Manifolds, Tensors, and Forms

An Introduction for Mathematicians and Physicists

Paul Renteln

• Publisher: Cambridge University Press
• Year: 2013
• Print ISBN:
  9781107042193
• eBook ISBN:
  9781107597075
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

Providing a succinct yet comprehensive treatment of the essentials of modern differential geometry and topology, this book's clear prose and informal style make it accessible to advanced undergraduate and graduate students in mathematics and the physical sciences. The text covers the basics of multilinear algebra, differentiation and integration on manifolds, Lie groups and Lie algebras, homotopy and de Rham cohomology, homology, vector bundles, Riemannian and pseudo-Riemannian geometry, and degree theory. It also features over 250 detailed exercises, and a variety of applications revealing fundamental connections to classical mechanics, electromagnetism (including circuit theory), general relativity and gauge theory. Solutions to the problems are available for instructors at www.cambridge.org/9781107042193.

Table of contents

• Cover
• Manifolds, Tensors, and Forms
• Title Page
• Copyright Page
• Contents
• Preface
• 1 Linear algebra
• 2 Multilinear algebra
• 3 Differentiation on manifolds
• 4 Homotopy and de Rham cohomology
• 5 Elementary homology theory
• 6 Integration on manifolds
• 7 Vector bundles
• 8 Geometric manifolds
• 9 The degree of a smooth map
• Appendix A Mathematical background
• Appendix B The spectral theorem
• Appendix C Orientations and top-dimensional forms
• Appendix D Riemann normal coordinates
• Appendix E Holonomy of an infinitesimal loop
• Appendix F Frobenius' theorem
• Appendix G The topology of electrical circuits
• Appendix H Intrinsic and extrinsic curvature
• References
• Index