" 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 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. Six of the seven chapters of Act V can be read completely independently from the rest of the book. 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. 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. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner. Using 235 hand-drawn diagrams, Needham deploys Newton’s geometrical methods to provide geometrical explanations of the classical results.
In the first four acts, Tristan Needham puts the geometry back into differential geometry. Visual Differential Geometry and Forms fulfills two principal goals.
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