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Undergraduate Lecture Notes in Physics Siegfried Hess Tensors for Physics Undergraduate Lecture Notes in Physics Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topicsthroughoutpureandappliedphysics.Eachtitleintheseriesissuitableasabasisfor undergraduateinstruction,typicallycontainingpracticeproblems,workedexamples,chapter summaries, andsuggestions for further reading. ULNP titles mustprovide at least oneof thefollowing: (cid:129) Anexceptionally clear andconcise treatment of astandard undergraduate subject. (cid:129) Asolidundergraduate-levelintroductiontoagraduate,advanced,ornon-standardsubject. (cid:129) Anovel perspective or anunusual approachto teaching asubject. ULNPespeciallyencouragesnew,original,andidiosyncraticapproachestophysicsteaching at theundergraduate level. ThepurposeofULNPistoprovideintriguing,absorbingbooksthatwillcontinuetobethe reader’spreferred reference throughout theiracademic career. Series editors Neil Ashby ProfessorEmeritus, University of Colorado, Boulder, CO,USA William Brantley Professor, FurmanUniversity, Greenville, SC,USA Michael Fowler Professor, University of Virginia, Charlottesville, VA, USA Morten Hjorth-Jensen Professor, University of Oslo, Oslo,Norway Michael Inglis Professor, SUNY Suffolk CountyCommunity College, LongIsland, NY,USA Heinz Klose ProfessorEmeritus, Humboldt University Berlin, Germany HelmySherif Professor, University of Alberta, Edmonton, AB,Canada More information about this series at http://www.springer.com/series/8917 Siegfried Hess Tensors for Physics 123 Siegfried Hess InstituteforTheoretical Physics Technical University Berlin Berlin Germany ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate LectureNotes inPhysics ISBN 978-3-319-12786-6 ISBN 978-3-319-12787-3 (eBook) DOI 10.1007/978-3-319-12787-3 LibraryofCongressControlNumber:2015936466 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 dissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis 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 hereinorforanyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia (www.springer.com) Preface Tensors are needed in Physics to describe anisotropies and orientational behavior. While every physics student knows what a vector is, there is often an uneasiness about the notion tensor. In lectures, I used to tell students: “you can be a good physicist without knowing much about tensors, but when you learn how to handle tensors and what they are good for, you will have a considerable advantage. And hereisyourchancetolearnabouttensorsasamathematicaltoolandtogetfamiliar with their applications to physics.” This book is, up to Chap. 14, largely based on the two books: Siegfried Hess, Vektor- und Tensor-Rechnung, which, in turn, was based on lectures for first-year physics students, and Siegfried Hess and Walter Köhler, Formeln zur Tensor-Rechnung, a collection of computational rules and formulas needed in more advanced theory. Both books were published by Palm and Enke, Erlangen, Germany in 1980, reprinted in 1982, but are out of print since many years. Here,theemphasis isonCartesian tensorsin3D.Theapplications oftensorsto be presented are strongly influenced by my presentations of the standard four courses in Theoretical Physics: Mechanics, Quantum Mechanics, Electrodynamics and Optics, Thermodynamics and Statistical Physics, and by my research experi- enceinthekinetictheoryofgasesofparticles withspinandofrotatingmolecules, intransport,orientationalandopticalphenomenaofmolecularfluids,liquidcrystals and colloidal dispersions, in hydrodynamics and rheology, as well as in the elastic and plastic properties of solids. The original publications cited, in particular in the second part of the book, show a wide range of applications of tensors. An outlook to4DisprovidedinChap.18,wheretheMaxwellequationsofelectrodynamicsare formulated in the appropriate four-dimensional form. While learning the mathematics, first- and second-year students may skip the applications involving physics they are not yet familiar with, however, brief introductions to basic physics are given at many places in the book. Exercises are found throughout the book, answers and solutions are given at the end. v vi Preface Here, I wish to express my gratitude to Prof. Ludwig Waldmann (1913–1980), who introduced me to Cartesian Tensors, quite some time ago, when I was a student. I thank my master- and PhD-students, postdocs, co-workers, and col- leagues for fruitful cooperation on research projects, where tensors played a key role. I am grateful to Springer for publishing this Tensor book in the series Undergraduate Lecture Notes in Physics, and I thank Adelheid Duhm, Project CoordinatoratProductionPhysicsBooksofSpringerinHeidelbergforherdiligent editorial work. Berlin Siegfried Hess Contents Part I A Primer on Vectors and Tensors 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Preliminary Remarks on Vectors. . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Norm and Distance. . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.3 Vectors for Classical Physics . . . . . . . . . . . . . . . . . . 6 1.1.4 Vectors for Special Relativity. . . . . . . . . . . . . . . . . . 7 1.2 Preliminary Remarks on Tensors. . . . . . . . . . . . . . . . . . . . . . 7 1.3 Remarks on History and Literature . . . . . . . . . . . . . . . . . . . . 8 1.4 Scope of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Coordinate System and Position Vector. . . . . . . . . . . . . . . . . 11 2.1.1 Cartesian Components. . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Length of the Position Vector, Unit Vector . . . . . . . . 12 2.1.3 Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.4 Spherical Polar Coordinates . . . . . . . . . . . . . . . . . . . 14 2.2 Vector as Linear Combination of Basis Vectors . . . . . . . . . . . 14 2.2.1 Orthogonal Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.2 Non-orthogonal Basis . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Linear Transformations of the Coordinate System. . . . . . . . . . 16 2.3.1 Translation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.2 Affine Transformation. . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Rotation of the Coordinate System . . . . . . . . . . . . . . . . . . . . 19 2.4.1 Orthogonal Transformation . . . . . . . . . . . . . . . . . . . 19 2.4.2 Proper Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Definitions of Vectors and Tensors in Physics . . . . . . . . . . . . 22 2.5.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5.2 What is a Tensor?. . . . . . . . . . . . . . . . . . . . . . . . . . 23 vii viii Contents 2.5.3 Multiplication by Numbers and Addition of Tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5.4 Remarks on Notation. . . . . . . . . . . . . . . . . . . . . . . . 24 2.5.5 Why the Emphasis on Tensors? . . . . . . . . . . . . . . . . 24 2.6 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6.1 Parity Operation. . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6.2 Parity of Vectors and Tensors. . . . . . . . . . . . . . . . . . 26 2.6.3 Consequences for Linear Relations . . . . . . . . . . . . . . 27 2.6.4 Application: Linear and Nonlinear Susceptibility Tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.7 Differentiation of Vectors and Tensors with Respect to a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.7.1 Time Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.7.2 Trajectory and Velocity. . . . . . . . . . . . . . . . . . . . . . 29 2.7.3 Radial and Azimuthal Components of the Velocity. . . 30 2.8 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Symmetry of Second Rank Tensors, Cross Product . . . . . . . . . . . 33 3.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Symmetric and Antisymmetric Parts . . . . . . . . . . . . . 33 3.1.2 Isotropic, Antisymmetric and Symmetric Traceless Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1.3 Trace of a Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1.4 Multiplication and Total Contraction of Tensors, Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.5 Fourth Rank Projections Tensors . . . . . . . . . . . . . . . 36 3.1.6 Preliminary Remarks on “Antisymmetric Part and Vector” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.7 Preliminary Remarks on the Symmetric Traceless Part. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Dyadics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2.1 Definition of a Dyadic Tensor . . . . . . . . . . . . . . . . . 37 3.2.2 Products of Symmetric Traceless Dyadics . . . . . . . . . 38 3.3 Antisymmetric Part, Vector Product . . . . . . . . . . . . . . . . . . . 40 3.3.1 Dual Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.2 Vector Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4 Applications of the Vector Product . . . . . . . . . . . . . . . . . . . . 43 3.4.1 Orbital Angular Momentum. . . . . . . . . . . . . . . . . . . 43 3.4.2 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4.3 Motion on a Circle . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.4.4 Lorentz Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4.5 Screw Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Contents ix 4 Epsilon-Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1 Definition, Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1.1 Link with Determinants. . . . . . . . . . . . . . . . . . . . . . 47 4.1.2 Product of Two Epsilon-Tensors. . . . . . . . . . . . . . . . 48 4.1.3 Antisymmetric Tensor Linked with a Vector . . . . . . . 50 4.2 Multiple Vector Products. . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2.1 Scalar Product of Two Vector Products. . . . . . . . . . . 50 4.2.2 Double Vector Products. . . . . . . . . . . . . . . . . . . . . . 50 4.3 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.3.1 Angular Momentum for the Motion on a Circle . . . . . 51 4.3.2 Moment of Inertia Tensor . . . . . . . . . . . . . . . . . . . . 52 4.4 Dual Relation and Epsilon-Tensor in 2D . . . . . . . . . . . . . . . . 53 4.4.1 Definitions and Matrix Notation. . . . . . . . . . . . . . . . 53 5 Symmetric Second Rank Tensors. . . . . . . . . . . . . . . . . . . . . . . . . 55 5.1 Isotropic and Symmetric Traceless Parts . . . . . . . . . . . . . . . . 55 5.2 Principal Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2.1 Principal Axes Representation . . . . . . . . . . . . . . . . . 56 5.2.2 Isotropic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2.3 Uniaxial Tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2.4 Biaxial Tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2.5 Symmetric Dyadic Tensors . . . . . . . . . . . . . . . . . . . 59 5.3 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.3.1 Moment of Inertia Tensor of Molecules. . . . . . . . . . . 60 5.3.2 Radius of Gyration Tensor. . . . . . . . . . . . . . . . . . . . 62 5.3.3 Molecular Polarizability Tensor . . . . . . . . . . . . . . . . 63 5.3.4 Dielectric Tensor, Birefringence . . . . . . . . . . . . . . . . 63 5.3.5 Electric and Magnetic Torques. . . . . . . . . . . . . . . . . 64 5.4 Geometric Interpretation of Symmetric Tensors. . . . . . . . . . . . 65 5.4.1 Bilinear Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.4.2 Linear Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.4.3 Volume and Surface of an Ellipsoid . . . . . . . . . . . . . 67 5.5 Scalar Invariants of a Symmetric Tensor . . . . . . . . . . . . . . . . 69 5.5.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.5.2 Biaxiality of a Symmetric Traceless Tensor . . . . . . . . 69 5.6 Hamilton-Cayley Theorem and Consequences. . . . . . . . . . . . . 71 5.6.1 Hamilton-Cayley Theorem. . . . . . . . . . . . . . . . . . . . 71 5.6.2 Quadruple Products of Tensors. . . . . . . . . . . . . . . . . 72 5.7 Volume Conserving Affine Transformation . . . . . . . . . . . . . . 73 5.7.1 Mapping of a Sphere onto an Ellipsoid . . . . . . . . . . . 73 5.7.2 Uniaxial Ellipsoid. . . . . . . . . . . . . . . . . . . . . . . . . . 73