Applicable Differential Geometry

An introduction to geometrical topics used in applied mathematics and theoretical physics.

M. Crampin (Author), F. A. E. Pirani (Author)

9780521231909, Cambridge University Press

Paperback, published 26 March 1987

404 pages
22.9 x 15.2 x 2.3 cm, 0.59 kg

This is an introduction to geometrical topics that are useful in applied mathematics and theoretical physics, including manifolds, metrics, connections, Lie groups, spinors and bundles, preparing readers for the study of modern treatments of mechanics, gauge fields theories, relativity and gravitation. The order of presentation corresponds to that used for the relevant material in theoretical physics: the geometry of affine spaces, which is appropriate to special relativity theory, as well as to Newtonian mechanics, is developed in the first half of the book, and the geometry of manifolds, which is needed for general relativity and gauge field theory, in the second half. Analysis is included not for its own sake, but only where it illuminates geometrical ideas. The style is informal and clear yet rigorous; each chapter ends with a summary of important concepts and results. In addition there are over 650 exercises, making this a book which is valuable as a text for advanced undergraduate and postgraduate students.

The background: vector calculus
1. Affine spaces
2. Curves, functions and derivatives
3. Vector fields and flows
4. Volumes and subspaces: exterior algebra
5. Calculus of forms
6. Frobenius's theorem
7. Metrics on affine spaces
8. Isometrics
9. Geometry of surfaces
10. Manifolds
11. Connections
12. Lie groups
13. The tangent and cotangent bundles
14. Fibre bundles
15. Connections revisited.

Subject Areas: Differential & Riemannian geometry [PBMP]