DOCSLIB.ORG

Intro-Chap VI

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Intro-Chap VI Pre-metric Electromagnetism By D. H. Delphenich For every good man chopping at the roots of evil there are a thousand more hacking at the leaves. Henry David Thoreau Leaf-hacking, being more labor-intensive than root-chopping, is more likely to get funding. David Henry Delphenich CONTENTS Page Table of Figures………………………………………………………………... iv Introduction…………………………………………………………………….. 1 1. The unification of electricity and magnetism 1 2. The evolution of geometrical optics into wave optics 3 3. Geometrization of gravity 7 4. Attempts at unifying electromagnetism with gravitation 10 5. Rise of quantum electrodynamics 15 6. Spacetime topology and electromagnetism 19 7. Pre-metric electromagnetism 20 8. Summary of contents 23 Chapter I – Calculus of exterior differential forms…………………………… 29 1. Multilinear algebra 29 2. Exterior algebra 32 3. Exterior derivative 41 4. Divergence operator 43 5. Lie derivative 44 6. Integration of differential forms 47 7. Relationship to vector calculus 49 Chapter II – Topology of differentiable manifolds……………………………… 54 1. Differentiable manifolds 54 2. Differential forms on manifolds 67 3. Differentiable singular cubic chains 68 4. Integration of differential forms 74 5. De Rham’s theorem 75 6. Hodge theory 78 7. Time-space splittings of the spacetime manifold 82 Chapter III – Static electric and magnetic fields…………………………………. 86 1. Electric charge 87 2. Electric field strength and flux 90 3. Electric excitation (displacement) 93 4. Electrostatic field equations 95 5. Electrostatic potential functions 96 6. Magnetic charge and flux 97 7. Magnetic excitation (induction) 100 8. Magnetostatic field equations 102 9. Magnetostatic potential 1-forms 103 10. Field-source duality 104 ii Pre-metric electromagnetism Chapter IV – Dynamic electromagnetic fields…………………………………….. 107 1. Electromagnetic induction 107 2. Conservation of charge 110 3. Pre-metric Maxwell equations 112 4. Electromagnetic potential 1-forms 114 Chapter V – Electromagnetic constitutive laws……………………………………. 120 1. Electromagnetic fields in macroscopic matter 120 2. Linear constitutive laws 121 3. Examples of local linear media 125 4. Some physical phenomena due to magneto-electric couplings 133 5. Nonlinear constitutive laws 135 6. Effective quantum constitutive laws 138 Chapter VI – Partial differential equations on manifolds………………………… 148 1. Differential operators on vector bundles 148 2. Jet manifolds 154 3. Exterior differential systems 157 4. Boundary-value problems 159 5. Initial-value problems 161 6. Distributions on differential forms 165 7. Fundamental solutions 170 8. Fourier transforms 175 Chapter VII – The interaction of fields and sources………………………………. 180 1. Construction of fields from sources 181 2. Lorentz force 192 3. Interaction of fields 195 4. Interaction of currents 197 Chapter VIII – Electromagnetic waves ……………………………………………. 201 1. Electromagnetic waves 201 2. Characteristics 213 3. Examples of dispersion laws 217 4. Speed of wave propagation 225 Chapter IX – Geometrical optics…………………………………………………… 231 1. The generalized eikonal equation 232 2. Bicharacteristics – null geodesics 233 3. Parallel translation 239 4. Huygens’s principle 243 5. Diffraction 254 Chapter X – The calculus of variations and electromagnetism…………………… 258 1. Electromagnetic energy 259 2. The calculus of variations in terms of vector bundles 267 Contents iii 3. Variational formulation of electromagnetic problems 272 4. Motion of a charge in an electromagnetic field 275 5. Fermat’s principle 287 Chapter XI – Symmetries of electromagnetism…………………………………… 291 1. Transformation groups 292 2. Symmetries of action functionals 300 3. Examples of Noether symmetries 303 4. Symmetries of electromagnetic action functionals 308 5. Symmetries of systems of differential equations 315 Chapter XII – Projective geometry and electromagnetism………………………… 330 1. Elementary projective geometry 332 2. Projective geometry and mechanics 342 3. The Plücker-Klein embedding 353 4. Projective geometry and electromagnetism 356 Chapter XIII – Complex relativity and line geometry………………………………. 362 1. Complex structures on real vector spaces 363 2. Isomorphic representations of the Lorentz group 367 3. General relativity in terms of Lorentzian frames 381 4. General relativity in terms of complex orthonormal frames 398 5. Discussion 408 Index………………………………………………………………………………… 411 LIST OF FIGURES 1. The boundary of a boundary of a 2-cube. 70 2. The representation of a circle as a 1-chain with boundary zero. 70 3. Two-dimensional spaces that are described by 2-chains. 