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

Visual Differential Geometry and Forms is an inviting, intuitive, and visual exploration of differential geometry and forms that provides a geometrical explanation of classical results and an undergraduate introduction to differential forms. It includes proofs of the Global Gauss-Bonnet theorem, Gausss Theorema Egregium, Riemann curvature tensor, and Einsteins field equation, as well as the unification of all integral theorems of vector calculus, Maxwells equations, de Rham cohomology, differential geometry via Cartans method of moving frames, and the calculation of the Riemann tensor using curvature 2-forms.

Format: Paperback / softback
Length: 584 pages
Publication date: 13 July 2021
Publisher: Princeton University Press

Visual Differential Geometry and Forms is a captivating and insightful exploration of differential geometry and forms, achieving two primary objectives. In the first four acts, Tristan Needham reintroduces geometry into differential geometry, employing 235 hand-drawn diagrams to provide geometrical explanations of classical results. In the fifth act, he offers the first undergraduate introduction to differential forms, presenting 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, establishing a stunning connection between local geometry and global topology.
A simple, geometrical proof of Gausss renowned Theorema Egregium.
A comprehensive geometrical treatment of the Riemann curvature tensor of an n-manifold.
A detailed geometrical treatment of Einsteins field equation, describing gravity as curved spacetime (General Relativity), with its implications for gravitational waves, black holes, and cosmology.

The final act delves into topics such 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.

Weight: 1004g
Dimension: 178 x 252 x 33 (mm)
ISBN-13: 9780691203706