Cauchy's Cours D'analyse
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Jean-Baptiste Charles Joseph Bélanger (1790-1874), the Backwater Equation and the Bélanger Equation
THE UNIVERSITY OF QUEENSLAND DIVISION OF CIVIL ENGINEERING REPORT CH69/08 JEAN-BAPTISTE CHARLES JOSEPH BÉLANGER (1790-1874), THE BACKWATER EQUATION AND THE BÉLANGER EQUATION AUTHOR: Hubert CHANSON HYDRAULIC MODEL REPORTS This report is published by the Division of Civil Engineering at the University of Queensland. Lists of recently-published titles of this series and of other publications are provided at the end of this report. Requests for copies of any of these documents should be addressed to the Civil Engineering Secretary. The interpretation and opinions expressed herein are solely those of the author(s). Considerable care has been taken to ensure accuracy of the material presented. Nevertheless, responsibility for the use of this material rests with the user. Division of Civil Engineering The University of Queensland Brisbane QLD 4072 AUSTRALIA Telephone: (61 7) 3365 3619 Fax: (61 7) 3365 4599 URL: http://www.eng.uq.edu.au/civil/ First published in 2008 by Division of Civil Engineering The University of Queensland, Brisbane QLD 4072, Australia © Chanson This book is copyright ISBN No. 9781864999211 The University of Queensland, St Lucia QLD JEAN-BAPTISTE CHARLES JOSEPH BÉLANGER (1790-1874), THE BACKWATER EQUATION AND THE BÉLANGER EQUATION by Hubert CHANSON Professor, Division of Civil Engineering, School of Engineering, The University of Queensland, Brisbane QLD 4072, Australia Ph.: (61 7) 3365 3619, Fax: (61 7) 3365 4599, Email: [email protected] Url: http://www.uq.edu.au/~e2hchans/ REPORT No. CH69/08 ISBN 9781864999211 Division of Civil Engineering, The University of Queensland August 2008 Jean-Baptiste BÉLANGER (1790-1874) (Courtesy of the Bibliothèque de l'Ecole Nationale Supérieure des Ponts et Chaussées) Abstract In an open channel, the transition from a high-velocity open channel flow to a fluvial motion is a flow singularity called a hydraulic jump. -
GYRODYNAMICS Introduction to the Dynamics of Rigid Bodies
2 GYRODYNAMICS Introduction to the dynamics of rigid bodies Introduction. Though Newton wrote on many topics—and may well have given thought to the odd behavior of tops—I am not aware that he committed any of that thought to writing. But by Euler was active in the field, and it has continued to bedevil the thought of mathematical physicists. “Extended rigid bodies” are classical abstractions—alien both to relativity and to quantum mechanics—which are revealed to our dynamical imaginations not so much by commonplace Nature as by, in Maxwell’s phrase, the “toys of Youth.” That such toys behave “strangely” is evident to the most casual observer, but the detailed theory of their behavior has become notorious for being elusive, surprising and difficult at every turn. Its formulation has required and inspired work of wonderful genius: it has taught us much of great worth, and clearly has much to teach us still. Early in my own education as a physicist I discovered that I could not understand—or, when I could understand, remained unpersuaded by—the “elementary explanations” of the behavior of tops & gyros which are abundant in the literature. So I fell into the habit of avoiding the field, waiting for the day when I could give to it the time and attention it obviously required and deserved. I became aware that my experience was far from atypical: according to Goldstein it was in fact a similar experience that motivated Klein & Sommerfeld to write their 4-volume classic, Theorie des Kreisels (–). In November I had occasion to consult my Classical Mechanics II students concerning what topic we should take up to finish out the term. -
Augustin-Louis Cauchy - Wikipedia, the Free Encyclopedia 1/6/14 3:35 PM Augustin-Louis Cauchy from Wikipedia, the Free Encyclopedia
