Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts
Author(s): Tristan Needham (Author)
- Publisher: Princeton University Press
- Publication Date: July 13, 2021
- Edition: First Edition
- Language: English
- Print length: 530 pages
- ISBN-10: 0691203709
- ISBN-13: 9780691203706
Book Description
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.
Editorial Reviews
Review
“Finalist for the PROSE Award in Mathematics, Association of American Publishers”
“Needham proposes to provide a truly geometric ‘visual’ explication of differential geometry, and he succeeds brilliantly. I know nothing like it in the literature.”
—Frank Morgan, EMS Magazine
“[The] book offers a truly unique and original take on differential geometry, and it amply deserves inclusion within the pantheon of textbook deities.”
—Eric Poisson, Notices of the AMS
“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, but this is the best that I have come across so far and I will continue to enjoy learning from it (and almost certainly teaching from it) over the coming years, I am sure.”
—Jonathan Shock, Mathemafrica
“[Proactively] rethinks the way this important area of mathematics should be considered and taught.” ―
MathSciNet
“The book is a remarkable and highly original approach to the basic stem of differential geometry. And that mathematical trunk has roots and branches in so many other unexpected yet related subjects, each of which can be equally well approached from the same geometrical point of view.”
—Adhemar Bultheel, MAA Reviews
“[
Visual Differential Geometry and Forms] 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.”—Sean M. Eli & Krešmir Josić, SIAM Review
Review
“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.”—Ed Catmull, former president of Pixar and Disney Animation Studios
“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.”—Bernard Schutz, Cardiff University
“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.”—Michael Berry, University of Bristol, Europe
“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.”
—John Baez, University of California, Riverside
“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.”—Tevian Dray, Oregon State University
“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.”—Loring Tu, author of Introductory Lectures on Equivariant Cohomology
“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.”—David A. Singer, Case Western Reserve University
About the Author
Tristan Needham, Emeritus Professor of Mathematics, University of San Francisco
Tristan Needham (son of the distinguished social anthropologist Rodney Needham) grew up in Oxford, England. He studied physics at Merton College, Oxford, before moving to the Mathematical Institute, where he enjoyed the great privilege of studying black holes under the supervision of Sir Roger Penrose. Tristan received his DPhil in 1987 and joined the faculty of the University of San Francisco in 1989. His continuing mission is to seek out new intuitive forms of understanding, and new visualizations.
He is also the author of
Visual Complex Analysis: 25th Anniversary Edition.
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