Hardback : HK$1,236.00
An inviting, intuitive, and visualexploration 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.
An inviting, intuitive, and visualexploration 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.
Tristan Needham is professor of mathematics at the University of San Francisco. He is the author of Visual Complex Analysis.
"Finalist for the PROSE Award in Mathematics, Association of
American Publishers"
"Needham proposes to provide a truly geometric 'visual' explication
of differential geometry, and he succeeds brilliantly. I know
nothing like it in the literature."---Frank Morgan, EMS
Magazine
"[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
"[Visual Differential Geometry and Forms] is a book that stands out
among its peers. It is fun to read and provides a unique and
intuitive approach to differential geometry. The author’s passion
for the subject is evident throughout the book. Although Needham’s
approach is unorthodox, it is rewarding, and complements the
exposition found in standard textbooks."---Sean M. Eli & Krešmir
Josić, SIAM Review
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