Visual Differential Geometry and Forms:
A Mathematical Drama in Five Acts

“Oh my God! I wish I had this book when I was in college. Not only does it explain important mathematical principles better, but it explains principles I didn’t even know were principles.”

“I love this book: profound mathematics, accessibly presented in plain relaxed English, caring in equal measure for both rigor and intuition. Visual Differential Geometry and Forms is elegant, beautifully typeset, deftly illustrated, and lovingly crafted. Needham masterfully puts geometry back in charge of mathematics, and he shows us that geometry still has some new things to tell us.”

“VDGF is a book that stands out among its peers. It is fun to read and provides a unique and intuitive approach to differential geometry. The author's passion for the subject is evident throughout the book. Although Needham's approach is unorthodox, it is rewarding and complements the exposition found in standard textbooks.”

“Needham proposes to provide a truly geometric “visual” explication of differential
geometry, and he succeeds brilliantly. I know of nothing like it in the literature. ... many of Needham’s geometric explanations were wonderfully new for me. ... The writing style is magnificent, and the scholarship is impeccable. The treatment of general relativity,
perhaps the most striking application of differential geometry, is splendid. ... Through it all flows a generous geometric spirit, a yearning after geometry fulfilled. This magnificent book, one of a kind, merits the close attention and tender appreciation of all true scholars and lovers of geometry.”

“Here are five hundred eloquent pages packed with mathematical wisdom and deeply rooted in history. Visual Differential Geometry and Forms is visual indeed, with parallel transport, curvature, and geodesics depicted through pomelos, durians, squashes, pumpkins, potatoes, and toothpicks. I wish I’d had Needham’s book when I was a student.”

“This unorthodox and highly original treatise on differential geometry eschews ‘rigor’ in the pedantic sense of the word—instead, it presents geometric proofs that are illuminating and instructive. Needham's book makes a great supplement to a standard textbook on differential geometry.”

“The result is a truly unique textbook, and a remarkably successful one at that. There is really nothing like it out there. ... Invariably the diagrams are beautiful to behold ... the book offers a truly unique and original take on differential geometry, and it amply deserves inclusion within the pantheon of textbook deities.”

“Needham has done quite a remarkable job of proving Gauss’s theorem [Egregium] in a way that provides intuition and geometric insight rather than mere logical rigor. ... equally important is Needham’s remarkable ability to refine these ideas to their simple, intuitive cores and to visualize anything and everything, be it with his painstakingly clear diagrams or his demonstrations with...fruits and vegetables. ... The Pseudosphere and the Hyperbolic Plane [Chapter 5] is wonderful, building up from hand-built models of the pseudosphere to Beltrami’s construction of the upper half plane model... I applaud Needham’s choice to focus on what he’s passionate about rather than trying for an impossible completeness. And his insistence on actually trying to explain [general] relativity, rather than merely hand-waving in its direction as a motivation for studying curvature, parallel transport, and all the rest, seems inextricably tied to his boundless enthusiasm and expositional confidence, both of which are evident throughout. His enthusiasm, in particular, is quite infectious... ...this book is very definitely a sibling to Needham’s previous book, Visual Complex Analysis (VCA). As in VCA, many arguments are based on a “Newtonian form of geometric reasoning” making liberal use of ultimate equality... I found this style to be quite effective, developing much better intuition than other approaches to this material,... ”

“The book is a remarkable and highly original approach to the basic stem of differential geometry. ... professional mathematicians, even those who are already somewhat familiar with the subject, will appreciate the book even more [than students], not because of the results, but because of the original approach. ... Needham throws away all the usual textbooks to re-think the material almost from scratch, using his geometrical glasses and following the footsteps of the giants of the past.”

“The tale of geometry—from ancient Greek struggles with the parallel postulate to Einstein's amazing discovery that space and time are curved—is rich, fascinating, and very much still in progress. Textbooks on geometry often present a dry, abbreviated version of this wonderful story. Not so with Needham's book: here, he narrates everything in detail while explaining the math with hundreds of pictures.”

“A tour de force! This comprehensive treatment of introductory differential geometry actually emphasizes geometry, while providing clear explanations of the underlying concepts. Best of all are the many original figures. Recommended for motivated novices and experts alike as a guide to what differential geometry is really about.”

“It’s the kind of book that you want by your bedside ... it takes you into a world of adventure in new ideas. Studying physics, some two decades ago at The University of Bristol, ... this book would have answered all of my prayers. This is a valuable and beautifully created guide to what can at first seem a confusing area of mathematical physics. There are other contenders that try to teach this subject [of Differential Forms], but this is the best that I have come across”

“Presenting a geometric exposition of differential geometry, this eloquently written book has a very clear and logical organization. The figures are nothing short of magnificent.”