Michael J. Crowe, a History of Vector Analysis
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Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E
BOOK REVIEW Einstein’s Italian Mathematicians: Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E. Rowe Einstein’s Italian modern Italy. Nor does the author shy away from topics Mathematicians: like how Ricci developed his absolute differential calculus Ricci, Levi-Civita, and the as a generalization of E. B. Christoffel’s (1829–1900) work Birth of General Relativity on quadratic differential forms or why it served as a key By Judith R. Goodstein tool for Einstein in his efforts to generalize the special theory of relativity in order to incorporate gravitation. In This delightful little book re- like manner, she describes how Levi-Civita was able to sulted from the author’s long- give a clear geometric interpretation of curvature effects standing enchantment with Tul- in Einstein’s theory by appealing to his concept of parallel lio Levi-Civita (1873–1941), his displacement of vectors (see below). For these and other mentor Gregorio Ricci Curbastro topics, Goodstein draws on and cites a great deal of the (1853–1925), and the special AMS, 2018, 211 pp. 211 AMS, 2018, vast secondary literature produced in recent decades by the world that these and other Ital- “Einstein industry,” in particular the ongoing project that ian mathematicians occupied and helped to shape. The has produced the first 15 volumes of The Collected Papers importance of their work for Einstein’s general theory of of Albert Einstein [CPAE 1–15, 1987–2018]. relativity is one of the more celebrated topics in the history Her account proceeds in three parts spread out over of modern mathematical physics; this is told, for example, twelve chapters, the first seven of which cover episodes in [Pais 1982], the standard biography of Einstein. -
Vectors and Beyond: Geometric Algebra and Its Philosophical
dialectica Vol. 63, N° 4 (2010), pp. 381–395 DOI: 10.1111/j.1746-8361.2009.01214.x Vectors and Beyond: Geometric Algebra and its Philosophical Significancedltc_1214 381..396 Peter Simons† In mathematics, like in everything else, it is the Darwinian struggle for life of ideas that leads to the survival of the concepts which we actually use, often believed to have come to us fully armed with goodness from some mysterious Platonic repository of truths. Simon Altmann 1. Introduction The purpose of this paper is to draw the attention of philosophers and others interested in the applicability of mathematics to a quiet revolution that is taking place around the theory of vectors and their application. It is not that new math- ematics is being invented – on the contrary, the basic concepts are over a century old – but rather that this old theory, having languished for many decades as a quaint backwater, is being rediscovered and properly applied for the first time. The philosophical importance of this quiet revolution is not that new applications for old mathematics are being found. That presumably happens much of the time. Rather it is that this new range of applications affords us a novel insight into the reasons why vectors and their mathematical kin find application at all in the real world. Indirectly, it tells us something general but interesting about the nature of the spatiotemporal world we all inhabit, and that is of philosophical significance. Quite what this significance amounts to is not yet clear. I should stress that nothing here is original: the history is quite accessible from several sources, and the mathematics is commonplace to those who know it. -
Lecture Notes for Felix Klein Lectures