71 4. Magnetic hysteresis. 135 5. Initial-value problems for ordinary and partial differential equations. 161 6. The fundamental solution for d/dx . 171 7. A typical 4-cycle that intersects the hypersurface φ(x) = 0. 216 8. The general Fresnel quartic (biaxial case). 219 9. Fresnel quartic (uniaxial case). 220 10. The construction of the evolved momentary wave surfaces by means of Huygens’s principle. 250 11. Geodesics in the geometrical optics approximation and in the diffracted case 257 12. Bringing a current loop in from infinity in an external magnetic field. 264 13. Klein’s hierarchy of geometries. 331 14. The projective line as an affine line plus a point at infinity. 336 15. An example of a perspectivity in the projective plane. 338 16. The effect of a converging lens on parallel lines. 343 17. The projection of a helix from homogeneous to inhomogeneous coordinates. 344 18. The planes defined by an isotropic F. 360 Introduction The term “pre-metric electromagnetism” refers to the formulation of the mathematical theory of electromagnetism in a manner that not only does not assume the existence of a Lorentzian metric on the spacetime manifold to begin with, but also exhibits the appearance of such a geometrical structure as a natural consequence of investigating the manner in which electromagnetic waves propagate through that medium. Since the Lorentzian metric that appears – at least, under restricted conditions – is commonly taken to account for the presence of gravitation in the spacetime medium in the viewpoint of general relativity, one sees that in order to properly define the context of pre-metric electromagnetism one must really discuss not only electricity and magnetism, but also gravity, as well. Hence, before we embark upon the detailed discussion of the mathematical and physical bases for the theory of pre-metric electromagnetism, we shall briefly recall the conceptual evolution of the three relevant physical phenomena of electricity, magnetism, and gravity. 1. The unification of electricity and magnetism 1. In ancient times, it is unlikely that man ever suspected that the natural phenomena associated with electricity, such as lightning and static electricity on animal furs, could possibly be associated with magnetic phenomena, which were discovered somewhat later in history, and most likely in the Iron Age in the context of lodestones, which are magnetite deposits that have become magnetized from long-term exposure to the Earth’s magnetic field. However, when Europe emerged from the Dark Ages into the Renaissance the science of electricity gradually gave way to the development of batteries 2, wires, and currents, on the one hand, with the development of compasses for marine navigation on the other. It was only a matter of time before the early experimenters, primarily Hans Christian Oersted (1777-1851) and Michael Faraday (1791-1867), noticed that electrical currents could produce measurable magnetic fields around conductors, while time- varying magnetic fields could conversely induce electrical currents in current loops. Actually, this is not a precise converse, since a time-constant electrical current can induce a time-constant magnetic field, while a time-constant magnetic field will not induce an electrical current of any sort. This reciprocal set of phenomena was called electromagnetic induction . One of Faraday’s other innovations in the name of electromagnetism was the introduction of the concept of invisible force fields distributed in space that accounted to the forces of attraction or repulsion that “test” electric charges and magnetic dipoles experienced when placed at each point. Although nowadays the concept of vector fields 1 For a comprehensive historical timeline of the theories of electricity and the ether, one should confer the tomely two-volume treatise of E. T. Whittaker [ 1]. 2 Apparently, the concept of a battery had been developed in a rudimentary form by ancient cultures, as pottery that seemed to involve weak acids and metal electrodes has been unearthed by archeologists. 