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

Tristan Needham

An inviting, intuitive, and visual exploration of differential geometry and forms

Ed Catmull
Former President of
Pixar and Disney Animation Studios

“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.”

Bernard Schutz FRS
Cardiff University

“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.”

Frank Morgan
Williams College

“With many explanations that were wonderfully new for me, this book brilliantly succeeds in providing a truly geometric visual explication of differential geometry. Needham’s writing style is magnificent, and the scholarship is impeccable. The treatment of General Relativity, perhaps the most striking application of differential geometry, is splendid. I know of nothing like this work in the literature.”

Michael Berry
University of Bristol Europe

“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.”

Loring Tu
Author of Introductory Lectures on Equivariant Cohomology

“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.”

reviewed by
Adhemar Bultheel

“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.”

John Baez
University of California Riverside

“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.”

Tevian Dray
Oregon State University

“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.”

reviewed by
Jonathan Shock

“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”

David A. Singer
Case Western Reserve University

“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.”

Publisher's Blurb

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 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 V on Forms elucidates 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.

Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.

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