Preprint typeset in JHEP style - HYPER VERSION Lecture Notes for Felix Klein Lectures Gregory W. Moore Abstract: These are lecture notes for a series of talks at the Hausdorff Mathematical Institute, Bonn, Oct. 1-11, 2012 December 5, 2012 Contents 1. Prologue 7 2. Lecture 1, Monday Oct. 1: Introduction: (2,0) Theory and Physical Mathematics 7 2.1 Quantum Field Theory 8 2.1.1 Extended QFT, defects, and bordism categories 9 2.1.2 Traditional Wilsonian Viewpoint 12 2.2 Compactification, Low Energy Limit, and Reduction 13 2.3 Relations between theories 15 3. Background material on superconformal and super-Poincar´ealgebras 18 3.1 Why study these? 18 3.2 Poincar´eand conformal symmetry 18 3.3 Super-Poincar´ealgebras 19 3.4 Superconformal algebras 20 3.5 Six-dimensional superconformal algebras 21 3.5.1 Some group theory 21 3.5.2 The (2, 0) superconformal algebra SC(M1,5|32) 23 3.6 d = 6 (2, 0) super-Poincar´e SP(M1,5|16) and the central charge extensions 24 3.7 Compactification and preserved supersymmetries 25 3.7.1 Emergent symmetries 26 3.7.2 Partial Topological Twisting 26 3.7.3 Embedded Four-dimensional = 2 algebras and defects 28 N 3.8 Unitary Representations of the (2, 0) algebra 29 3.9 List of topological twists of the (2, 0) algebra 29 4. Four-dimensional BPS States and (primitive) Wall-Crossing 30 4.1 The d = 4, = 2 super-Poincar´ealgebra SP(M1,3|8). 30 N 4.2 BPS particle representations of four-dimensional = 2 superpoincar´eal- N gebras 31 4.2.1 Particle representations 31 4.2.2 The half-hypermultiplet 33 4.2.3 Long representations 34 -
Recent Books on Vector Analysis. 463
1921.] RECENT BOOKS ON VECTOR ANALYSIS. 463 RECENT BOOKS ON VECTOR ANALYSIS. Einführung in die Vehtoranalysis, mit Anwendungen auf die mathematische Physik. By R. Gans. 4th edition. Leip zig and Berlin, Teubner, 1921. 118 pp. Elements of Vector Algebra. By L. Silberstein. New York, Longmans, Green, and Company, 1919. 42 pp. Vehtor analysis. By C. Runge. Vol. I. Die Vektoranalysis des dreidimensionalen Raumes. Leipzig, Hirzel, 1919. viii + 195 pp. Précis de Calcul géométrique. By R. Leveugle. Preface by H. Fehr. Paris, Gauthier-Villars, 1920. lvi + 400 pp. The first of these books is for the most part a reprint of the third edition, which has been reviewed in this BULLETIN (vol. 21 (1915), p. 360). Some small condensation of results has been accomplished by making deductions by purely vector methods. The author still clings to the rather common idea that a vector is a certain triple on a certain set of coordinate axes, so that his development is rather that of a shorthand than of a study of expressions which are not dependent upon axes. The second book is a quite brief exposition of the bare fundamentals of vector algebra, the notation being that of Heaviside, which Silberstein has used before. It is designed primarily to satisfy the needs of those studying geometric optics. However the author goes so far as to mention a linear vector operator and the notion of dyad. The exposition is very simple and could be followed by high school students who had had trigonometry. The third of the books mentioned is an elementary exposition of vectors from the standpoint of Grassmann. -
Felix Klein and His Erlanger Programm D
WDS'07 Proceedings of Contributed Papers, Part I, 251–256, 2007. ISBN 978-80-7378-023-4 © MATFYZPRESS Felix Klein and his Erlanger Programm D. Trkovsk´a Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. The present paper is dedicated to Felix Klein (1849–1925), one of the leading German mathematicians in the second half of the 19th century. It gives a brief account of his professional life. Some of his activities connected with the reform of mathematics teaching at German schools are mentioned as well. In the following text, we describe fundamental ideas of his Erlanger Programm in more detail. References containing selected papers relevant to this theme are attached. Introduction The Erlanger Programm plays an important role in the development of mathematics in the 19th century. It is the title of Klein’s famous lecture Vergleichende Betrachtungen ¨uber neuere geometrische Forschungen [A Comparative Review of Recent Researches in Geometry] which was presented on the occasion of his admission as a professor at the Philosophical Faculty of the University of Erlangen in October 1872. In this lecture, Felix Klein elucidated the importance of the term group for the classification of various geometries and presented his unified way of looking at various geometries. Klein’s basic idea is that each geometry can be characterized by a group of transformations which preserve elementary properties of the given geometry. Professional Life of Felix Klein Felix Klein was born on April 25, 1849 in D¨usseldorf,Prussia. Having finished his study at the grammar school in D¨usseldorf,he entered the University of Bonn in 1865 to study natural sciences. -
