Mathematical Engineering. Series editors Claus Hillermeier, Neubiberg, Germany Jörg Schröder, Essen, Germany Bernhard Weigand, Stuttgart, Germany

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1 Mathematical Engineering Series editors Claus Hillermeier, Neubiberg, Germany Jörg Schröder, Essen, Germany Bernhard Weigand, Stuttgart, Germany

2 More information about this series at

3 Hung Nguyen-Schäfer Jan-Philip Schmidt Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers Second Edition

4 Hung Nguyen-Schäfer EM-motive GmbH A Joint Company of Daimler and Bosch Ludwigsburg, Germany Jan-Philip Schmidt Interdisciplinary Center for Scientific Computing (IWR) University of Heidelberg Heidelberg, Germany ISSN ISSN (electronic) Mathematical Engineering ISBN ISBN (ebook) DOI / Library of Congress Control Number: 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

5 In memory of Gregorio Ricci-Curbastro ( ) and Tullio Levi-Civita ( ), who invented Tensor Calculus, for which Elwin Bruno Christoffel ( ) had prepared the ground; Carl Friedrich Gauss ( ) and Bernhard Riemann ( ), who invented Differential Geometry

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7 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 Schr odinger 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 cooperation 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-Schäfer vii

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9 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, incorporating 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 coordinate 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 mechanical 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 mathematicians 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, Heisenberg, 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 mechanix

10 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), continuum 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 Heidelberg, Germany Hung Nguyen-Schäfer Jan-Philip Schmidt

11 Contents 1 General Basis and Bra-Ket Notation Introduction to General Basis and Tensor Types General Basis in Curvilinear Coordinates Orthogonal Cylindrical Coordinates Orthogonal Spherical Coordinates Eigenvalue Problem of a Linear Coupled Oscillator Notation of Bra and Ket Properties of Kets Analysis of Bra and Ket Bra and Ket Bases Gram-Schmidt Scheme of Basis Orthonormalization Cauchy-Schwarz and Triangle Inequalities Computing Ket and Bra Components Inner Product of Bra and Ket Outer Product of Bra and Ket Ket and Bra Projection Components on the Bases Linear Transformation of Kets Coordinate Transformations Hermitian Operator Applying Bra and Ket Analysis to Eigenvalue Problems References Tensor Analysis Introduction to Tensors Definition of Tensors An Example of a Second-Order Covariant Tensor Tensor Algebra General Bases in General Curvilinear Coordinates Metric Coefficients in General Curvilinear Coordinates Tensors of Second Order and Higher Orders xi

12 xii Contents Tensor and Cross Products of Two Vectors in General Bases Rules of Tensor Calculations Coordinate Transformations Transformation in the Orthonormal Coordinates Transformation of Curvilinear Coordinates in E N Examples of Coordinate Transformations Transformation of Curvilinear Coordinates in R N Tensor Calculus in General Curvilinear Coordinates Physical Component of Tensors Derivatives of Covariant Bases Christoffel Symbols of First and Second Kind Prove That the Christoffel Symbols Are Symmetric Examples of Computing the Christoffel Symbols Coordinate Transformations of the Christoffel Symbols Derivatives of Contravariant Bases Derivatives of Covariant Metric Coefficients Covariant Derivatives of Tensors Riemann-Christoffel Tensor Ricci s Lemma Derivative of the Jacobian Ricci Tensor Einstein Tensor References Elementary Differential Geometry Introduction Arc Length and Surface in Curvilinear Coordinates Unit Tangent and Normal Vector to Surface The First Fundamental Form The Second Fundamental Form Gaussian and Mean Curvatures Riemann Curvature Gauss-Bonnet Theorem Gauss Derivative Equations Weingarten s Equations Gauss-Codazzi Equations Lie Derivatives Vector Fields in Riemannian Manifold Lie Bracket Lie Dragging Lie Derivatives Torsion and Curvature in a Distorted and Curved Manifold