Augustin-Louis Cauchy - Wikipedia, the free encyclopedia 1/6/14 3:35 PM Augustin-Louis Cauchy From Wikipedia, the free encyclopedia Baron Augustin-Louis Cauchy (French: [oɡystɛ̃ Augustin-Louis Cauchy lwi koʃi]; 21 August 1789 – 23 May 1857) was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic Cauchy around 1840. Lithography by Zéphirin principle of the Belliard after a painting by Jean Roller. generality of algebra exploited by earlier Born 21 August 1789 authors. He defined Paris, France continuity in terms of Died 23 May 1857 (aged 67) infinitesimals and gave Sceaux, France several important Nationality French theorems in complex Fields Mathematics analysis and initiated the Institutions École Centrale du Panthéon study of permutation École Nationale des Ponts et groups in abstract Chaussées algebra. A profound École polytechnique mathematician, Cauchy Alma mater École Nationale des Ponts et exercised a great Chaussées http://en.wikipedia.org/wiki/Augustin-Louis_Cauchy Page 1 of 24 Augustin-Louis Cauchy - Wikipedia, the free encyclopedia 1/6/14 3:35 PM influence over his Doctoral Francesco Faà di Bruno contemporaries and students Viktor Bunyakovsky successors. His writings Known for See list cover the entire range of mathematics and mathematical physics. "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)."[1] Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks. He was a devout Roman Catholic, strict Bourbon royalist, and a close associate of the Jesuit order. -
Leonhard Euler - Wikipedia, the Free Encyclopedia Page 1 of 14
Leonhard Euler - Wikipedia, the free encyclopedia Page 1 of 14 Leonhard Euler From Wikipedia, the free encyclopedia Leonhard Euler ( German pronunciation: [l]; English Leonhard Euler approximation, "Oiler" [1] 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[2] He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. [3] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Portrait by Emanuel Handmann 1756(?) Euler, read Euler, he is the master of us all." [4] Born 15 April 1707 Euler was featured on the sixth series of the Swiss 10- Basel, Switzerland franc banknote and on numerous Swiss, German, and Died Russian postage stamps. The asteroid 2002 Euler was 18 September 1783 (aged 76) named in his honor. He is also commemorated by the [OS: 7 September 1783] Lutheran Church on their Calendar of Saints on 24 St. Petersburg, Russia May – he was a devout Christian (and believer in Residence Prussia, Russia biblical inerrancy) who wrote apologetics and argued Switzerland [5] forcefully against the prominent atheists of his time. -
An Appreciation and Discussion of Paul Germain’S “The Method of Virtual Power in the Mechanics of Continuous Media I: Second-Gradient Theory”
NISSUNA UMANA INVESTIGAZIONE SI PUO DIMANDARE VERA SCIENZIA S’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI LEONARDO DA VINCI vol. 8 no. 2 2020 Mathematics and Mechanics of Complex Systems MARCELO EPSTEIN AND RONALD E. SMELSER AN APPRECIATION AND DISCUSSION OF PAUL GERMAIN’S “THE METHOD OF VIRTUAL POWER IN THE MECHANICS OF CONTINUOUS MEDIA I: SECOND-GRADIENT THEORY” msp MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMS Vol. 8, No. 2, 2020 dx.doi.org/10.2140/memocs.2020.8.191 ∩ MM AN APPRECIATION AND DISCUSSION OF PAUL GERMAIN’S “THE METHOD OF VIRTUAL POWER IN THE MECHANICS OF CONTINUOUS MEDIA I: SECOND-GRADIENT THEORY” MARCELO EPSTEIN AND RONALD E. SMELSER Paul Germain’s 1973 article on the method of virtual power in continuum me- chanics has had an enormous impact on the modern development of the disci- pline. In this article we examine the historical context of the ideas it contains and discuss their continuing importance. Our English translation of the French original appears elsewhere in this volume (MEMOCS 8:2 (2020), 153–190). Introduction Among the many contributions of Paul Germain (1920–2009) to mechanics, this classical 1973 article[1973a] on the method of virtual power in continuum mechan- ics stands out for its enormous impact on the modern development of the discipline, as evidenced by hundreds of citations and by its direct or indirect influence in establishing a paradigm of thought for succeeding generations. In this article we examine the historical antecedents of the ideas contained in the article and discuss their continuing relevance. The article was published in French in the Journal de Mécanique. -