2 Pre-metric electromagnetism seems rather commonplace and above dispute, nevertheless, in its day the ideas of invisible lines of force apparently seemed rather mystical and dubious. It is important to note that the key to defining such fields is the association of a purely dynamical notion – namely, force – with more esoteric ones, in the form of electrical and magnetic fields. In addition to this key development, one must also observe the evolution of the theory of electrostatic forces from the early measurements by Charles Augustin de Coulomb (1736-1806) and the formulation of the empirical law that bears his name to its formulation in terms of potential theory by Laplace, Poisson, and the host of contributors to the theory of boundary-value problems in the Laplace or Poisson equation, such as Green, Cauchy, Dirichlet, Neumann, Robin, and many more. It was also found that although the static magnetic field was
Load more

Recommended publications
  • Ch Pter 1 Electricity and Magnetism Fundamentals
    CH PTER 1 ELECTRICITY AND MAGNETISM FUNDAMENTALS PART ONE 1. Who discovered the relationship between magnetism and electricity that serves as the foundation for the theory of electromagnetism? A. Luigi Galvani B. Hans Christian Oersted C.Andre Ampere D. Charles Coulomb 2. Who demonstrated the theory of electromagnetic induction in 1831? A. Michael Faraday B.Andre Ampere C.James Clerk Maxwell D. Charles Coulomb 3. Who developed the electromagnetic theory of light in 1862? A. Heinrich Rudolf Hertz B. Wilhelm Rontgen C. James Clerk Maxwell D. Andre Ampere 4. Who discovered that a current-carrying conductor would move when placed in a magnetic field? A. Michael Faraday B.Andre Ampere C.Hans Christian Oersted D. Gustav Robert Kirchhoff 5. Who discovered the most important electrical effects which is the magnetic effect? A. Hans Christian Oersted B. Sir Charles Wheatstone C.Georg Ohm D. James Clerk Maxwell 6. Who demonstrated that there are magnetic effects around every current-carrying conductor and that current-carrying conductors can attract and repel each other just like magnets? A. Luigi Galvani B.Hans Christian Oersted C. Charles Coulomb D. Andre Ampere 7. Who discovered superconductivity in 1911? A. Kamerlingh Onnes B.Alex Muller C.Geory Bednorz D. Charles Coulomb 8. The magnitude of the induced emf in a coil is directly proportional to the rate of change of flux linkages. This is known as A. Joule¶s Law B. Faraday¶s second law of electromagnetic induction C.Faraday¶s first law of electromagnetic induction D. Coulomb¶s Law 9. Whenever a flux inking a coil or current changes, an emf is induced in it.
    [Show full text]
  • Advanced Magnetism and Electromagnetism
    ELECTRONIC TECHNOLOGY SERIES ADVANCED MAGNETISM AND ELECTROMAGNETISM i/•. •, / .• ;· ... , -~-> . .... ,•.,'. ·' ,,. • _ , . ,·; . .:~ ~:\ :· ..~: '.· • ' ~. 1. .. • '. ~:;·. · ·!.. ., l• a publication $2 50 ADVANCED MAGNETISM AND ELECTROMAGNETISM Edited by Alexander Schure, Ph.D., Ed. D. - JOHN F. RIDER PUBLISHER, INC., NEW YORK London: CHAPMAN & HALL, LTD. Copyright December, 1959 by JOHN F. RIDER PUBLISHER, INC. All rights reserved. This book or parts thereof may not be reproduced in any form or in any language without permission of the publisher. Library of Congress Catalog Card No. 59-15913 Printed in the United States of America PREFACE The concepts of magnetism and electromagnetism form such an essential part of the study of electronic theory that the serious student of this field must have a complete understanding of these principles. The considerations relating to magnetic theory touch almost every aspect of electronic development. This book is the second of a two-volume treatment of the subject and continues the attention given to the major theoretical con­ siderations of magnetism, magnetic circuits and electromagnetism presented in the first volume of the series.• The mathematical techniques used in this volume remain rela­ tively simple but are sufficiently detailed and numerous to permit the interested student or technician extensive experience in typical computations. Greater weight is given to problem solutions. To ensure further a relatively complete coverage of the subject matter, attention is given to the presentation of sufficient information to outline the broad concepts adequately. Rather than attempting to cover a large body of less important material, the selected major topics are treated thoroughly. Attention is given to the typical practical situations and problems which relate to the subject matter being presented, so as to afford the reader an understanding of the applications of the principles he has learned.
