Authors:
Introduces differentiable manifolds using a theoretical physics approach
Includes applications to differential geometry and general relativity
Expands on the first edition with additional examples, more exercises, new topics, and a complete solutions manual
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Table of contents (8 chapters)
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Front Matter
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Back Matter
About this book
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 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, connections, Riemannian manifolds, Lie groups, and Hamiltonian mechanics. 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.
This second edition greatly expands upon the first by including more examples, additional exercises, and new topics, such as the moment map and fiber bundles. Detailed solutions to every exercise are also provided.
Differentiable Manifolds is addressed to advanced undergraduate or beginning graduate students in mathematics or physics. Prerequisites include multivariable calculus, linear algebra, differential equations, and a basic knowledge of analytical mechanics
Review of the first edition:
This book presents an introduction to differential geometry and the calculus on manifolds with a view on some of its applications in physics. … The present author has succeeded in writing a book which has its own flavor and its own emphasis, which makes it certainly a valuable addition to the literature on the subject. Frans Cantrijn, Mathematical Reviews
Keywords
- differentiable manifolds
- Differentiable manifolds physics
- differential forms algebra
- Riemannian manifolds
- metric tensor
- Lie groups physics and geometry
- Hamiltonian classical mechanics
- Lie algebras physics
- Fiber bundles physics
- Euler equations
- Lie derivatives
- time-dependent formalism
- Vector field
- Tensor field
- Lie derivatives
Authors and Affiliations
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Instituto de Ciencias, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico
Gerardo F. Torres del Castillo
About the author
Bibliographic Information
Book Title: Differentiable Manifolds
Book Subtitle: A Theoretical Physics Approach
Authors: Gerardo F. Torres del Castillo
DOI: https://doi.org/10.1007/978-3-030-45193-6
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2020
Hardcover ISBN: 978-3-030-45192-9Published: 24 June 2020
Softcover ISBN: 978-3-030-45195-0Published: 24 June 2021
eBook ISBN: 978-3-030-45193-6Published: 23 June 2020
Edition Number: 2
Number of Pages: X, 444
Number of Illustrations: 136 b/w illustrations, 2 illustrations in colour
Topics: Differential Geometry, Mathematical Methods in Physics, Topological Groups, Lie Groups, Classical Mechanics