Mathematical Language and Mathematical Progress by Eberhard Knobloch
Mathematical language and mathematical progress By Eberhard Knobloch In 1972 the space probe Pioneer 10 was launched, carrying a plaque which contains the first message of mankind to leave our solar system1: The space probe is represented by a circular segment and a rectangle designed on the same scale as that of the man and the woman. This is not true of the solar system: sun and planets are represented by small circles and points. We do not know whether there are other intelligent beings living on stars in the universe, or whether they will even understand the message. But the non-verbal, the symbolical language of geometry seemed to be more appropriate than any other language. The grammar of any ordinary language is so complicated that still every computer breaks down with regards to it. 1 Karl Märker, "Sind wir allein im Weltall? Kosmos und Leben", in: Astronomie im Deutschen Museum, Planeten - Sterne - Welteninseln, hrsg. von Gerhard Hartl, Karl Märker, Jürgen Teichmann, Gudrun Wolfschmidt (München: Deutsches Museum, 1993), p. 217. [We reproduce the image of the plaque from: http://en.wikipedia.org/wiki/Pioneer_plaque ; Karine Chemla] 1 The famous Nicholas Bourbaki wrote in 19482: "It is the external form which the mathematician gives to his thought, the vehicle which makes it accessible to others, in short, the language suited to mathematics; this is all, no further significance should be attached to it". Bourbaki added: "To lay down the rules of this language, to set up its vocabulary and to clarify its syntax, all that is indeed extremely useful." But it was only the least interesting aspect of the axiomatic method for him. -
La Notion De Couple En Mécanique : Réhabiliter Poinsot
La notion de couple en mécanique : réhabiliter Poinsot par Ivor Grattan-Guinness Historien des mathématiques et philosophe des sciences Professeur émérite à l’université du Middlesex (UK) Figure 1: Louis Poinsot, gravure de Boilly. Jules Boilly (1796-1874) est un peintre et lithographe spécialisé dans la gravure de personnalités, dont les membres de l’Institut (P.S. Girard, J.D. Cassini, Alexis Bouvard). Il était le fils d’un peintre plus renommé, Léopold Boilly (1761-1845), auteur de peintures de genre et de portraits, notamment de Sadi Carnot. I - PRÉLIMINAIRES En 1803, Louis Poinsot publia un traité de statique, à caractère révolutionnaire puisqu’il posait clairement le sujet non seulement en termes de forces mais aussi en terme de « couples » (c’est son expression), c’est-à-dire des paires de forces non colinéaires égales en amplitude et en direction mais en sens opposés. Plus tard, il adapta cette notion pour induire en dynamique une relation 1 nouvelle entre mouvement linéaire et mouvement de rotation. Le présent article résume ces développements et examine leur réception, qui fut lente parmi ses contemporains mathématiciens et quasi inexistante parmi les « historiens » de la mécanique plus tard. 1. Les organisations Un beau jour, lors des années révolutionnaires II ou III, période à présent plus connue sous le nom d’année 1794, un adolescent orphelin, étudiant au collège Louis le Grand à Paris, tomba sur un prospectus annonçant la création d’une nouvelle institution d’enseignement supérieur. Intrigué, il se porta candidat et fut accepté, ce qui détermina la suite de sa longue carrière. Cette institution constituait un des deux projets du gouvernement français pour résoudre l’une des crises sociales causées par cinq années de révolution et de ruptures. -
Lagrange's Theory of Analytical Functions and His Ideal of Purity of Method
