Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts
Original price was: $113,51.$19,99Current price is: $19,99.
- 100% Satisfaction Guaranteed!
- Immediate Digital Delivery
- Download Risk-Free
✔ Digital file type(s): １𝐏𝐃𝐅
FROM THE AUTHOR:
All legitimate copies of VDGF produced by Princeton University Press are crisply printed on high-quality paper. If you obtain a shoddily printed copy, it’s a fake: please return it and purchase a genuine PUP copy.
An inviting, intuitive, and visual exploration of differential geometry and forms
Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton’s geometrical methods to provide new geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to Differential Forms that treats advanced topics in an intuitive and geometrical manner.
Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss’s famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein’s field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology.
The final act provides an intuitive, geometrical introduction to Differential Forms, elucidating such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell’s equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan’s method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms.
Six of the seven chapters of Act V can be read completely independently from the rest of the book, providing a self-contained introduction to Differential Forms.
Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be understood and taught.
35 reviews for Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts
You must be logged in to post a review.
Related products
-
Contemporary Mathematics for Business & Consumers, 9th (MindTap Course List)
Original price was: $289,99.$19,99Current price is: $19,99. -
Essential Calculus Skills Practice Workbook with Full
Original price was: $33,00.$16,99Current price is: $16,99. -
Everything Is Predictable: How Bayesian Statistics Explain Our World
Original price was: $59,00.$15,99Current price is: $15,99.
Ian –
Received this yesterday (yes, the day of publication). Can’t stop picking it up and browsing. REAL mathematics. My first university destroyed my interest in maths with a purist course – axioms, theorems, proofs – ground through painfully.
There is a chasm in differential geometry between the curves and surfaces level, then the differential forms level, which I have struggled to get over, even with Loring Tu’s fine book on manifolds. Needham’s text has masses of real examples which I believe will bridge this gap. I did not know I did not know so much!
Looks even better than Visual Complex Analysis.
Squid –
One’s review is supposed to be of the contents, not the print quality. “ “When a wise man points at the moon the imbecile examines the finger”
Lost in Siberia –
The moon will be there. And may be seen with or without a finger pointing. But if that finger has an obvious wound or hideous blemish, which might be cancerous, you do want to call for attention to that.
A. Boros –
Beautifully illustrated, topics lucidly explained. Absolutely recommend to all interested in differential geometry.
William Martin –
Great illustrations, Excellent explanations
Cesar Lopez Monsalvo –
This book will become a classic in no time. Tristan Needham’s approach to the subject is fresh and, indeed, extremely visual. His practical examples regarding transport on surfaces are incredibly clear. Jacobi fields become very intuitive objects. It is a great companion to any good text in differential geometry.
Alexandra Brosius –
I am very pleased with this book. The figures are black and white but extremely innovative. I think this masterpiece will become the modern MTW (Misner, Thorne, and Wheeler) of differential geometry. In other words: epic, legendary, enthusiastic without the MTW backdrop of taking the (presumably male) reader on any distasteful romantic conquests.
Back to the book at hand: very clever and insightful. I’ve looked at a few complex analysis and differential geometry books and have some familiarity with differential forms. I am definitely learning new things from this text.
The author’s genuine enthusiasm clearly shines through. Reading this book is unexpectedly delightful: like attending an impeccably planned opera while simultaneously listening to a professor spend an entire afternoon joyfully answering a question after a lecture.
Oh, and the index is very thorough! If what you are looking for is in the book, you will find a listing with every possible combination of ideas. I recommend reading the index to experience a thoughtful grouping of ideas in a new light. This is a 5-act book, but I’d call the index Act 6.
I have read the whole book at this point, and I will continue to browse for fun. I could also envision returning to pages with literal fruits and vegetables in figures: this book could serve as a kitchen laboratory manual for leisurely fruit peel curvature explorations. In addition, student-led math/science clubs could use the book as a starting point for developing public outreach demonstrations suitable for anyone old enough to chew fruit.
G W –
This book is very physical. Full of intuitions and motivations. The introduction of the Riemann tensor, Jacobi equation leading to the Einstein field equation is the best I have read. The starting point is also low, only requiring some basic calculus and linear algebra background. It was a real joy to read.
The only downside is that the book binding is really bad. Multiple pages are loose and off with very light use.
