Differentiable Manifolds

A Theoretical Physics Approach

  • Gerardo F. Torres del Castillo

Table of contents

  1. Front Matter
    Pages I-VIII
  2. Gerardo F. Torres del Castillo
    Pages 1-28
  3. Gerardo F. Torres del Castillo
    Pages 29-48
  4. Gerardo F. Torres del Castillo
    Pages 49-65
  5. Gerardo F. Torres del Castillo
    Pages 67-91
  6. Gerardo F. Torres del Castillo
    Pages 93-114
  7. Gerardo F. Torres del Castillo
    Pages 115-160
  8. Gerardo F. Torres del Castillo
    Pages 161-200
  9. Gerardo F. Torres del Castillo
    Pages 201-253
  10. Back Matter
    Pages 255-275

About this book

Introduction

This textbook gives a concise introduction to the theory of differentiable manifolds, focusing on their applications to differential equations, differential geometry, and Hamiltonian mechanics.

The work’s first three chapters introduce the basic concepts of the theory, such as differentiable maps, tangent vectors, vector and tensor fields, differential forms, local one-parameter groups of diffeomorphisms, and Lie derivatives. These tools are subsequently employed in the study of differential equations (Chapter 4), connections (Chapter 5), Riemannian manifolds (Chapter 6), Lie groups (Chapter 7), and Hamiltonian mechanics (Chapter 8). Throughout, the book contains examples, worked out in detail, as well as exercises intended to show how the formalism is applied to actual computations and to emphasize the connections among various areas of mathematics.

Differentiable Manifolds is addressed to advanced undergraduate or beginning graduate students in mathematics or physics. Prerequisites include multivariable calculus, linear algebra, differential equations, and (for the last chapter) a basic knowledge of analytical mechanics.

Keywords

Euler equations Hamiltonian mechanics Lie derivatives Riemannian manifolds differentiable manifolds differential forms time-dependent formalism

Authors and affiliations

  • Gerardo F. Torres del Castillo
    • 1
  1. 1.Universidad Autónoma de Puebla, Instituto de CienciasCiudad UniversitariaPueblaMexico

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-8176-8271-2
  • Copyright Information Springer Science+Business Media, LLC 2012
  • Publisher Name Birkhäuser Boston
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-8176-8270-5
  • Online ISBN 978-0-8176-8271-2
  • About this book