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Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers Mathematical Engineering Hung Nguyen-Schäfer Jan-Philip Schmidt Tensor Analysis and Elementary Diff erential Geometry for Physicists and Engineers Second Edition Mathematical Engineering Series editors Claus Hillermeier, Neubiberg, Germany Jo¨rg Schro¨der, Essen, Germany Bernhard Weigand, Stuttgart, Germany More information about this series at http://www.springer.com/series/8445 Hung Nguyen-Scha¨fer • Jan-Philip Schmidt Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers Second Edition Hung Nguyen-Scha¨fer Jan-Philip Schmidt EM-motive GmbH Interdisciplinary Center for Scientific Computing (IWR) A Joint Company of Daimler University of Heidelberg and Bosch Heidelberg, Germany Ludwigsburg, Germany ISSN 2192-4732 ISSN 2192-4740 (electronic) Mathematical Engineering ISBN 978-3-662-48495-1 ISBN 978-3-662-48497-5 (eBook) DOI 10.1007/978-3-662-48497-5 Library of Congress Control Number: 2016940970 © Springer-Verlag Berlin Heidelberg 2014, 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Berlin Heidelberg In memory of Gregorio Ricci-Curbastro (1853–1925) and Tullio Levi-Civita (1873–1941), who invented Tensor Calculus, for which Elwin Bruno Christoffel (1829–1900) had prepared the ground; Carl Friedrich Gauss (1777–1855) and Bernhard Riemann (1826–1866), who invented Differential Geometry ThiS is a FM Blank Page Preface to the Second Edition In the second edition, all chapters are carefully revised in which typos are corrected and many additional sections are included. Furthermore, two new chapters that deal with Cartan differential forms and applications of tensors and Dirac notation to quantum mechanics are added in this book. Contrary to tensors, Cartan differential forms based on exterior algebra in Chap. 4 using the wedge product are an approach to multivariable calculus that is independent of coordinates. Therefore, they are very useful methods for differential geometry, topology, and theoretical physics in multidimensional manifolds. In Chap. 6, the quantum entanglement of a composite system that consists of two entangled subsystems has alternatively been interpreted by means of symmetries. Both mathematical approaches of Dirac matrix and wave formulations are used to analyze and calculate the expectation values, probability density operators, and wave functions for nonrelativistic and relativistic particles in a composite system using time-dependent Schrodinger€ equation (TDSE), Klein–Gordon equation, and Dirac equation as well. I would like to thank Mrs. Eva Hestermann-Beyerle and Mrs. Birgit Kollmar- Thoni at Springer Heidelberg for their invaluable suggestions and excellent coop- eration to publish this second edition successfully. Finally, my special thanks go to my wife for her understanding, patience, and endless support as I wrote this book in my leisure and vacation time. Ludwigsburg, Germany Hung Nguyen-Scha¨fer vii ThiS is a FM Blank Page Preface to the First Edition This book represents a joint effort by a research engineer and a mathematician. The initial idea for it arose from our many years of experience in the automotive industry, from advanced research development, and of course from our common research interest in applied mathematics, physics, and engineering. The main reason for this cooperation is the fact that mathematicians generally approach problems using mathematical rigor, but which need not always be practically applicable; at the same time, engineers usually deal with problems involving applied mathematics, which must ultimately work in the “real world” of industry. Having recognized that what mathematicians consider rigor can be more like rigor mortis for engineers and physicists, this joint effort proposes a compromise between the mathematical rigors and less rigorous applied mathematics, incorpo- rating different points of view. Our main aim is to bridge the mathematical gap between where physics and engineering mathematics end and where tensor analysis begins, which we do with the help of a powerful and user-friendly tool often employed in computational methods for physical and engineering problems in any general curvilinear coordi- nate system. However, tensor analysis has certain strict rules and conventions that must unconditionally be adhered to. This book is intended to support research scientists and practicing engineers in various fields who use tensor analysis and differential geometry in the context of applied physics and electrical and mechan- ical engineering. Moreover, it can also be used as a textbook for graduate students in applied physics and engineering. Tensor analysis and differential geometry were pioneered by great mathemati- cians in the late nineteenth century, chiefly Curbastro, Levi-Civita, Christoffel, Ricci, Gauss, Riemann, Weyl, and Minkowski, and later promoted by well-known theoretical physicists in the early twentieth century, mainly Einstein, Dirac, Hei- senberg, and Fermi, working on relativity and quantum mechanics. Since then, tensor analysis and differential geometry have taken on an increasingly important role in the mathematical language used in the modern physics of quantum mechan- ix x Preface to the First Edition ics and general relativity and in many applied sciences fields. They have also been applied to computational mechanical and electrical engineering in classical mechanics, aero- and vibroacoustics, computational fluid dynamics (CFD), contin- uum mechanics, electrodynamics, and cybernetics. Approaching the topics of tensors and differential geometry in a mathematically rigorous way would require an immense amount of effort, which would not be practical for working engineers and applied physicists. As such, we decided to present these topics in a comprehensive and approachable way that will show readers how to work with tensors and differential geometry and to apply them to modeling the physical and engineering world. This book also includes numerous examples with solutions and concrete calculations in order to guide readers through these complex topics step by step. For the sake of simplicity and keeping the target audience in mind, we deliberately neglect certain aspects of mathematical rigor in this book, discussing them informally instead. Therefore, those readers who are more mathematically interested should consult the recommended literature. We would like to thank Mrs. Hestermann-Beyerle and Mrs. Kollmar-Thoni at Springer Heidelberg for their helpful suggestions and valued cooperation during the preparation of this book. Ludwigsburg, Germany Hung Nguyen-Scha¨fer Heidelberg, Germany Jan-Philip Schmidt Contents 1 General Basis and Bra-Ket Notation ......................... 1 1.1 Introduction to General Basis and Tensor Types . 1 1.2 General Basis in Curvilinear Coordinates . 2 1.2.1 Orthogonal Cylindrical Coordinates . ............ 5 1.2.2 Orthogonal Spherical Coordinates ................. 8 1.3 Eigenvalue Problem of a Linear Coupled Oscillator . ........ 11 1.4 Notation of Bra and Ket . ............................ 15 1.5 Properties of Kets . ............. 15 1.6 Analysis of Bra and Ket . ........................... 16 1.6.1 Bra and Ket Bases ............................ 16 1.6.2 Gram-Schmidt Scheme of Basis Orthonormalization . 18 1.6.3 Cauchy-Schwarz and Triangle Inequalities . ....... 19 1.6.4 Computing Ket and Bra Components . ............. 19 1.6.5 Inner Product of Bra and Ket . ................... 20 1.6.6 Outer Product of Bra and Ket . 22 1.6.7 Ket and Bra Projection Components on the Bases . 23 1.6.8 Linear Transformation of Kets ................... 23 1.6.9 Coordinate Transformations . .................. 25 1.6.10 Hermitian Operator . .......... 29 1.7 Applying Bra and Ket Analysis to Eigenvalue Problems . .... 31 References . 34 2 Tensor Analysis ........................................ 35 2.1 Introduction to Tensors . 35 2.2 Definition of Tensors . .............................. 36 2.2.1 An Example of a Second-Order Covariant Tensor ..... 38 2.3 Tensor Algebra . ................................... 38 2.3.1 General Bases in General Curvilinear Coordinates ..... 38 2.3.2 Metric Coefficients in General Curvilinear Coordinates . 47 2.3.3 Tensors of Second Order and Higher Orders ......... 51 xi xii Contents 2.3.4 Tensor and Cross Products of Two Vectors in General Bases ..................................... 55 2.3.5 Rules of Tensor Calculations
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