Tristan Needham

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About Tristan Needham
OCTOBER 2021 UPDATE: "Visual Differential Geometry and Forms" now has a website: VDGF.space. It contains both a STATIC INITIAL ERRATA (corrected in the new printing) and a DYNAMIC ERRATA, listing errors that were reported *after* PUP froze the current version.
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Tristan Needham (son of the distinguished social anthropologist Rodney Needham) grew up in Oxford, England, where he attended the Dragon School (with Stephen Wolfram and Hugh Laurie).
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 current focus is Differential Geometry, but Complex Analysis, General Relativity, and the history of science are abiding loves. His continuing mission is to seek out new intuitive forms of understanding, and new visualizations.
His book "Visual Complex Analysis" (Oxford University Press) won first prize in the National Jesuit Book Award Competition. An earlier paper received the Mathematical Association of America's Carl B. Allendoerfer Award.
His new book, "Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts", was published by Princeton University Press on July 13, 2021.
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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.