ESL –
Haven’t gotten to read through yet — but looks excellent and it’s the author of Visual Complex Analysis… 🙂
Will update when I’ve read through the book — a process I’m excitedly anticipating.
Kaneisland –
I just received this paperback and started reading. The explanations are very clear with beautiful pictures as his “Visual Complex Analysis” book published in 1990s was. It is very pleasurable to read this book. I actually took his math class long time ago at University of San Francisco. His math class with visualization changed my entire perspective toward mathematics.
Ross Mcgowan –
Just starting out on this book. Only covered the first chapter. I will write a fuller review once I’m deepr into the book.
I have always been of the persuasion that professional mathematicians get a ‘handle’ , ‘hook’ on a subject by analogies, once these anaolgies hit there limits then these original thought processes are rightly replaced by the core mathematics.
What I like about this book and the sister book on complex analysis is that the author gives us the analogies along with the mathematics and equal weight are placed on both and they both go hand in hand.
Having had mathematics thrown at me wrapped up in a muddy ball at University and failing miserably at it. I have since spent a large part of my adult life being an ‘ amatuer mathematician’, taking MY time to comprehend subjects MY way and the two books by the author are the closest I have come to finding someone who thinks like me (or I think like them).
Great book. I’m looking forward to continuing on the journey with this book.
Simon Templar –
L’ Autore con questo testo ha bissato il precedente capolavoro “Visual Complex Analysis”. Orientato principalmente alle applicazioni fisiche della relatività generale, il testo propone l’argomento secondo un approccio intuitivo ma senza perdere mai di vista il rigore matematico. Personalmente ho collocato questo splendido volume accanto allo Spivak e al Do Carmo.
Leeber Cohen –
This text will be potential great interest to three different groups of readers. The first group of readers will be interested in the history and biographies of the mathematicians and physicists who developed and used differential geometry and forms. As stated the text can be read at this level with a basic understanding of calculus and geometry. The many places in the text left as excercises and the problems can be skipped. Get an idea off what differential geometry and forms are about. The excellent drawings and numerous drawings make the book a pleasure to read. Clearly 5 stars.
If you are a self learner interested in learning differential geometry and forms, the book is more problematic. The subject of tensors and differential forms is well covered in the fifth section. If you are familiar with contravariant, covector, and duality from previous study of tensors, this section of the book will help you understand df, dx,dy, and dz as one forms. There is nice introduction to external derivatives, wedge products, Maxwell’s equations, spacetime and curve space. Peter Collier’s recent publication on this subject is a much easier read but the math by Needham is much more rigorous. As the author suggests section 5 (act 5) of the book can be read separately. You might want to read Fleisch’s Maxwell’s Equations and Vectors texts before reading Act 5. You also should consider Guidry’s Modern General Realtivity. The review of tensors and Jacobians in Chapter 3 of Guidry’s text is superb.
There are some issues for self learners in the first four acts of Needham’s text. There should be a online site to see the suggested excercises in the text. The chapters with multiple unsolved problems are frustrating. They are perfect for advanced undergraduate math or physics majors. This would allow the text to be used in a classroom setting with teaching assistants. That being said the differential geometry explored in these chapters is beautiful. It is rewarding to try to understand the geometry and compare it with the results from vector calculus.
I will leave it to the physics and math experts to review the text as a classroom standard or supplemental text. The text clearly is 5 stars at many levels. A second version written for self learners would be wonderful. If you are a self learner, you may give the book a lower score. A book with solved problems for self learners would be beautiful. The author Tristan Needham deserves our thanks for his beautiful text.
John P. Rickert –
Ordinarily, I wait until I’ve read all or a lot of a book before I rate it. I am not very far in but am more than happy to give the book 5 stars for the great benefits and insights I have already received from it. Initially, I thought this might be a book with some nice illustrations while not all that substantive on the mathematics, and I was willing to buy it even then. (I’m reminded of that great canard that every book on topology either ends with the definition of a Möbius strip or is a personal communication to J.H.C. Whitehead.) Here, what a pleasant surprise to see the breadth and depth of the topics! And yet, written so naturally and agreeably that this book should be accessible to undergraduates. An outstanding example of what mathematics and mathematical exposition should be.