Lagrange’s Theory of Analytical Functions and his Ideal of Purity of Method Giovanni Ferraro, Marco Panza To cite this version: Giovanni Ferraro, Marco Panza. Lagrange’s Theory of Analytical Functions and his Ideal of Purity of Method. Archive for History of Exact Sciences, Springer Verlag, 2012, 66, pp.95-197. hal-00614606 HAL Id: hal-00614606 https://hal.archives-ouvertes.fr/hal-00614606 Submitted on 12 Aug 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LAGRANGE’S THEORY OF ANALYTICAL FUNCTIONS AND HIS IDEAL OF PURITY OF METHOD GIOVANNI FERRARO AND MARCO PANZA ABSTRACT. We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and concentrate on power-series expansions, on the algorithm for derivative functions, and the remainder theorem—especially the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method—the generality and internal organization of the discipline. -
La Notion De Couple En Mécanique : Réhabiliter Poinsot
Bibnum Textes fondateurs de la science Physique La notion de couple en mécanique : réhabiliter Poinsot Ivor Grattan-Guiness Traducteur : Alexandre Moatti Édition électronique URL : http://journals.openedition.org/bibnum/725 ISSN : 2554-4470 Éditeur FMSH - Fondation Maison des sciences de l'homme Référence électronique Ivor Grattan-Guiness, « La notion de couple en mécanique : réhabiliter Poinsot », Bibnum [En ligne], Physique, mis en ligne le 01 janvier 2013, consulté le 19 avril 2019. URL : http:// journals.openedition.org/bibnum/725 © BibNum La notion de couple en mécanique : réhabiliter Poinsot par Ivor Grattan-Guinness Historien des mathématiques et philosophe des sciences Professeur émérite à l’université du Middlesex (UK) Figure 1: Louis Poinsot, gravure de Boilly. Jules Boilly (1796-1874) est un peintre et lithographe spécialisé dans la gravure de personnalités, dont les membres de l’Institut (P.S. Girard, J.D. Cassini, Alexis Bouvard). Il était le fils d’un peintre plus renommé, Léopold Boilly (1761-1845), auteur de peintures de genre et de portraits, notamment de Sadi Carnot. I - PRÉLIMINAIRES En 1803, Louis Poinsot publia un traité de statique, à caractère révolutionnaire puisqu’il posait clairement le sujet non seulement en termes de forces mais aussi en terme de « couples » (c’est son expression), c’est-à-dire des paires de forces non colinéaires égales en amplitude et en direction mais en sens opposés. Plus tard, il adapta cette notion pour induire en dynamique une relation 1 nouvelle entre mouvement linéaire et mouvement de rotation. Le présent article résume ces développements et examine leur réception, qui fut lente parmi ses contemporains mathématiciens et quasi inexistante parmi les « historiens » de la mécanique plus tard. -
God, King, and Geometry: Revisiting the Introduction to Cauchy's Cours D'analyse
God, King, and Geometry: Revisiting the Introduction to Cauchy's Cours d'Analyse Michael J. Barany Program in History of Science, Princeton University, 129 Dickinson Hall, Princeton, New Jersey 08544, United States Abstract This article offers a systematic reading of the introduction to Augustin-Louis Cauchy's landmark 1821 mathematical textbook, the Cours d'analyse. Despite its emblematic status in the history of mathematical analysis and, indeed, of modern mathematics as a whole, Cauchy's introduction has been more a source for suggestive quotations than an object of study in its own right. Cauchy's short mathematical metatext offers a rich snapshot of a scholarly paradigm in transition. A close reading of Cauchy's writing reveals the complex modalities of the author's epistemic positioning, particularly with respect to the geometric study of quantities in space, as he struggles to refound the discipline on which he has staked his young career. Keywords: Augustin-Louis Cauchy, Cours d'analyse, History of Analysis, Mathematics and Politics 2000 MSC: 01-02, 65-03, 97-03 1. Introduction Despite its emblematic status in the history of mathematical analysis and, indeed, of modern mathematics as a whole, the introduction to Augustin-Louis Cauchy's 1821 Cours d'analyse has been, in the historical literature, more a source for suggestive quotations than an object of study in its own right. In his introduction, Cauchy definitively outlines what were to be the foundations of his new rigorous mathematics, invoking both specific mathematical practices and their underlying philosophical principles. His text is thus a fecund encapsulation of the mathematical and epistemological work which would make him \the man who taught rigorous analysis to all of Europe" (Grabiner, 1981, p. -
