Introduction to Tensor Analysis and the Calculus of Moving Surfaces

Introduction to Tensor Analysis and the Calculus of Moving Surfaces Pavel Grinfeld Introduction to Tensor Analysis and the Calculus of Moving Surfaces 123 Pavel Grinfeld Department of Mathematics Drexel University Philadelphia, PA, USA ISBN 978-1-4614-7866-9 ISBN 978-1-4614-7867-6 (eBook) DOI 10.1007/978-1-4614-7867-6 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2013947474 Mathematics Subject Classifications (2010): 4901, 11C20, 15A69, 35R37, 58A05, 51N20, 51M05, 53A05, 53A04 © Springer Science+Business Media New York 2013 This work is subject to copyright. 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Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface The purpose of this book is to empower the reader with a magnificent new perspective on a wide range of fundamental topics in mathematics. Tensor calculus is a language with a unique ability to express mathematical ideas with utmost utility, transparency, and elegance. It can help students from all technical fields see their respective fields in a new and exciting way. If calculus and linear algebra are central to the reader’s scientific endeavors, tensor calculus is indispensable. This particular textbook is meant for advanced undergraduate and graduate audiences. It envisions a time when tensor calculus, once championed by Einstein, is once again a common language among scientists. A plethora of older textbooks exist on the subject. This book is distinguished from its peers by the thoroughness with which the underlying essential elements are treated. It focuses a great deal on the geometric fundamentals, the mechanics of change of variables, the proper use of the tensor notation, and the interplay between algebra and geometry. The early chapters have many words and few equations. The definition of a tensor comes only in Chap. 6—when the reader is ready for it. Part III of this book is devoted to the calculus of moving surfaces (CMS). One of the central applications of tensor calculus is differential geometry, and there is probably not one book about tensors in which a major portion is not devoted to manifolds. The CMS extends tensor calculus to moving manifolds. Applications of the CMS are extraordinarily broad. The CMS extends the language of tensors to physical problems with moving interfaces. It is an effective tool for analyzing boundary variations of partial differential equations. It also enables us to bring the calculus of variations within the tensor framework. While this book maintains a reasonable level of rigor, it takes great care to avoid a formalization of the subject. Topological spaces and groups are not mentioned. Instead, this book focuses on concrete objects and appeals to the reader’s geometric intuition with respect to such fundamental concepts as the Euclidean space, surface, length, area, and volume. A few other books do a good job in this regard, including [2, 8, 31, 46]. The book [42] is particularly concise and offers the shortest path to the general relativity theory. Of course, for those interested in relativity, Hermann v vi Preface Weyl’s classic Space, Time, Matter [47] is without a rival. For an excellent book with an emphasis on elasticity, see [40]. Along with eschewing formalism, this book also strives to avoid vagueness associated with such notions as the infinitesimal differentials dxi . While a number of fundamental concepts are accepted without definition, all subsequent elements of the calculus are derived in a consistent and rigorous way. The description of Euclidean spaces centers on the basis vectors Zi . These important and geometrically intuitive objects are absent from many textbooks. Yet, their use greatly simplifies the introduction of a number of concepts, including i i k the metric tensor Zij D Zi Zj and Christoffel symbol jk D Z @Zj =@Z . Furthermore, the use of vector quantities goes a long way towards helping the student see the world in a way that is independent of Cartesian coordinates. The notation is of paramount importance in mastering the subject. To borrow a sentence from A.J. McConnell [31]: “The notation of the tensor calculus is so much an integral part of the calculus that once the student has become accustomed to its peculiarities he will have gone a long way towards solving the difficulties of the theory itself.” The introduction of the tensor technique is woven into the presentation of the material in Chap. 4. As a result, the framework is described in a natural context that makes the effectiveness of the rules and conventions apparent. This is unlike most other textbooks which introduce the tensor notation in advance of the actual content. In spirit and vision, this book is most similar to A.J. McConnell’s classic Applications of Tensor Calculus [31]. His concrete no-frills approach is perfect for the subject and served as an inspiration for this book’s style. Tullio Levi-Civita’s own The Absolute Differential Calculus [28] is an indispensable source that reveals the motivations of the subject’s co-founder. Since a heavy emphasis in placed on vector-valued quantities, it is important to have good familiarity with geometric vectors viewed as objects on their own terms rather than elements in Rn. A number of textbooks discuss the geometric nature of vectors in great depth. First and foremost is J.W. Gibbs’ classic [14], which served as a prototype for later texts. Danielson [8] also gives a good introduction to geometric vectors and offers an excellent discussion on the subject of differentiation of vector fields. The following books enjoy a good reputation in the modern differential geometry community: [3, 6, 23, 29, 32, 41]. Other popular textbooks, including [38, 43]are known for taking the formal approach to the subject. Virtually all books on the subject focus on applications, with differential geometry front and center. Other common applications include analytical dynamics, continuum mechanics, and relativity theory. Some books focus on particular applications. A case in point is L.V. Bewley’s Tensor Analysis of Electric Circuits And Machines [1]. Bewley envisioned that the tensor approach to electrical engineering would become a standard. Here is hoping his dream eventually comes true. Philadelphia, PA Pavel Grinfeld Contents 1 Why Tensor Calculus? .................................................... 1 Part I Tensors in Euclidean Spaces 2 Rules of the Game .......................................................... 11 2.1 Preview .............................................................. 11 2.2 The Euclidean Space ............................................... 11 2.3 Length, Area, and Volume .......................................... 13 2.4 Scalars and Vectors ................................................. 14 2.5 The Dot Product .................................................... 14 2.5.1 Inner Products and Lengths in Linear Algebra ......... 15 2.6 The Directional Derivative ......................................... 15 2.7 The Gradient ........................................................ 17 2.8 Differentiation of Vector Fields ................................... 18 2.9 Summary ............................................................ 19 3 Coordinate Systems and the Role of Tensor Calculus .................. 21 3.1 Preview .............................................................. 21 3.2 Why Coordinate Systems? ......................................... 21 3.3 What Is a Coordinate System? ..................................... 23 3.4 Perils of Coordinates................................................ 24 3.5 The Role of Tensor Calculus ....................................... 27 3.6 A Catalog of Coordinate Systems .................................. 27 3.6.1 Cartesian Coordinates ................................... 28 3.6.2 Affine Coordinates .....................................

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