Visual Differential Geometry and Forms

A Mathematical Drama in Five Acts

Tristan Needham

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Hardback

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English

Princeton University Press
13 July 2021

Calculus; Differential & Riemannian geometry

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 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.

By: Tristan Needham
Imprint: Princeton University Press
Country of Publication: United States
Dimensions: Height: 254mm, Width: 178mm,
ISBN: 9780691203690
ISBN 10: 0691203695
Pages: 536
Publication Date: 13 July 2021
Audience: College/higher education , Professional and scholarly , Primary , Undergraduate
Format: Hardback
Publisher's Status: Active

Tristan Needham is professor of mathematics at the University of San Francisco. He is the author of Visual Complex Analysis.

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

Finalist for the PROSE Award in Mathematics, Association of American Publishers Tristan Needham's goal is to give a natural and intuitive proof of Gauss's Theorema Egregium. The usual proof of this result is infamously inscrutable and depends on apparently miraculous cancellations, making this a noble and ambitious goal, in which Needham largely succeeds. . . . [What] seems particularly special to me about Needham's book is not just the author's unique voice or the visual presentation of the material, but the fact that it conveys a faithful sense of how (many) research mathematicians actually think about differential geometry. ---Clayton Shonkwiler, American Mathematical Monthly 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 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