History of Mathematics in the Higher Education Curriculum
History of Mathematics in the Higher Education Curriculum Mathematical Sciences HE Curriculum Innovation Project Innovation Curriculum HE Sciences Mathematical Edited by Mark McCartney History of Mathematics in the Higher Education Curriculum Edited by Mark McCartney A report by the working group on History of Mathematics in the Higher Education Curriculum, May 2012. Supported by the Maths, Stats and OR Network, as part of the Mathematical Sciences Strand of the National HE STEM Programme, and the British Society for the History of Mathematics (BSHM). Working group members: Noel-Ann Bradshaw (University of Greenwich; BSHM Treasurer); Snezana Lawrence (Bath Spa University; BSHM Education Officer); Mark McCartney (University of Ulster; BSHM Publicity Officer); Tony Mann (University of Greenwich; BSHM Immediate Past President); Robin Wilson (Pembroke College, Oxford; BSHM President). History of Mathematics in the Higher Education Curriculum Contents Contents Introduction 5 Teaching the history of mathematics at the University of St Andrews 9 History in the undergraduate mathematics curriculum – a case study from Greenwich 13 Teaching History of Mathematics at King’s College London 15 History for learning Analysis 19 History of Mathematics in a College of Education Context 23 Teaching the history of mathematics at the Open University 27 Suggested Resources 31 History of Mathematics in the Higher Education Curriculum Introduction Introduction Mathematics is usually, and of course correctly, presented ‘ready-made’ to students, with techniques and applications presented systematically and in logical order. However, like any other academic subject, mathematics has a history which is rich in astonishing breakthroughs, false starts, misattributions, confusions and dead-ends. This history gives a narrative and human context which adds colour and context to the discipline. -
Lessons from the Avalanche of Numbers: Big Data in Historical Perspective
I/S: A JOURNAL OF LAW AND POLICY FOR THE INFORMATION SOCIETY Lessons from the Avalanche of Numbers: Big Data in Historical Perspective MEG LETA AMBROSE, JD, PHD* Abstract: The big data revolution, like many changes associated with technological advancement, is often compared to the industrial revolution to create a frame of reference for its transformative power, or portrayed as altogether new. This article argues that between the industrial revolution and the digital revolution is a more valuable, yet overlooked period: the probabilistic revolution that began with the avalanche of printed numbers between 1820 and 1840. By comparing the many similarities between big data today and the avalanche of numbers in the 1800s, the article situates big data in the early stages of a prolonged transition to a potentially transformative epistemic revolution, like the probabilistic revolution. The widespread changes in and characteristics of a society flooded by data results in a transitional state that creates unique challenges for policy efforts by disrupting foundational principles relied upon for data protection. The potential of a widespread, lengthy transition also places the law in a pivotal position to shape and guide big data-based inquiry through to whatever epistemic shift may lie ahead. * Assistant Professor, Communication, Culture & Technology, Georgetown University. Many thanks to my Governing Algorithms course and Samantha Fried, whose thesis Quantify This: Statistics, The State, and Governmentality (on file at CCT, available upon request by emailing [email protected]) directed me to many of the wonderful historical secondary sources in this article. Additional thanks to John Grant at Palantir, Professor Paul Ohm, Solon Barocas and all the wonderful participants at the 2014 Privacy Law Scholars Conference for their insightful comments and suggestions.