13 Contents xiii Killing Vector Fields Invariant Time Derivatives on Moving Surfaces Invariant Time Derivative of an Invariant Field Invariant Time Derivative of Tensors Tangent, Cotangent Bundles and Manifolds Levi-Civita Connection on Manifolds References Differential Forms Introduction Definitions of Spaces on the Manifold Differential k-forms The Notation ω X Exterior Derivatives Interior Product Pullback Operator of Differential Forms Pushforward Operator of Differential Forms The Hodge Star Operator Star Operator in Vector Calculus and Differential Forms Star Operator and Inner Product Star Operator in the Minkowski Spacetime References Applications of Tensors and Differential Geometry Nabla Operator in Curvilinear Coordinates Gradient, Divergence, and Curl Gradient of an Invariant Gradient of a Vector Divergence of a Vector Divergence of a Second-Order Tensor Curl of a Covariant Vector Laplacian Operator Laplacian of an Invariant Laplacian of a Contravariant Vector Applying Nabla Operators in Spherical Coordinates Gradient of an Invariant Divergence of a Vector Curl of a Vector The Divergence Theorem Gauss and Stokes Theorems Green s Identities First Green s Identity Second Green s Identity Differentials of Area and Volume

14 xiv Contents Calculating the Differential of Area Calculating the Differential of Volume Governing Equations of Computational Fluid Dynamics Continuity Equation Navier-Stokes Equations Energy (Rothalpy) Equation Basic Equations of Continuum Mechanics Cauchy s LawofMotion Principal Stresses of Cauchy s Stress Tensor Cauchy s Strain Tensor Constitutive Equations of Elasticity Laws Maxwell s Equations of Electrodynamics Maxwell s Equations in Curvilinear Coordinate Systems Maxwell s Equations in the Four-Dimensional Spacetime The Maxwell s Stress Tensor The Poynting s Theorem Einstein Field Equations Schwarzschild s Solution of the Einstein Field Equations Schwarzschild Black Hole References Tensors and Bra-Ket Notation in Quantum Mechanics Introduction Quantum Entanglement and Nonlocality Alternative Interpretation of Quantum Entanglement The Hilbert Space State Vectors and Basis Kets The Pauli Matrices Combined State Vectors Expectation Value of an Observable Probability Density Operator Density Operator of a Pure Subsystem Density Operator of an Entangled Composite System Heisenberg s Uncertainty Principle The Wave-Particle Duality De Broglie Wavelength Formula The Compton Effect Double-Slit Experiments with Electrons The Schr odinger Equation Time Evolution in Quantum Mechanics The Schr odinger and Heisenberg Pictures Time-Dependent Schr odinger Equation (TDSE) Discussions of the Schr odinger Wave Functions

15 Contents xv 6.13 The Klein-Gordon Equation The Dirac Equation References Appendix A: Relations Between Covariant and Contravariant Bases Appendix B: Physical Components of Tensors Appendix C: Nabla Operators Appendix D: Essential Tensors Appendix E: Euclidean and Riemannian Manifolds Appendix F: Probability Function for the Quantum Interference Appendix G: Lorentz and Minkowski Transformations in Spacetime Appendix H: The Law of Large Numbers in Statistical Mechanics Mathematical Symbols in This Book Further Reading Index

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17 About the Authors Hung Nguyen-Schäfer is a senior technical manager in development of electric machines for hybrid and electric vehicles at EM-motive GmbH, a joint company of Daimler and Bosch in Germany. He received B.Sc. and M.Sc. in mechanical engineering with nonlinear vibrations in fluid mechanics from the University of Karlsruhe (KIT), Germany, in 1985 and a Ph.D. degree in nonlinear thermo- and fluid dynamics from the same university in He joined Bosch Company and worked as a technical manager on many development projects. Between 2007 and 2013, he was in charge of rotordynamics, bearings, and design platforms of automotive turbochargers at Bosch Mahle Turbo Systems in Stuttgart. He is also the author of three other professional engineering books Aero and Vibroacoustics of Automotive Turbochargers, Springer (2013), Rotordynamics of Automotive Turbochargers in Springer Tracts in Mechanical Engineering, Second Edition, Springer (2015), and Computational Design of Rolling Bearings, Springer (2016). Jan-Philip Schmidt is a mathematician. He studied mathematics, physics, and economics at the University of Heidelberg, Germany. He received a Ph.D. degree in mathematics from the University of Heidelberg in His doctoral thesis was funded by a research fellowship from the Heidelberg Academy of Sciences, in collaboration with the Interdisciplinary Center for Scientific Computing (IWR) at the University of Heidelberg. His academic working experience comprises several research visits in France and Israel, as well as project works at the Max-Planck- Institute for Mathematics in the Sciences (MPIMIS) in Leipzig and at the Max- Planck-Institute for Molecular Genetics (MPIMG) in Berlin. He also worked as a research associate in the AVACS program at Saarland University, Cluster of Excellence (MMCI). xvii