Introduction Sally Shuttleworth and Geoffrey Cantor
1 Introduction Sally Shuttleworth and Geoffrey Cantor “Reviews are a substitute for all other kinds of reading—a new and royal road to knowledge,” trumpeted Josiah Conder in 1811.1 Conder, who sub- sequently became proprietor and editor of the Eclectic Review, recognized that periodicals were proliferating, rapidly increasing in popularity, and becoming a major sector in the market for print. Although this process had barely begun at the time Conder was writing, the number of peri- odical publications accelerated considerably over the ensuing decades. According to John North, who is currently cataloguing the wonderfully rich variety of British newspapers and periodicals, some 125,000 titles were published in the nineteenth century.2 Many were short-lived, but others, including the Edinburgh Review (1802–1929) and Punch (1841 on), possess long and honorable histories. Not only did titles proliferate, but, as publishers, editors, and proprietors realized, the often-buoyant market for periodicals could be highly profitable and open to entrepreneurial exploitation. A new title might tap—or create—a previously unexploited niche in the market. Although the expensive quarterly reviews, such as the Edinburgh Review and the Quarterly, have attracted much scholarly attention, their circulation figures were small (they generally sold only a few thousand copies), and their readership was predominantly upper middle class. By contrast, the tupenny weekly Mirror of Literature is claimed to have achieved an unprecedented circulation of 150,000 when it was launched in 1822. A few later titles that were likewise cheap and aimed at a mass readership also achieved circulation figures of this magnitude. -
Simple Circuit Theory and the Solution of Two Electricity Problems from The
Simple circuit theory and the solution of two electricity problems from the Victorian Age A C Tort ∗ Departamento de F´ısica Te´orica - Instituto de F´ısica Universidade Federal do Rio de Janeiro Caixa Postal 68.528; CEP 21941-972 Rio de Janeiro, Brazil May 22, 2018 Abstract Two problems from the Victorian Age, the subdivision of light and the determination of the leakage point in an undersea telegraphic cable are discussed and suggested as a concrete illustrations of the relationships between textbook physics and the real world. Ohm’s law and simple algebra are the only tools we need to discuss them in the classroom. arXiv:0811.0954v1 [physics.pop-ph] 6 Nov 2008 ∗e-mail: [email protected]. 1 1 Introduction Some time ago, the present author had the opportunity of reading Paul J. Nahin’s [1] fascinating biog- raphy of the Victorian physicist and electrician Oliver Heaviside (1850-1925). Heaviside’s scientific life unrolls against a background of theoretical and technical challenges that the scientific and technological developments fostered by the Industrial Revolution presented to engineers and physicists of those times. It is a time where electromagnetic theory as formulated by James Clerk Maxwell (1831-1879) was un- derstood by only a small group of men, Lodge, FitzGerald and Heaviside, among others, that had the mathematical sophistication and imagination to grasp the meaning and take part in the great Maxwellian synthesis. Almost all of the electrical engineers, or electricians as they were called at the time, considered themselves as “practical men”, which effectively meant that most of them had a working knowledge of the electromagnetic phenomena spiced up with bits of electrical theory, to wit, Ohm’s law and the Joule effect. -
The Concept of Field in the History of Electromagnetism
