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A History Of The Calculus Of Variations From The 17h Throught The 19th Century
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Book Synopsis A History of the Calculus of Variations from the 17th through the 19th Century by : H. H. Goldstine
Download or read book A History of the Calculus of Variations from the 17th through the 19th Century written by H. H. Goldstine and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 427 pages. Available in PDF, EPUB and Kindle. Book excerpt: The calculus of variations is a subject whose beginning can be precisely dated. It might be said to begin at the moment that Euler coined the name calculus of variations but this is, of course, not the true moment of inception of the subject. It would not have been unreasonable if I had gone back to the set of isoperimetric problems considered by Greek mathemati cians such as Zenodorus (c. 200 B. C. ) and preserved by Pappus (c. 300 A. D. ). I have not done this since these problems were solved by geometric means. Instead I have arbitrarily chosen to begin with Fermat's elegant principle of least time. He used this principle in 1662 to show how a light ray was refracted at the interface between two optical media of different densities. This analysis of Fermat seems to me especially appropriate as a starting point: He used the methods of the calculus to minimize the time of passage cif a light ray through the two media, and his method was adapted by John Bernoulli to solve the brachystochrone problem. There have been several other histories of the subject, but they are now hopelessly archaic. One by Robert Woodhouse appeared in 1810 and another by Isaac Todhunter in 1861.
Book Synopsis The Early Period of the Calculus of Variations by : Paolo Freguglia
Download or read book The Early Period of the Calculus of Variations written by Paolo Freguglia and published by Birkhäuser. This book was released on 2016-06-27 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph explores the early development of the calculus of variations in continental Europe during the Eighteenth Century by illustrating the mathematics of its founders. Closely following the original papers and correspondences of Euler, Lagrange, the Bernoullis, and others, the reader is immersed in the challenge of theory building. We see what the founders were doing, the difficulties they faced, the mistakes they made, and their triumphs. The authors guide the reader through these works with instructive commentaries and complements to the original proofs, as well as offering a modern perspective where useful. The authors begin in 1697 with Johann Bernoulli’s work on the brachystochrone problem and the events leading up to it, marking the dawn of the calculus of variations. From there, they cover key advances in the theory up to the development of Lagrange’s δ-calculus, including: • The isoperimetrical problems • Shortest lines and geodesics • Euler’s Methodus Inveniendi and the two Additamenta Finally, the authors give the readers a sense of how vast the calculus of variations has become in centuries hence, providing some idea of what lies outside the scope of the book as well as the current state of affairs in the field. This book will be of interest to anyone studying the calculus of variations who wants a deeper intuition for the techniques and ideas that are used, as well as historians of science and mathematics interested in the development and evolution of modern calculus and analysis.
Book Synopsis A Historian Looks Back by : Judith V. Grabiner
Download or read book A Historian Looks Back written by Judith V. Grabiner and published by MAA. This book was released on 2010-10-14 with total page 307 pages. Available in PDF, EPUB and Kindle. Book excerpt: An inspiring collection of a historian's work on the history of mathematics.
Book Synopsis A History of the Progress of the Calculus of Variations During the Nineteenth Century by : Isaac Todhunter
Download or read book A History of the Progress of the Calculus of Variations During the Nineteenth Century written by Isaac Todhunter and published by . This book was released on 1861 with total page 572 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences by : Ivor Grattan-Guinness
Download or read book Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences written by Ivor Grattan-Guinness and published by Routledge. This book was released on 2002-09-11 with total page 1788 pages. Available in PDF, EPUB and Kindle. Book excerpt: * Examines the history and philosophy of the mathematical sciences in a cultural context, tracing their evolution from ancient times up to the twentieth century * 176 articles contributed by authors of 18 nationalities * Chronological table of main events in the development of mathematics * Fully integrated index of people, events and topics * Annotated bibliographies of both classic and contemporary sources * Unique coverage of Ancient and non-Western traditions of mathematics
Book Synopsis A History of Analysis by : Hans Niels Jahnke
Download or read book A History of Analysis written by Hans Niels Jahnke and published by American Mathematical Soc.. This book was released on 2003 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: Analysis as an independent subject was created as part of the scientific revolution in the seventeenth century. Kepler, Galileo, Descartes, Fermat, Huygens, Newton, and Leibniz, to name but a few, contributed to its genesis. Since the end of the seventeenth century, the historical progress of mathematical analysis has displayed unique vitality and momentum. No other mathematical field has so profoundly influenced the development of modern scientific thinking. Describing this multidimensional historical development requires an in-depth discussion which includes a reconstruction of general trends and an examination of the specific problems. This volume is designed as a collective work of authors who are proven experts in the history of mathematics. It clarifies the conceptual change that analysis underwent during its development while elucidating the influence of specific applications and describing the relevance of biographical and philosophical backgrounds. The first ten chapters of the book outline chronological development and the last three chapters survey the history of differential equations, the calculus of variations, and functional analysis. Special features are a separate chapter on the development of the theory of complex functions in the nineteenth century and two chapters on the influence of physics on analysis. One is about the origins of analytical mechanics, and one treats the development of boundary-value problems of mathematical physics (especially potential theory) in the nineteenth century. The book presents an accurate and very readable account of the history of analysis. Each chapter provides a comprehensive bibliography. Mathematical examples have been carefully chosen so that readers with a modest background in mathematics can follow them. It is suitable for mathematical historians and a general mathematical audience.