    [Show full text]
  • Bibliography
    Bibliography [AK98] V. I. Arnold and B. A. Khesin, Topological methods in hydrodynamics, Springer- Verlag, New York, 1998. [AL65] Holt Ashley and Marten Landahl, Aerodynamics of wings and bodies, Addison- Wesley, Reading, MA, 1965, Section 2-7. [Alt55] M. Altman, A generalization of Newton's method, Bulletin de l'academie Polonaise des sciences III (1955), no. 4, 189{193, Cl.III. [Arm83] M.A. Armstrong, Basic topology, Springer-Verlag, New York, 1983. [Bat10] H. Bateman, The transformation of the electrodynamical equations, Proc. Lond. Math. Soc., II, vol. 8, 1910, pp. 223{264. [BB69] N. Balabanian and T.A. Bickart, Electrical network theory, John Wiley, New York, 1969. [BLG70] N. N. Balasubramanian, J. W. Lynn, and D. P. Sen Gupta, Differential forms on electromagnetic networks, Butterworths, London, 1970. [Bos81] A. Bossavit, On the numerical analysis of eddy-current problems, Computer Methods in Applied Mechanics and Engineering 27 (1981), 303{318. [Bos82] A. Bossavit, On finite elements for the electricity equation, The Mathematics of Fi- nite Elements and Applications IV (MAFELAP 81) (J.R. Whiteman, ed.), Academic Press, 1982, pp. 85{91. [Bos98] , Computational electromagnetism: Variational formulations, complementar- ity, edge elements, Academic Press, San Diego, 1998. [Bra66] F.H. Branin, The algebraic-topological basis for network analogies and the vector cal- culus, Proc. Symp. Generalised Networks, Microwave Research, Institute Symposium Series, vol. 16, Polytechnic Institute of Brooklyn, April 1966, pp. 453{491. [Bra77] , The network concept as a unifying principle in engineering and physical sciences, Problem Analysis in Science and Engineering (K. Husseyin F.H. Branin Jr., ed.), Academic Press, New York, 1977.
    [Show full text]
  • Chapter 7 Magnetism and Electromagnetism
    Chapter 7 Magnetism and Electromagnetism Objectives • Explain the principles of the magnetic field • Explain the principles of electromagnetism • Describe the principle of operation for several types of electromagnetic devices • Explain magnetic hysteresis • Discuss the principle of electromagnetic induction • Describe some applications of electromagnetic induction 1 The Magnetic Field • A permanent magnet has a magnetic field surrounding it • A magnetic field is envisioned to consist of lines of force that radiate from the north pole to the south pole and back to the north pole through the magnetic material Attraction and Repulsion • Unlike magnetic poles have an attractive force between them • Two like poles repel each other 2 Altering a Magnetic Field • When nonmagnetic materials such as paper, glass, wood or plastic are placed in a magnetic field, the lines of force are unaltered • When a magnetic material such as iron is placed in a magnetic field, the lines of force tend to be altered to pass through the magnetic material Magnetic Flux • The force lines going from the north pole to the south pole of a magnet are called magnetic flux (φ); units: weber (Wb) •The magnetic flux density (B) is the amount of flux per unit area perpendicular to the magnetic field; units: tesla (T) 3 Magnetizing Materials • Ferromagnetic materials such as iron, nickel and cobalt have randomly oriented magnetic domains, which become aligned when placed in a magnetic field, thus they effectively become magnets Electromagnetism • Electromagnetism is the production
    [Show full text]
  • The Principle of Equivalence As a Criterion of Identity Ryan Samaroo
    Forthcoming in Synthese The Principle of Equivalence as a Criterion of Identity Ryan Samaroo Somerville College, Woodstock Road, Oxford, OX2 6HD, United Kingdom [email protected] Dedicated to the memory of William (Bill) Demopoulos Abstract In 1907 Einstein had the insight that bodies in free fall do not “feel” their own weight. This has been formalized in what is called “the principle of equivalence.” The principle motivated a critical analysis of the Newtonian and special-relativistic concepts of inertia, and it was indispensable to Einstein’s development of his theory of gravitation. A great deal has been written about the principle. Nearly all of this work has focused on the content of the principle and whether it has any content in Einsteinian gravitation, but more remains to be said about its methodological role in the development of the theory. I argue that the principle should be understood as a kind of foundational principle known as a criterion of identity. This work extends and substantiates a recent account of the notion of a criterion of identity by William Demopoulos. Demopoulos argues that the notion can be employed more widely than in the foundations of arithmetic and that we see this in the development of physical theories, in particular space-time theories. This new account forms the basis of a general framework for applying a number of mathematical theories and for distinguishing between applied mathematical theories that are and are not empirically constrained. 1. Introduction “The principle of equivalence,” which Einstein originally used to refer to one particular statement, has come to refer to a number of interrelated principles in the theory of gravitation.1 1 Einstein’s first formulation of the principle can be found in his article “On the Relativity Principle and the Conclusions Drawn from It” (1907, p.