The concept of field in the history of electromagnetism Giovanni Miano Department of Electrical Engineering University of Naples Federico II ET2011-XXVII Riunione Annuale dei Ricercatori di Elettrotecnica Bologna 16-17 giugno 2011 Celebration of the 150th Birthday of Maxwell’s Equations 150 years ago (on March 1861) a young Maxwell (30 years old) published the first part of the paper On physical lines of force in which he wrote down the equations that, by bringing together the physics of electricity and magnetism, laid the foundations for electromagnetism and modern physics. Statue of Maxwell with its dog Toby. Plaque on E-side of the statue. Edinburgh, George Street. Talk Outline ! A brief survey of the birth of the electromagnetism: a long and intriguing story ! A rapid comparison of Weber’s electrodynamics and Maxwell’s theory: “direct action at distance” and “field theory” General References E. T. Wittaker, Theories of Aether and Electricity, Longam, Green and Co., London, 1910. O. Darrigol, Electrodynamics from Ampère to Einste in, Oxford University Press, 2000. O. M. Bucci, The Genesis of Maxwell’s Equations, in “History of Wireless”, T. K. Sarkar et al. Eds., Wiley-Interscience, 2006. Magnetism and Electricity In 1600 Gilbert published the “De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure” (On the Magnet and Magnetic Bodies, and on That Great Magnet the Earth). ! The Earth is magnetic ()*+(,-.*, Magnesia ad Sipylum) and this is why a compass points north. ! In a quite large class of bodies (glass, sulphur, …) the friction induces the same effect observed in the amber (!"#$%&'(, Elektron). Gilbert gave to it the name “electricus”. -
Université Du Québec À Montréal Edwin B. Wilson Aux Origines De L
UNIVERSITÉ DU QUÉBEC À MONTRÉAL EDWIN B. WILSON AUX ORIGINES DE L’ÉCONOMIE MATHÉMATIQUE DE PAUL SAMUELSON: ESSAIS SUR L’HISTOIRE ENTREMÊLÉE DE LA SCIENCE ÉCONOMIQUE, DES MATHÉMATIQUES ET DES STATISTIQUES AUX ÉTATS-UNIS, 1900-1940 THÈSE PRÉSENTÉE COMME EXIGENCE PARTIELLE DU DOCTORAT EN ÉCONOMIQUE PAR CARVAJALINO FLOREZ JUAN GUILLERMO DÉCEMBRE 2016 UNIVERSITÉ DU QUÉBEC À MONTRÉAL EDWIN B. WILSON AT THE ORIGIN OF PAUL SAMUELSON'S MATHEMATICAL ECONOMICS: ESSAYS ON THE INTERWOVEN HISTORY OF ECONOMICS, MATHEMATICS AND STATISTICS IN THE U.S., 1900-1940 THESIS PRESENTED AS PARTIAL REQUIREMENT OF DOCTOR OF PHILOSPHY IN ECONOMIC BY CARVAJALINO FLOREZ JUAN GUILLERMO DECEMBER 2016 ACKNOWLEDGMENTS For the financial aid that made possible this project, I am thankful to the Université du Québec À Montréal (UQAM), the Fonds de Recherche du Québec Société et Culture (FRQSC), the Social Science and Humanities Research Council of Canada (SSHRCC) as well as the Centre Interuniversitaire de Recherche sur la Science et la Technologie (CIRST). I am also grateful to archivists for access to various collections and papers at the Harvard University Archives (Edwin Bidwell Wilson), David M. Rubenstein Rare Book & Manuscript Library, Duke University (Paul A. Samuelson and Lloyd Metzler), Library of Congress, Washington, D.C. (John von Neumann and Oswald Veblen) and Yale University Library (James Tobin). I received encouraging and intellectual support from many. I particularly express my gratitude to the HOPE Center, at Duke University, where I spent the last semester of my Ph.D. and where the last chapter of this thesis was written. In particular, I thank Roy Weintraub for all his personal support, for significantly engaging with my work and for his valuable research advice. -
The Philosophy of Religion Contents
The Philosophy of Religion Course notes by Richard Baron This document is available at www.rbphilo.com/coursenotes Contents Page Introduction to the philosophy of religion 2 Can we show that God exists? 3 Can we show that God does not exist? 6 If there is a God, why do bad things happen to good people? 8 Should we approach religious claims like other factual claims? 10 Is being religious a matter of believing certain factual claims? 13 Is religion a good basis for ethics? 