Book Synopsis Direct Methods in the Calculus of Variations by : Bernard Dacorogna
Download or read book Direct Methods in the Calculus of Variations written by Bernard Dacorogna and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent years there has been a considerable renewal of interest in the clas sical problems of the calculus of variations, both from the point of view of mathematics and of applications. Some of the most powerful tools for proving existence of minima for such problems are known as direct methods. They are often the only available ones, particularly for vectorial problems. It is the aim of this book to present them. These methods were introduced by Tonelli, following earlier work of Hilbert and Lebesgue. Although there are excellent books on calculus of variations and on direct methods, there are recent important developments which cannot be found in these books; in particular, those dealing with vector valued functions and relaxation of non convex problems. These two last ones are important in appli cations to nonlinear elasticity, optimal design . . . . In these fields the variational methods are particularly effective. Part of the mathematical developments and of the renewal of interest in these methods finds its motivations in nonlinear elasticity. Moreover, one of the recent important contributions to nonlinear analysis has been the study of the behaviour of nonlinear functionals un der various types of convergence, particularly the weak convergence. Two well studied theories have now been developed, namely f-convergence and compen sated compactness. They both include as a particular case the direct methods of the calculus of variations, but they are also, both, inspired and have as main examples these direct methods.
Book Synopsis Apollonius: Conics Books V to VII by : Gerald J. Toomer
Download or read book Apollonius: Conics Books V to VII written by Gerald J. Toomer and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 978 pages. Available in PDF, EPUB and Kindle. Book excerpt: With the publication of this book I discharge a debt which our era has long owed to the memory of a great mathematician of antiquity: to pub lish the /llost books" of the Conics of Apollonius in the form which is the closest we have to the original, the Arabic version of the Banu Musil. Un til now this has been accessible only in Halley's Latin translation of 1710 (and translations into other languages entirely dependent on that). While I yield to none in my admiration for Halley's edition of the Conics, it is far from satisfying the requirements of modern scholarship. In particular, it does not contain the Arabic text. I hope that the present edition will not only remedy those deficiencies, but will also serve as a foundation for the study of the influence of the Conics in the medieval Islamic world. I acknowledge with gratitude the help of a number of institutions and people. The John Simon Guggenheim Memorial Foundation, by the award of one of its Fellowships for 1985-86, enabled me to devote an unbroken year to this project, and to consult essential material in the Bodleian Li brary, Oxford, and the Bibliotheque Nationale, Paris. Corpus Christi Col lege, Cambridge, appointed me to a Visiting Fellowship in Trinity Term, 1988, which allowed me to make good use of the rich resources of both the University Library, Cambridge, and the Bodleian Library.
Book Synopsis Near the Horizon: An Invitation to Geometric Optics by : Henk W. Broer
Download or read book Near the Horizon: An Invitation to Geometric Optics written by Henk W. Broer and published by American Mathematical Soc.. This book was released on 2016-12-31 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: Near the Horizon starts out by considering several optical phenomena that can occur when the sun is near the horizon. One can sometimes see objects that are actually below the horizon. Sometimes there seems to be a dark strip in the middle of the solar disk. These are a result of the way that the atmosphere affects the geometry of light rays. Broer starts his book with the Fermat principle (rays of light take least-time paths) and deduces from it laws for refraction and reflection; by expressing these as conservation laws, he can handle both the case of inhomogeneous layers of air and the case of continuous variation in the refraction index. A surprising application is the brachistochrone problem, in which the path of fastest descent is determined by studying how a light ray would behave in a “flat earth” atmosphere whose refraction index is determined by the gravitational potential. This leads to a very interesting chapter on the cycloid and its properties. The final chapters move from the elementary theory to a more sophisticated version in which the Fermat Principle leads to a Riemannian metric whose geodesics are the paths of light rays. This gives us an optics which is geometric in a new sense, and serves as a nice demonstration of the physical applicability of Riemannian geometry.
