Differentiable Manifolds: A Theoretical Physics Approach 2012th Edition by Gerardo F. Torres del Castillo (PDF)

Description

This textbook delves into the theory behind differentiable manifolds while exploring various physics applications along the way. Included throughout the book are a collection of exercises of varying degrees of difficulty. Differentiable Manifolds is intended for graduate students and researchers interested in a theoretical physics approach to the subject. Prerequisites include multivariable calculus, linear algebra, and differential equations and a basic knowledge of analytical mechanics.

User’s Reviews

Editorial Reviews: Review From the reviews:“The purpose of this book is to present some fundamental notions of differentiable geometry of manifolds and some applications in physics. The topics developed in the book are of interest of advanced undergraduate and graduate students in mathematics and physics. The author succeeded to connect differential geometry with mechanics. The computations are clearly explained and the theory is supported by several examples. Throughout the book there is a large collection of exercises … which help the reader to fix the obtained knowledge.” (Marian Ioan Munteanu, Zentralblatt MATH, Vol. 1237, 2012)“This book presents an introduction to differential geometry and the calculus on manifolds with a view on some of its applications in physics. … The book is primarily oriented towards advanced undergraduate and graduate students in mathematics and 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, Issue 2012 k) From the Back Cover 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. About the Author Gerardo Torres del Castillo has published two books previously, both in Birkhauser’s Progress in Mathematical Physics series: 3-D Spinors, Spin-Weighted Functions and their Applications, and Spinors in Four-Dimensional Spaces. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I have found this to be an excellent book for understanding such concepts as vector fields, tensor fields, one parameter groups of transformations, Lie derivatives, exterior derivatives, connections on Riemannian manifolds and so on. All the concepts are illustrated with detailed examples. These concepts are necessary for a proper understanding of manifolds. Perhaps the one thing I would like to have seen which the author probably did not have space for would be proof of Stokes theorem.However I am very happy with this book. It is suitable for self study and is at the level of advanced undergraduate or beginning graduate students in mathematics or physics.

⭐I read this book as a self-study to learn differentiable manifolds, and I really recommend this book to anyone interested to learn these beautiful topics in a quite short time.Typically, one of the difficulties with manifolds is that all the concepts can be developed in a completely abstract way without even requiring to do any computation. Although this is good in later stages when one has a good grasp of the underlying theoretical machinery, it might be quite frustrating in earlier stages of learning. This book completely fills the gap. One can have small by small progress by learning small parts of the book and practicing some differential calculations to get introduced to the mathematical concepts. I believe this gives a good working knowledge of underlying ideas.

⭐Introducing the concepts related to theoretical manifolds in an easy, yet rigorous way, this book helps the reader to become proficient in some important areas of theoretical physics like classical mechanics and general relativity. It is useful both for mathematics and physics students: it allows the first to ascertain the power of advanced mathematical tools in applied sciences and enhances the latter’s understanding of the mathematical methods of modern physics. A five-star introductory book.

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