14 1 Introduction to the philosophy of religion Why study the philosophy of religion? If you are religious: to deepen your understanding of your religion; to help you to apply your religion to real-life problems. Whether or not you are religious: to understand important strands in our cultural history; to understand one of the foundations of modern ethical debate; to see the origins of types of philosophical argument that get used elsewhere. The scope of the subject We shall focus on the philosophy of religions like Christianity, Islam and Judaism. Other religions can be quite different in nature, and can raise different questions. The questions in the contents list indicate the scope of the subject. Reading You do not need to do extra reading, but if you would like to do so, you could try either one of these two books: Brian Davies, An Introduction to the Philosophy of Religion. Oxford University Press, third edition, 2003. Chad Meister, Introducing Philosophy of Religion. Routledge, 2009. 2 Can we show that God exists? What sorts of demonstration are there? Proofs in the strict sense: logical and mathematical proof Demonstrations based on external evidence Demonstrations based on inner experience How strong are these different sorts of demonstration? Which ones could other people reject, and on what grounds? What sort of thing could have its existence shown in each of these ways? What might we want to show? That God exists That it is reasonable to believe that God exists The ontological argument Greek onta, things that exist. -
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! Revising Space Time Geometry: A Proposal for a New Romance in Many Dimensions Renate Quehenberger* Quantum Cinema - a digital Vision (PEEK) Department of Mediatheory at the University of Applied Arts, Vienna August 25, 2012 Abstract The ontological positioning of our existence, deeply connected with the hierarchy problem concerning the dimensions of space and time, is one of the major problems for our understanding of the “ultimate” nature of reality. The resulting problem is that mathematical concepts that need so-called “extra-dimensions” have been widely disregarded owing to a lack of physical interpretation. This article reviews conventions, imaginations and assumptions about the non-imaginative by starting at the origins of the of 4D and 5D space-time concepts and proposing a new geometrical approach via a hyper- Euclidian path for a mentally accessible vision of continuous AND discrete complex space configuration in dimensions up to higher order. e-mail address: [email protected] 1 RCZ Quehenberger “Time and Space... It is not nature which imposes them upon us, it is we who impose them upon nature because we find them convenient.” [Henri Poincaré, 1905] Introduction The general physicists’ assertion is that we are living in a three-dimensional world with one time- dimension and cosmic space is curved in the fourth dimension is based on a relativistic world view that has gained general acceptance. In fact this is a post-Copernicanian world view which does not pay attention to the consequences of Max Planck’s revolution and the development in quantum physics of the past 100 years. -
Spacetime Algebra As a Powerful Tool for Electromagnetism
Spacetime algebra as a powerful tool for electromagnetism Justin Dressela,b, Konstantin Y. Bliokhb,c, Franco Norib,d aDepartment of Electrical and Computer Engineering, University of California, Riverside, CA 92521, USA bCenter for Emergent Matter Science (CEMS), RIKEN, Wako-shi, Saitama, 351-0198, Japan cInterdisciplinary Theoretical Science Research Group (iTHES), RIKEN, Wako-shi, Saitama, 351-0198, Japan dPhysics Department, University of Michigan, Ann Arbor, MI 48109-1040, USA Abstract We present a comprehensive introduction to spacetime algebra that emphasizes its prac- ticality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in stud- ies of the single photon wavefunction. The complex structure of spacetime also underpins the emergence of electromagnetic waves, circular polarizations, the normal variables for canonical quantization, the distinction between electric and magnetic charge, complex spinor representations of Lorentz transformations, and the dual (electric-magnetic field exchange) symmetry that produces helicity conservation in vacuum fields. This latter symmetry manifests as an arbitrary global phase of the complex field, motivating the use of a complex vector potential, along with an associated transverse and gauge-invariant bivector potential, as well as complex (bivector and scalar) Hertz potentials.