Titling my review in honor as "Topology Illustrated" is probably the best elevator pitch description I could give for this new text, a vitally needed introduction to the geometric side of differential manifolds that is as colorful - and about as overly large - as Saveliev's tome. And like that volume, it could very well be used as a doorstop if you are OK with your doors constantly swinging shut due to all the times when you will want to pick it back up to use it.
This book covers anywhere from the tangent plane and similar geometric ideas (I FINALLY had my "aha moment" about cotangent bundles!) through differential forms and what used to be called the "absolute differential calculus" (exterior and covariant differentiation - exterior differentiation is given multiple perspectives thoroughly, the wedge product which is oddly concatinated as "wedgeproduct" here, push-forwards and pull-backs, integration of forms) and even badly needed elementary introductions to advanced ideas (Poincare Lemma, general manifolds outside of subsets of R^n, bundles, atlases including patching and partitions of unity).
Illustrations fill the pages and the text relies on them, which is probably my top reason for tilting my cap to "Topology Illustrated." This is to differential geometry what that book is to differential topology: an illustrated introduction to a topic that has very little illustrations; I may have enough illustrations in my considerable library on differential geometry to cover the sheer amount contained in this one book, but I am not sure.
One last pat on the back: the Appendices. The first is probably the most thorough and honest direct attempt to link differential forms and tensors without slouching too far into overly complicated multilinear algebra - it may be the *only* attempt I've seen to do so, though I have not delved far enough into Bishop's book to see if it is done there.
The second appendix I have not completed yet, but at a glance includes de Rahm, homotopy, Darboux's Theorem, the nearly uniformly absent (at this level) topic of symplectic manifolds, geometric mechanics and potential theory. I know the author doesn't want to "double the size of the book" with this material, but - much like feedback Bachman and Weintraub encountered in their first editions - I'm going to guess he's going to get enough readers motivated toward "filling the advanced gap" to suggest to Fortney that maybe he should do just that - fatten the book by expanding the stubs he has written in Appendix B. These need illustrations and elementary treatments, Fortney has proven himself to be the person for the job, and it might be the piece to make a second edition into that badly needed link between introductory and advanced tomes on this topic.
The book is not without its flaws. It contains the (in my opinion) confusing algebraic formalism a la Spivak's infamously forcefed introduction in his otherwiese excellent and historic introduction, though it does not unpack as quickly as that book. Typos abound - especially toward the end and during longer expositions, Springer regulars will recognize this text as joining the army of first editions whose editors seemingly just plain fell asleep on the job, so it has more of a "Undergraduate Texts in Mathematics" feel to it than the more immediately clean "Springer Undergraduate Mathematics Series."
As warned about thoroughly in Bachman, "counting pierced sheets" is not a good way to visualize integration of forms; plane fields are a much better generalization anyway, but despite all that, it is done in this book. Fortney points out (correctly) that there are some forms where this IS valid - though not specifying clearly and accessibly which ones should be used and which should be avoided= and also points out that such pictures are necessarily nearest-integer estimations of form integrals, unless you visualize a "partially pierced" sheet. But, alas, we see this awkward old method trotted out again anyway, potentially because poor physics majors may encounter it in a certain famous tome on relativity, among numerous other places.
Oh, and with due respect to Sternberg, who knows far more about this topic than I ever will, tangent plans are *not* "attached" to manifolds like Post-It notes at their corresponding points, nor are they "translated subspaces" replacing the origin with the studied point like some kind of affine plane, though this undoubtedly helps tie in Calculus WHEN carefully presented. Rather, they are *entirely new vector spaces.* There's too much to risk with confusing the first-time student with this analogy, much more so than the "planar shish-kebab" picture of integration mentioned above.
In general, though, the book fills the need for consolidation of ideas of differential forms along with illustrations that are - excepting the above critiques, along with a few others - accurate and helpful visualizations of a mysterious entity that seems to work like magic with its ability to tie all the disparate ideas of Calculus III together. In this respect it is the first and only of its kind with illustrations, and in the respect of introductory texts, it is among a group of very few recent releases that dare to dabble in advanced ideas.
As another comparison, this does what Shashahani's recent graduate text does for advanced material - it shows pictures of the ideas mostly as illustrations of surfaces in R^3 for you to generalize from, without falling into the temptation to turn the text entirely in that direction like Thorpe did, gradient normals and all.
Does it fill the need to go beyond recent texts like Vector Calculus vs. Vector Analysis or A Geometric Approach to Differential Forms to become the badly needed link to graduate and research-level material in Conlon? No. There is still a hole here, one that O'Neill or Weintraub tried to bridge from the beginner's side and one that Janich and Loring Tu's near-perfect standard try to bridge over from the *advanced* side.
Instead, this is more on the level - and, with hopefully upcoming edits, quality - of Walchap or O'Neill or Grinfeld's Tensor Calculus, all three of which should absolutely be purchased by the beginner along with this volume for a short library that gets started with the topic. The suggested reading index of this book, by the way, is a great place to start building a large list!
EDIT: Fixed some typos and made some sentences clearer. I also want to add a note to Springer to PLEASE, PLEASE PLEASE edit your texts before publishing them. It is easy to catch spelling and grammar errors, and things like fixing indices on the formulas with Taylor's Series in the Exterior Differentiation chapter ought to be a cinch, as well. We can't just rely on the authors to do it all themselves after writing the whole book or collect and check hundreds of emails and update the errors in between editions. Also, though I appreciate your choice of paper better than the dimestore comic quality pulp that makes using Vector Calculus vs. Vector Analysis akin to running nails down a chalkboard, and though I applaud your improvements in binding your paperbacks following the terrible job with Lang's Algebra, the cover of this book creaks like a B-movie crypt. I shouldn't have to wish that WD-40 works on books. This printing is - pun intended - an atlas, and it makes me almost wish I had waited on a paperback version and used the Ebook instead.
EDIT 9/13/2020: CREEEEEEEEEEEAAAAAAAAAAAK! Goes my copy. It sounds like opening a crypt after heavy use at this point. The more I refer to this, the more I think this is the companion to Dineen's Multivariable Calculus and Geometry and Bachman's A Geometric Approach to Differential Forms, and it is much easier to read than Spivak's Calculus on Manifolds if you have not had algebra or analysis yet. It has more content than Harold Edwards' text and far more than Bressoud.
Remember this is a book on GETTING STARTED on Differential Forms and Manifolds. If you want a nice, compact early-stage volume to use as a quick reference or as a place to move on from this book without having to take Topology or something (i.e. the usual "Linear Algebra and Calculus" famously bad minimal requirement) then buy McInerny's "First Steps in Differential Geometry" (this is not the classic kind of DG; he means more of the kind that you can reach via the Loring Tu topological route, which is sort of like going up Everest on the Eastern face).
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Readability: I found the text very comfortable to read, with thorough explanations and examples.
I have not yet completed the process of learning from the book but it has been greatly useful and unquestionably valuable in attempting to procure an understanding of the subject material.
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standard silvaReviewed in Brazil on December 22, 2021
Ótima introdução ao assunto, sem pré-requisitos. O texto é autossuficiente.
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- MR J R STERNReviewed in the United Kingdom on August 15, 2020
Aid to intuitive understanding of important concept
