Differential Forms: A Complement to Vector Calculus 1st Edition by Steven H. Weintraub (PDF)

Description

This text is one of the first to treat vector calculus using differential forms in place of vector fields and other outdated techniques. Geared towards students taking courses in multivariable calculus, this innovative book aims to make the subject more readily understandable. Differential forms unify and simplify the subject of multivariable calculus, and students who learn the subject as it is presented in this book should come away with a better conceptual understanding of it than those who learn using conventional methods.

User’s Reviews

Editorial Reviews: About the Author Steven H. Weintraub is a Professor of Mathematics at Lehigh University. He received his Ph.D. from Princeton University, spent many years at Louisiana State University, and has been at Lehigh since 2001. He has visited UCLA, Rutgers, Oxford, Yale, Gottingen, Bayreuth, and Hannover. Professor Weintraub is a member of the American Mathematical Society and currently serves as an Associate Secretary of the AMS. He has written more than 50 research papers on a wide variety of mathematical subjects, and ten other books.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐By now (2017) there are numerous texts on the market from which to learn the topic of differential forms.Many years ago (1976) my first introduction to the topic of differential forms was inspired by the textbook Gravitation, by Misner, Thorne, and Wheeler. Steven Weinberg’s text, Gravitation and Cosmology, was valuable in this regard, also. . Regrettably, neither of those textbooks satisfied my need for an introductory account. I very much enjoy both of those physics textbooks. Both are (still) useful. However, I remain dubious as to the efficacy of physics textbooks trying to present mathematics piecemeal–or, of mathematics books trying to teach physics piecemeal ! Of course, many of my favorite textbooks do just that. (Alternating between the physical and the mathematical ). Weintraub has taken an alternative route, as he writes: “…looking at this material more mathematically actually simplifies it….” (preface). Acquire the book, and see if you agree. Quite frankly, I agree with that statement. Thus, Weintraub’s path is decidedly more mathematical and less physical and yet a simpler approach to differential forms !(1) Firstly, the book is quite computational: Therefore, the examples and student exercises are as concrete as required for manipulative proficiency. For the physicist or engineer, this text will provide much in the way of manipulative skill. Read: “…vector calculus only works in R^3…best to stick to differential forms themselves, which are naturally defined in all dimensions.” (page 31). The second chapter introduces Manifolds: “…we can integrate differential forms over oriented manifolds, this is the reason we are interested in them here.” A 25-page heuristic introduction which paves the way for the following chapter of ‘One-Forms.’ We arrive to the third chapter: This chapter places the previous two in perspective. We read: “The requirement that differential forms be alternating comes from the fact that we need to keep track of orientation.” (page 68). Meet pull-backs: ” a tricky idea,” (page 73). Twenty pages of pull-backs and push-forwards (pages 73-93) are enlightening, presenting examples and providing for excellent practice exercises. A bonus, of pedagogic utility, are the answers supplied for the exercises. A useful complement, indeed. Now, we have embraced the simpler aspects (that is, chapters one, two and three)–it is time to get serious:(2) That is, chapter four: integration. Read: “…line integrals are nothing new, this is one instance of our basic theme, that vector calculus all comes from differential forms.” (page 109). Integrals: from line, to surface (flux, page 130)to volume (3-form over a solid body–page 160). Seventy-five pages presented as lucid as possible. And, we reinforce pull-backs (from the previous chapter) with another go-around here, reading: ” We will now reformulate integration in these terms.You will see that we have been using pull-backs in disguise, now we’re removing the disguise.” (page 163). An excellent pedagogic strategy !(3) Fifth chapter: Generalized Stokes theorem. Again, concrete and computational supersedes theoretical, providing for refreshing alternative to the usual abstraction. In fact, Weintraub points out “…we have tried to write this book in such a way that you can skip the proofs without loss of continuity…” (preface). The author has achieved another of his stated goals. Final Chapter (six) surveys more advanced offerings (cohomology) and reinforces the previous textual material at an ever more advanced vantage. Again, excellent pedagogy ! Not only are we given a brief forward look (cohomology), but, a backward glance (of earlier material) at an increased level of abstraction.(4) My only quibble with the text is the absence of a bibliography and a rather abbreviated subject index. Otherwise, this text, ” A complement to vector calculus” delivers exactly what it promises. A bridge to more advanced offerings, this introductory mathematics text is difficult to ignore.Highly recommended !

⭐I used the book once to teach an early course on differential forms. The book reads well, for the most part. On the one hand, I found the book somewhat advanced: the proofs of some theorems are quite long especially if weighed against their importance. On the other hand, the book seemed a bit too elementary because the exercises require no proofs but only straightforward computations. The discussion of orientation of manifolds is too informal in my view even though the last, “advanced” chapter makes up for that somewhat.

⭐Fortunately there are several books, at an introductory level suitable for undergraduate students, on how differential forms constitute a “new” powerful mathematical technique that surpasses the outdated vector calculus. This book by Steven H. Weintraub is a very good example among others — such as: (i) “Advanced Calculus: A Differential Forms Approach” by Harold M. Edwards (Birkhäuser, Boston, 1994); (ii) “Vector Calculus, Linear Algebra, and Differential Forms” by John H. Hubbard and Barbara Burke Hubbard (Prentice Hall, NJ, 2nd ed., 2002).As far as I know, it was in “Gravitation” — by Charles W. Misner, Kip S. Thorne and John Archibald Wheeler (Freeman, San Francisco, 1973) — that a pictorial representation of forms was clearly presented to physicists for the first time. These authors went even further, explaining how “forms illuminate electromagnetism, and electromagnetism illuminates forms” (p. 105).However, until now, it seems that in engineering forms have been disregarded — despite early attempts by George A. Deschamps (see, e.g., his paper “Electromagnetism and differential forms”, Proc. IEEE, Vol. 69, pp. 676-679, 1981), not to mention Harley Flanders’s book (“Differential Forms with Applications to the Physical Sciences”, Dover, NY, 1989). Perhaps the book by Ismo V. Lindell (“Differential Forms in Electromagnetics”, IEEE Press/Wiley, NJ, 2004) will be able to change this sad scenario.It seems that the difficulty lies mainly in the fact that a proper understanding of k-forms, as antisymmetric (0,k) tensors in differentiable manifolds, requires the study of technical demanding subjects such as de Rham cohomology. However, this book shows that it is possible to make an introduction to forms without mastering such concepts in topological and smooth manifolds — although there is an extensive bibliography on this subject out there (the books by John M. Lee on manifolds are my favorite).For more advanced readers, the book by Friedrich H. Hehl and Yuri N. Obukhov on the “Foundations of Classical Electrodynamics” (Birkhäuser, Boston, 2003) is, in my opinion, the most elegant exposition on the relation between electromagnetism and forms.

⭐The language of differential forms presented at the level of this student-friendly text provides a refreshing outlook on vector analysis. And with a view towards more advanced courses, this book hints at the remarkable computational prowess it bears on differential geometry at large.In light of the author’s heuristic approach, the book does well in setting the stage for the applications he has in mind (casting Stokes’ theorem in its true form, for example).One should then go on to read books like Do Carmo, written in a similar vein, but this time, delineating the algebraic machinery needed to set up the theory in a more rigourous framework.Have fun!

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