    [Show full text]
  • Lecture 8: Magnets and Magnetism Magnets
    Lecture 8: Magnets and Magnetism Magnets •Materials that attract other metals •Three classes: natural, artificial and electromagnets •Permanent or Temporary •CRITICAL to electric systems: – Generation of electricity – Operation of motors – Operation of relays Magnets •Laws of magnetic attraction and repulsion –Like magnetic poles repel each other –Unlike magnetic poles attract each other –Closer together, greater the force Magnetic Fields and Forces •Magnetic lines of force – Lines indicating magnetic field – Direction from N to S – Density indicates strength •Magnetic field is region where force exists Magnetic Theories Molecular theory of magnetism Magnets can be split into two magnets Magnetic Theories Molecular theory of magnetism Split down to molecular level When unmagnetized, randomness, fields cancel When magnetized, order, fields combine Magnetic Theories Electron theory of magnetism •Electrons spin as they orbit (similar to earth) •Spin produces magnetic field •Magnetic direction depends on direction of rotation •Non-magnets → equal number of electrons spinning in opposite direction •Magnets → more spin one way than other Electromagnetism •Movement of electric charge induces magnetic field •Strength of magnetic field increases as current increases and vice versa Right Hand Rule (Conductor) •Determines direction of magnetic field •Imagine grasping conductor with right hand •Thumb in direction of current flow (not electron flow) •Fingers curl in the direction of magnetic field DO NOT USE LEFT HAND RULE IN BOOK Example Draw magnetic field lines around conduction path E (V) R Another Example •Draw magnetic field lines around conductors Conductor Conductor current into page current out of page Conductor coils •Single conductor not very useful •Multiple winds of a conductor required for most applications, – e.g.
    [Show full text]
  • Magnetic Induction
    Online Continuing Education for Professional Engineers Since 2009 Basic Electric Theory PDH Credits: 6 PDH Course No.: BET101 Publication Source: US Dept. of Energy “Fundamentals Handbook: Electrical Science – Vol. 1 of 4, Module 1, Basic Electrical Theory” Pub. # DOE-HDBK-1011/1-92 Release Date: June 1992 DISCLAIMER: All course materials available on this website are not to be construed as a representation or warranty on the part of Online-PDH, or other persons and/or organizations named herein. All course literature is for reference purposes only, and should not be used as a substitute for competent, professional engineering council. Use or application of any information herein, should be done so at the discretion of a licensed professional engineer in that given field of expertise. Any person(s) making use of this information, herein, does so at their own risk and assumes any and all liabilities arising therefrom. Copyright © 2009 Online-PDH - All Rights Reserved 1265 San Juan Dr. - Merritt Island, FL 32952 Phone: 321-501-5601 DOE-HDBK-1011/1-92 JUNE 1992 DOE FUNDAMENTALS HANDBOOK ELECTRICAL SCIENCE Volume 1 of 4 U.S. Department of Energy FSC-6910 Washington, D.C. 20585 Distribution Statement A. Approved for public release; distribution is unlimited. Department of Energy Fundamentals Handbook ELECTRICAL SCIENCE Module 1 Basic Electrical Theory Basic Electrical Theory TABLE OF CONTENTS TABLE OF CONTENTS LIST OF FIGURES .................................................. iv LIST OF TABLES ..................................................