Prasad Boddupalli –
It is a pity that Princeton University Press released such a poorly printed book. I suspect that it is a print-on-demand book, printed in a cheap press in India. But priced in dollars.
As for the contents it is a very good book to study differential geometry. However, good grounding in geometry and at least single variable calculus is required to understand the subject. Else, one would end up consulting geometry and calculus books at every step.
S Wood –
If you are studying Differential Geometry this is an excellent companion to have. Whilst it isn’t written as a textbook it provides invaluable, in depth content any student would appreciate.
Renato Orta –
Absolutely a wonderful book. It should be studied by all serious students of mathematics.
Fabrizio –
Ottimo libro
TSR –
It surely is another classic in the making. A difficult subject made accessible via heavy and quite stark illustrations.
Someone commented on print quality. They must have ordered a photocopy. Mine’s just fine.
A reader from –
It’s so disappointing that superlative content which other reviews cover has been released by Princeton University Press as a print-on-demand paperback glue binding.
Gary –
Tristan Needham has written a thoughtful exposition “…in a friendly and informal way.” (page 484). Allow me to back up for a moment (that is, backing up through time). My initial exposure to differential forms came via the “telephone” book of Misner, Thorne and Wheeler: Gravitation (exposure coming late 1970’s). I felt then (and still do) that their exposition of differential forms is discursive and a bit confusing (upon completing Needham’s book, my opinion did not change). Yet, after completing Needham’s book, many of the exercises in Arnold’s monograph were almost trivial to complete (Arnold’s pages 163-200). Thus, my view: mathematicians offer the best expositions of mathematical topics and physicists should not attempt to “teach” mathematics. A few of my observations regards Needham’s exposition:(1) Look at page 461 (boxed formulas # 38.60). It seems to me that a student who is unable to derive and to retain the trivial derivatives need not peruse this book. So, not everything needs be displayed, even if one is at undergraduate level. Along that line of thought, look at optional section # 34.8, of Faraday and Maxwell (pages 381-385). Compare this to Misner, Thorne and Wheeler (pages 73, 78-81). The similarities should be apparent and Needham does refer to MTW for elaboration.(2) Interestingly enough, the best part of this book are the exercises. For example, the final exercise in the book, Cosmic Curvature (#28, page 474), is provided a hint for its solution: “…by following the footsteps of our calculation of the Schwarzschild Black Hole, write down….” So, if you have followed those previous steps, this exercise is nearly painless. The mechanical proof of Snell’s law is nice (#14.1-14.4, page 87). Mark Levi’s book The Mathematical Mechanic also presents a “mechanical interpretation of Snell’s law.” (2009, page 58). An exercise, Minding’s Theorem, provides an opportunity in solving routine differential equations (# 7.1-7.3, page 336). Geometrical exercises for linear algebra are delightful (page 221, # 12.1-12.7, Visual Linear Algebra). You got linear algebra earlier than that (pages 154-158) and later than that (pages 361, 364, 365). In brief: All of the exercises are enlightening ! Exercises occur after each Act: That is a long stretch to wait. I am comfortable if I tackle exercises earlier rather than later. Happily those occur within the prose (example: ” Using these symmetries, you may confirm…” page 295).(3) Needham overlaps and enhances connections found in his previous book: Visual Complex Analysis. That book got me into trouble during a 1998 course of German (I was studying Needham’s book in class, rather than German !). I do not know if Needham’s 1997 book made headway into college curricula (but, it should have). I say the same here and now regards differential forms…(4) But, I fear that if great books written by Buck (1956, Advanced Calculus), Fleming (1965, Functions of Several Variables), Edwards (1965, Advanced Calculus), O’Neill (1966, Differential Geometry), Sternberg (1988, A Course in Mathematics), Bressoud (1991, Second-Year Calculus), Weintraub (1997, Differential Forms) and Hubbard & Hubbard (1999, Vector Calculus) have not made their impact upon the teaching of differential forms, then, nothing will (I hope Needham’s book will be enough to turn the corner). Needham is excellent preparation for Arnold’s advanced Mathematical Methods of Classical Mechanics and Darling’s advanced Differential Forms and Connections (do not eschew Burke’s Applied Differential Geometry, 1985).(5) The subject index is detailed (ten pages) and there is a helpful bibliography. Also, the “double boxed” expression (page 429) will come as little surprise if you have followed the lucid exposition of De Rham cohomology group of the torus (page 428). I am inclined to add a reference to supplement chapter 30 (pages 307- 333), that is, adding Baez ‘s The Meaning of Einstein’s Equation (see ArXiv.org).