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

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

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Mathematics

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maveriks

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7/17/2021 6:36:48 AM

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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 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 an n-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 Forms provocatively rethinks the way this important area of mathematics should be considered and taught.

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Tags: Acts Differential Drama Five Forms Geometry Mathematical Visual

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