, although the review is accurate). It's not really worth the exorbitant amount of money it's going for on the used market! It IS very complete, and has many examples, but it is written in a "math-technical" way that makes for very slow going, and it is somewhat "old-fashioned". Definitely, this is a "maths" type of book, rather than a "physics" type. I left the library not sorry to just let it go: it would take forever to actually complete the whole book. It could be useful as a reference, or if you have 3-6 months of free time to devote to it (and you actually like that kind of "maths" writing).

On the other hand if you are looking for a much more accessible and intuitive book, which you could actually complete, then I would recommend "Differential Forms: A Complement to Vector Calculus", by Steven Weintraub. At $79.95, it isn't cheap, but it's a real deal compared to von Westenholz! This is the book that made me understand differential forms, exterior differentiation, and their relation to E&M... There are other good books out there, but this is my single favorite introductory book on the subject.

While not the ideal candidate for a first book on the subject - although it does make an effort to justify its definitions - this is definitely the book to buy when you're looking to spend your money. It is complete, requiring some knowledge of basic topological concepts (nothing heavy), linear algebra and multivariate calculus. It contains a refresher chapter about differential calculus in R^n and then quickly moves to introduce manifolds, lie groups, bundles, connections, differential forms and so forth. Three things make this book shine: 1. the amount of material in it, 2. the very neat fashion in which it's organized, and 3. the amount of examples found in it. It explains the application of differential geometry to mechanics, electrodynamics, thermodynamics and even has a chapter on general relativity! In short, once you've completed an introductory book, this will definitely be a good choice for a second book.