    [Show full text]
  • Quelques Souvenirs Des Années 1925-1950 Cahiers Du Séminaire D’Histoire Des Mathématiques 1Re Série, Tome 1 (1980), P
    CAHIERS DU SÉMINAIRE D’HISTOIRE DES MATHÉMATIQUES GEORGES DE RHAM Quelques souvenirs des années 1925-1950 Cahiers du séminaire d’histoire des mathématiques 1re série, tome 1 (1980), p. 19-36 <http://www.numdam.org/item?id=CSHM_1980__1__19_0> © Cahiers du séminaire d’histoire des mathématiques, 1980, tous droits réservés. L’accès aux archives de la revue « Cahiers du séminaire d’histoire des mathématiques » im- plique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ - 19 - QUELQUES SOUVENIRS DES ANNEES 1925-1950 par Georges de RHAM Arrivé à la fin de ma carrière, je pense à son début. C’est dans ma année 1. en 1924, que j’ai décidé de me lancer dans les mathématiques. En 1921, ayant le bachot classique, avec latin et grec, attiré par la philosophie, j’hésitais d’entrer à la Facul- té des Lettres. Mais finalement je me décidai pour la Faculté des Sciences, avec à mon programme l’étude de la Chimie, de la Physique et surtout, pour finir, la Biologie. Je ne songeais pas aux Mathématiques, qui me semblaient un domaine fermé où je ne pourrais rien faire. Pourtant, pour comprendre des questions de Physique, je suis amené à ouvrir des livres de Mathématiques supérieures. J’entrevois qu’il y a là un domaine immense, qui ex- cité ma curiosité et m’intéresse à tel point qu’après cinq semestres à l’Université, j’ abandonne la Biologie pour aborder résolument les Mathématiques.
    [Show full text]
  • Constrained Optimization on Manifolds
    Constrained Optimization on Manifolds Von der Universit¨atBayreuth zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung von Juli´anOrtiz aus Bogot´a 1. Gutachter: Prof. Dr. Anton Schiela 2. Gutachter: Prof. Dr. Oliver Sander 3. Gutachter: Prof. Dr. Lars Gr¨une Tag der Einreichung: 23.07.2020 Tag des Kolloquiums: 26.11.2020 Zusammenfassung Optimierungsprobleme sind Bestandteil vieler mathematischer Anwendungen. Die herausfordernd- sten Optimierungsprobleme ergeben sich dabei, wenn hoch nichtlineare Probleme gel¨ostwerden m¨ussen. Daher ist es von Vorteil, gegebene nichtlineare Strukturen zu nutzen, wof¨urdie Op- timierung auf nichtlinearen Mannigfaltigkeiten einen geeigneten Rahmen bietet. Es ergeben sich zahlreiche Anwendungsf¨alle,zum Beispiel bei nichtlinearen Problemen der Linearen Algebra, bei der nichtlinearen Mechanik etc. Im Fall der nichtlinearen Mechanik ist es das Ziel, Spezifika der Struktur zu erhalten, wie beispielsweise Orientierung, Inkompressibilit¨atoder Nicht-Ausdehnbarkeit, wie es bei elastischen St¨aben der Fall ist. Außerdem k¨onnensich zus¨atzliche Nebenbedingungen ergeben, wie im wichtigen Fall der Optimalsteuerungsprobleme. Daher sind f¨urdie L¨osungsolcher Probleme neue geometrische Tools und Algorithmen n¨otig. In dieser Arbeit werden Optimierungsprobleme auf Mannigfaltigkeiten und die Konstruktion von Algorithmen f¨urihre numerische L¨osungbehandelt. In einer abstrakten Formulierung soll eine reelle Funktion auf einer Mannigfaltigkeit minimiert werden, mit der Nebenbedingung
    [Show full text]
  • The Universe of General Relativity, Springer 2005.Pdf
    Einstein Studies Editors: Don Howard John Stachel Published under the sponsorship of the Center for Einstein Studies, Boston University Volume 1: Einstein and the History of General Relativity Don Howard and John Stachel, editors Volume 2: Conceptual Problems of Quantum Gravity Abhay Ashtekar and John Stachel, editors Volume 3: Studies in the History of General Relativity Jean Eisenstaedt and A.J. Kox, editors Volume 4: Recent Advances in General Relativity Allen I. Janis and John R. Porter, editors Volume 5: The Attraction of Gravitation: New Studies in the History of General Relativity John Earman, Michel Janssen and John D. Norton, editors Volume 6: Mach’s Principle: From Newton’s Bucket to Quantum Gravity Julian B. Barbour and Herbert Pfister, editors Volume 7: The Expanding Worlds of General Relativity Hubert Goenner, Jürgen Renn, Jim Ritter, and Tilman Sauer, editors Volume 8: Einstein: The Formative Years, 1879–1909 Don Howard and John Stachel, editors Volume 9: Einstein from ‘B’ to ‘Z’ John Stachel Volume 10: Einstein Studies in Russia Yuri Balashov and Vladimir Vizgin, editors Volume 11: The Universe of General Relativity A.J. Kox and Jean Eisenstaedt, editors A.J. Kox Jean Eisenstaedt Editors The Universe of General Relativity Birkhauser¨ Boston • Basel • Berlin A.J. Kox Jean Eisenstaedt Universiteit van Amsterdam Observatoire de Paris Instituut voor Theoretische Fysica SYRTE/UMR8630–CNRS Valckenierstraat 65 F-75014 Paris Cedex 1018 XE Amsterdam France The Netherlands AMS Subject Classification (2000): 01A60, 83-03, 83-06 Library of Congress Cataloging-in-Publication Data The universe of general relativity / A.J. Kox, editors, Jean Eisenstaedt. p.