(6) I recall this: “…we can’t draw you a picture of a form…a k-form is something like the determinant.” (page 500, Hubbard & Hubbard, 1999,Vector Calculus, Linear Algebra and Differential Forms). Amusingly, you locate the word “determinant” only on page 153 of Needham’s exposition (Index, page 493). But, you do find determinants everywhere (for example: page 223).(7) I do find the brief note on connections to Dirac’s Bra-Ket notation to be circular reasoning (page 352).I do find expressions such as “supremely beautiful equations” to be entirely anthropomorphic (page 401).I do agree with Needham (page 475) that O’Neill’s text, Elementary Differential Geometry, is superlative.I do urge readers to revisit Chern’s fine ten-page exposition of Gauss-Bonnet Theorem (pages 162-171, Lectures on Differential Geometry). Chern is in contrast with Needham’s sprawling exposition. Compare !(8) Concluding: Although there is overlap with his Visual Complex Analysis book, this recent publication is every bit as informative and interesting (sometimes even inspiring). Needham is well within the reach of undergraduate students (as are many of the books I have listed above). You can have fun with his book.
Kaleberg –
This book is a wonderful introduction to differential geometry. Most books on the subject get right into the algebraic side of it, which is good and all, but makes it hard to build a physical intuition. This book works pictorially and gets one thinking in terms of movement and geometry rather than formula. Instead of dealing with the technical issues of limits, this book uses Newton’s idea of things being eventually equal as they get smaller. and reinforces this with a series of geometric demonstrations of proofs that would have been algebraically complex but are simple to visualize.
I haven’t taken a math course in forever, so I didn’t expect to get very far in this book, but the narrative was seductive. My freshman calculus returned slowly as I was reintroduced to curvature and finally got a visual understanding of how positive and negative curvature work. The historical approach was encouraging. If it took the likes of Gauss and Euler decades to figure something out, I had no reason for shame in having to reread a section a few times and spend a few days making sure I understood it.
The narrative carried me along when the book moved from those surfaces familiar from algebra to more general topological surfaces. It always seemed that there was so much freedom in the geometries of generalized surfaces that the mathematics must be inscrutable, but it turns out that curvature has all kinds of constraints. It isn’t just that a donut can be deformed into a coffee cup, it’s that the sum total of curvature of both surfaces are the same. Every inward bit of bending has to be balanced by the same bending out. Even better, there were multiple proofs, all geometric and well illustrated, of a number of important theorems. If one didn’t suit my learning style, another would. Visualizing bananas dripping honey and oddly shaped squashes embedded in three dimensional space really helped.
Then the narrative got deeper as the viewpoint shifted from the space outside the surface into the surface itself. If you’ve read the classic Flatland, you’d appreciate how much can be learned by changing one’s point of view. With diagrams and clear descriptions, the book explained how that now somewhat familiar curvature appeared to a creature living within the surface. Moving within a curved surface yields strange motions and peculiar rotations. As odd as they may seem, they all make sense, and they are much more constrained than they first appear. An accountant would appreciate so much of this book. Everything has to add up.
Then comes a bigger jump, from two dimensional curved surfaces to three dimensional curved surfaces, and few of us have a geometric intuition here. Having laid the groundwork, however, the book discusses the strange motions and peculiar rotations that we can experience living in three dimensional space. It is harder to visualize curved three dimensional space, but the book had readied us for this jump. I had to read the section on parallel transport three times before I felt I understood it properly, but in the end I did. Again, more illustrations, more theorems, more demonstrations. There is a bit more algebra here. Two dimensional curvature can be described with a single number, but higher dimensions require more numbers best arranged in matrices.
I really appreciated the section on general relativity and the way it treats gravity not as a force but as a property of curved space. Do the geodesics, the paths that appear to be straight to those dwelling within within the space, converge or diverge? That can be seen as a geometric question or as a question of attractive or repulsive force. The book had a really nice explanation of this and added a lot of depth to the idea of gravity as geometry.
