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

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An inviting, intuitive, and visualexploration of differential geometry and forms

Visual Differential Geometry and Formsfulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry.Using 235 hand-drawn diagrams, Needham deploys Newtons 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 Gausss famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of ann-manifold; and a detailed geometrical treatment of Einsteins 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 Maxwells equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartans 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 Formsprovocatively rethinks the way this important area of mathematics should be considered and taught.

Reviews

"Finalist for the PROSE Award in Mathematics, Association of American Publishers"
"[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

Author Bio

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