    [Show full text]
  • On Levi-Civita's Alternating Symbol, Schouten's Alternating Unit Tensors
    Mathematics Faculty Works Mathematics 2012 On Levi-Civita’s Alternating Symbol, Schouten’s Alternating Unit Tensors, CPT, and Quantization Evert Jan Post University of Houston Stan Sholar The Boeing Company Hooman Rahimizadeh Loyola Marymount University Michael Berg Loyola Marymount University, [email protected] Follow this and additional works at: https://digitalcommons.lmu.edu/math_fac Part of the Geometry and Topology Commons, and the Quantum Physics Commons Recommended Citation Sholar, S., H. Rahimizadeh, E. J. Post,“On Levi-Civita’s Alternating Symbol, Schouten’s Alternating Unit Tensors, CPT, and Quantization.” International Journal of Pure and Applied Mathematics, 78(6), 2012, 895-907. This Article is brought to you for free and open access by the Mathematics at Digital Commons @ Loyola Marymount University and Loyola Law School. It has been accepted for inclusion in Mathematics Faculty Works by an authorized administrator of Digital Commons@Loyola Marymount University and Loyola Law School. For more information, please contact [email protected]. International Journal of Pure and Applied Mathematics Volume 78 No. 6 2012, 895-907 AP ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu ijpam.eu ON LEVI-CIVITA’S ALTERNATING SYMBOL, SCHOUTEN’S ALTERNATING UNIT TENSORS, CPT, AND QUANTIZATION Evert Jan Post1, Stan Sholar2, Hooman Rahimizadeh3, Michael Berg4 § 1The University of Houston 4800, Calhoun Road, Houston, Texas, 77004, USA 2The Boeing Company 3Department of Mathematics Loyola Marymount University 1 LMU Drive, Los Angeles, CA 90045, USA Abstract: The purpose of the present article is to demonstrate that by adopt- ing a unifying differential geometric perspective on certain themes in physics one reaps remarkable new dividends in both microscopic and macroscopic domains.
    [Show full text]
  • Georges De Rham 1903--1990
    Georges de Rham 1903{1990 This is an English translation of the obituary notice by Beno Eckmann, appearing under the same title in Elemente der Mathematik 47(3) 118{122 (1992) (in German).∗ Original c 1992 Birkh¨auserVerlag and available at <http://eudml.org/doc/141534>. The news of the passing of Georges de Rham on 8 October 1990 has by now surely reached and moved all in the mathematical community. This journal would like to bid him farewell with the following communication. He was one of the important figures of mathematics in our century, his name and work belong to its enduring legacy, and the influence of his ideas on its development has by no means been exhausted yet. He was close to Elemente der Mathematik, which is broadly aimed at both instructors and students, and indeed for many years was among its contributors. Everything that was connected with teaching lay as close to his heart as research did. He was in the habit of saying, \Teaching, the conveying of essentials, making the beautiful intelligible and evident, that is what gives me joy; and instruction is always accompanied by interpretation." arXiv:1611.03806v1 [math.HO] 11 Nov 2016 He understood how to enlighten, in an unassuming yet memorable fashion, students at all levels about mathematics; perhaps they could unconsciously sense what a great mathematical power- house was at work here. The closer to him one became, whether as a novice or as a colleague, the more one was impressed by his personality: by his|there is no other way to say it—refined, yet not distant, bearing; by his charm, which came from the heart; by his unyielding straightforwardness and intensity; by his warmhearted friendship, loyalty, and readiness to help.
    [Show full text]