*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 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.

"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

"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."—Frank Morgan, Williams College

"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