### Details

ISBN: 9780691203706

Published: July 13, 2021

Publisher: Princeton University Press

Trim: 0.00 x 0.00 Inches

Pages: 584

# Visual Differential Geometry and Forms

## A Mathematical Drama in Five Acts

### Tristan Needham

### By Tristan Needham

**Format:**

Paperback

$45.00 / $60.99 CAN

### About the Book

**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 Newtonâ€™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 Gaussâ€™s famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an *n*-manifold; and a detailed geometrical treatment of Einsteinâ€™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 Maxwellâ€™s equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartanâ€™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.

### Author

**Tristan Needham**is professor of mathematics at the University of San Francisco. He is the author of

*Visual Complex Analysis*.