I was exhausted by the end of Act IV, but the journey had been worth it. I didn’t read the fifth act with the chapters on forms. Forms are a more modern, more abstract way of understanding those strange motions and peculiar rotations, but for now they are beyond me. Perhaps some day I will return to Visual Differential Geometry and finish with Forms, but my brain needs to recover a bit.
This book is definitely not for everyone. It assumes that you are at least familiar with vector calculus and some linear algebra. If you aren’t you will be lost, clear explanations and useful illustrations notwithstanding.
Also, a few quibbles, the book has some typographic and proof reading problems. The captions in a few of the illustrations seem to have been scrambled. There are an awful lot of variables distinguished by font or by markings above like carets and dashes and squiggles. Some of the block letter characters look all too similar to my aging eyes, and I would have appreciated a guide to the conventions of those markings, especially since they aren’t completely consistent throughout the chapters. A particular problem for me was the use of a comma that looked a lot like the prime marker indicating differentiation. More than once I had to stop and go over a derivation before realizing that an apparent derivative was just a comma.
All told, a great book. If you are differential geometry curious, think you have sufficient background and finally want to make sense of curvature, this is an excellent introduction.
Gustavo Sánchez Gómez –
De los mejores libros sobre GR que tengo!
Joachim Andres –
I do rarely give evaluations, however, this book is so outstanding with presenting a really difficult subject matter that I want to explicitly recommend it. Math textbooks even meant for a non-studying community almost always place rigour in the driver seat. This book, however, shows that developping first an intuitive understanding is the better approach. Working with graphs and pictures is so important as a basis and algebraic tools are by far better digestible then. I did not put my hands on differential geometry when I studied physics 30 years ago and I did not expect to get close to General Relativity some day. Now I see it is doable. In a future version of the book, Tensor Algebra could be replaced (or added as Act VI) by Geometric (Clifford) Algebra.
Pjotr –
Wunderbares Buch, mit leicht nachvollziehbaren Gedankengängen und Entwicklungen, selbst von komplexen Sachverhalten. Hätte ich wirklich gerne während des Studiums gehabt.
Lily Fore –
This book is incredible. It almost reads like fiction in that it doesn’t feel like work at all. I found it very hard to put down. There’s also lots of cool pictures that make things clear
Frank Fu –
A majority of the book are derivations in 3D and analogies using simple geometries in R3. I would recommend this to an undergraduate student looking to get into GR, but it’s not the best reference book.
Giorgi Melkadze –
I think this book should be recommended to math students as an excellent auxiliary tutorial and to physics student (who learn General Relativity) as a basic textbook in Differential Geometry.
VIDAL Diego –
No formula derivations, but only simple reasonings, as if modern geometry were only intuitive. a lot of design, but rather confuse and little explainments.
Ben Hsu –
Explains the subject well and does not require too much in prerequisites — the first two Acts can be read by someone with only some calculus and linear algebra. The best part are the exercises where the reader is encouraged to construct shapes out of paper or peel strips off a fruit. These remind me of the science books I read as a child. Those exercises are fun and instructive
E. Crandall –
I wish I had it when I took Differential Geometry years ago as an undergrad.
FennyL –
Geometry is put back into the development of differential geometry in this book.
Riccardo –
Ho acquistato questo libro prima dell’altro dello stesso autore “Visual Complex Analysis” che considero un piccolo capolavoro nel suo genere. Vale quindi la pena di leggere prima VCA e in seguito “Visual Differential Geomertry and Forms” che non gli è da meno. A mio avviso sono due libri da conservare in biblioteca sia da studenti sia da professionisti nell’ambito della ricerca scientifica.
Naturalmente non sono testi semplici inoltre il punto di vista è ben lungi dal punto di vista classico della geometria differenziale. Qui la trasformazione di Moebius la fa da padrone e la visione è veramente illuminante.
Siegbert Hagmann –
ein didaktisch äusserst gut geschriebenes Buch zur Differentialgeometrie
Colleen Farrelly –
The visualizations and fruit demonstrations were a welcome whimsical touch. The book is very thorough and presents material in an approachable manner. I would highly recommend this book for graduate students in math and physics who want an entertaining read chocked full of mathematical insight into differential geometry and some aspects of algebraic topology.