A Visual Introduction to Differential Forms and Calculus on Manifolds by Jon Pierre Fortney (PDF)

Description

This book explains and helps readers to develop geometric intuition as it relates to differential forms. It includes over 250 figures to aid understanding and enable readers to visualize the concepts being discussed. The author gradually builds up to the basic ideas and concepts so that definitions, when made, do not appear out of nowhere, and both the importance and role that theorems play is evident as or before they are presented. With a clear writing style and easy-to- understand motivations for each topic, this book is primarily aimed at second- or third-year undergraduate math and physics students with a basic knowledge of vector calculus and linear algebra.

User’s Reviews

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

⭐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).

⭐There is a very little to add to the comprehensive and excellent review of the first reviewer. But I’ll try.First, it is a beautiful book – outside and inside. The page size is bigger than the usual, hard cover and excellent binding as in the old good days (a real book), and it is visually very attractive, both the text and the many figures.Second, the writing style is clear, friendly and appealing. It is obvious that the author objective was clarity. It is a book aiming to the reader (while, unfortunately, most technical books merely demonstrate the knowledge of the author with no reader in mind).Accordingly, the author takes the baby steps approach, starts from a wide common ground and advances to a quite decent level with no shortcuts and with no taking for granted anything.BTW, beware that it is an introductory book – for example, I couldn’t find the Cartan structural equations or the Maurer-Cartan form, etc. For that case I prefer mat/physics oriented books such as: Differential Forms in Mathematical Physics by Von Westenholz (a relatively unknown book), or Analysis, Manifolds and Physics by Choquet-Bruhat, et al. But as an introductory, this book gives a deep insight and understanding, so I wish I had this book when I started learning forms.Personally, I had tensors approach background, so I’ve educated myself with reading numerous “classics” of the differential forms literature, understood the technical approach but remained a bit puzzled on the visual/geometrical interpretation (most importantly), until … I read this book.It is a work of love and a great book to read and to have in the library, highly recommended!

⭐Just one word, ‘Thank you professor Fortney’.This book is absolutely incredible. If you know some advanced Calculus, multivariable calculus and linear algebra and have absolutely no background in differential geometry, this is the place to start. This book is one unique book which can be read cover to cover. As a graduate student in applied math with little exposure to pure math, I would like to thank professor Fortney for writing this visual masterpiece. The expositions on differential forms, wedge products, manifolds, tensors is a lesson in mathematical pedagogy.I can go on and on describing how good this book is, best I can say, buy this book and you will be thankful to Professor Fortney forever.

⭐Excellent for self-study. Intuition with near-rigor — and distinct clarity about when it is being fully rigorous and when it is being near rigorous.Very reminiscent of Strogatz’s Non-linear Dynamics and Chaos, but with greater focus and thoroughness on topic and accordingly greater rigor — but with a similar willingness to mix formal and informal proof so as not to get too bogged in extreme edge cases. But cleanly set-up the student that wants to pursue fully general proofs.Excellent for self-study.I very much appreciated that the author covered the same topics from multiple angles. In particular he presented the exterior derivative from 4 angles – direct definition, derivation from axioms, definition relative to integration, and via visualization of common, but not wholly general cases as more often seen in physics. — The OkC of formal and less formal viewpoints strongly aided understanding and familiarity.

⭐Excellent introduction to differential forms. Very helpful in developing intuitive understanding of the concept. Very readable for a novice. Would highly recommend to beginners interested in differential geometry.

⭐Todo mundo, a estas alturas ya ha de estar familiarizado con los libros “Print on Demand” de editoriales como Springer Verlag, Birkhäuser, etc., la cual deja mucho que desear si se les compara con las ediciones anteriores al año 2011 cuando se cambió la legislación para bajar los estándares.En otra referencia dicen que este libro es como los anteriores. Es falso, no es como aquellas ediciones. Es cierto que las ilustraciones tienen una calidad media, están a color, lo cual no es usual en muchos de estos libros y que el tamaño del libro es poco usual. Sin embargo, la calidad del papel y de la impresión no es verdad que sea como la de las ediciones anteriores, así como la pasta y el empastado: es un “print on demand” disfrazado.Lo que es cierto, acerca del contenido del libro es que está muy bien concebido. Las ilustraciones están muy bien elegidas y en ese sentido sí creo que sea un libro muy bien logrado.

⭐

⭐I am extremely happy to get this book. I used to learn differentiable manifolds through several books but I were less able to visualize the things geometrically , because most books start with topological manifold and then differentiable manifolds in an abstract setting that left we readers abandoned. This book exposed all the basics of calculus on manifolds using baby manifold Rn, for n=1,2,3. This book also have enough geometric approach to convince fundamentals of differentiable manifolds. I understand this as the abc of differentiable manifolds. I am very thankful to “Jon Pierre Fortney’ who is the author of this episode for exposing the fundamentals of differentiable manifolds with great clearty with cartoon like pictures while explaining at different occasions.I also hope that authors of differentiable manifolds and geometry around the world would get great lesson about what kinds of things must keep in mind for better understanding of the readers while writing a book.

⭐Gostei muito do texto, o autor apresenta o assunto de uma maneira que pode ser entendida por todo aluno que haja dominado os fundamentos do Cálculo Vetorial.

⭐

⭐The book is verry good for self study, or for a first introduction to manifolds. Has a lot of examples completlly worked out.

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