Book Synopsis Exploring the Limits of Preclassical Mechanics by : Peter Damerow
Download or read book Exploring the Limits of Preclassical Mechanics written by Peter Damerow and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 428 pages. Available in PDF, EPUB and Kindle. Book excerpt: The question of when and how the basic concepts that characterize modern science arose in Western Europe has long been central to the history of science. This book examines the transition from Renaissance engineering and philosophy of nature to classical mechanics oriented on the central concept of velocity. For this new edition, the authors include a new discussion of the doctrine of proportions, an analysis of the role of traditional statics in the construction of Descartes' impact rules, and go deeper into the debate between Descartes and Hobbes on the explanation of refraction. They also provide significant new material on the early development of Galileo's work on mechanics and the law of fall.
Book Synopsis Elements of Classical and Geometric Optimization by : Debasish Roy
Download or read book Elements of Classical and Geometric Optimization written by Debasish Roy and published by CRC Press. This book was released on 2024-01-25 with total page 525 pages. Available in PDF, EPUB and Kindle. Book excerpt: This comprehensive textbook covers both classical and geometric aspects of optimization using methods, deterministic and stochastic, in a single volume and in a language accessible to non-mathematicians. It will help serve as an ideal study material for senior undergraduate and graduate students in the fields of civil, mechanical, aerospace, electrical, electronics, and communication engineering. The book includes: Derivative-based Methods of Optimization. Direct Search Methods of Optimization. Basics of Riemannian Differential Geometry. Geometric Methods of Optimization using Riemannian Langevin Dynamics. Stochastic Analysis on Manifolds and Geometric Optimization Methods. This textbook comprehensively treats both classical and geometric optimization methods, including deterministic and stochastic (Monte Carlo) schemes. It offers an extensive coverage of important topics including derivative-based methods, penalty function methods, method of gradient projection, evolutionary methods, geometric search using Riemannian Langevin dynamics and stochastic dynamics on manifolds. The textbook is accompanied by online resources including MATLAB codes which are uploaded on our website. The textbook is primarily written for senior undergraduate and graduate students in all applied science and engineering disciplines and can be used as a main or supplementary text for courses on classical and geometric optimization.
Book Synopsis Mathematical Conversations by : Robin Wilson
Download or read book Mathematical Conversations written by Robin Wilson and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 486 pages. Available in PDF, EPUB and Kindle. Book excerpt: Approximately fifty articles that were published in The Mathematical Intelligencer during its first eighteen years. The selection demonstrates the wide variety of attractive articles that have appeared over the years, ranging from general interest articles of a historical nature to lucid expositions of important current discoveries. Each article is introduced by the editors. "...The Mathematical Intelligencer publishes stylish, well-illustrated articles, rich in ideas and usually short on proofs. ...Many, but not all articles fall within the reach of the advanced undergraduate mathematics major. ... This book makes a nice addition to any undergraduate mathematics collection that does not already sport back issues of The Mathematical Intelligencer." D.V. Feldman, University of New Hamphire, CHOICE Reviews, June 2001.
Book Synopsis Fibonacci's De Practica Geometrie by : Barnabas Hughes
Download or read book Fibonacci's De Practica Geometrie written by Barnabas Hughes and published by Springer Science & Business Media. This book was released on 2007-12-15 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: Leonardo da Pisa, perhaps better known as Fibonacci (ca. 1170 – ca. 1240), selected the most useful parts of Greco-Arabic geometry for the book known as De Practica Geometrie. This translation offers a reconstruction of De Practica Geometrie as the author judges Fibonacci wrote it, thereby correcting inaccuracies found in numerous modern histories. It is a high quality translation with supplemental text to explain text that has been more freely translated. A bibliography of primary and secondary resources follows the translation, completed by an index of names and special words.
Book Synopsis Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences by : Ivor Grattan-Guiness
Download or read book Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences written by Ivor Grattan-Guiness and published by Routledge. This book was released on 2004-11-11 with total page 856 pages. Available in PDF, EPUB and Kindle. Book excerpt: First published in 2004. Routledge is an imprint of Taylor & Francis, an informa company.
Book Synopsis Nonlinear Functional Analysis and its Applications by : E. Zeidler
Download or read book Nonlinear Functional Analysis and its Applications written by E. Zeidler and published by Springer Science & Business Media. This book was released on 2013-12-11 with total page 675 pages. Available in PDF, EPUB and Kindle. Book excerpt: As long as a branch of knowledge offers an abundance of problems, it is full of vitality. David Hilbert Over the last 15 years I have given lectures on a variety of problems in nonlinear functional analysis and its applications. In doing this, I have recommended to my students a number of excellent monographs devoted to specialized topics, but there was no complete survey-type exposition of nonlinear functional analysis making available a quick survey to the wide range of readers including mathematicians, natural scientists, and engineers who have only an elementary knowledge of linear functional analysis. I have tried to close this gap with my five-part lecture notes, the first three parts of which have been published in the Teubner-Texte series by Teubner-Verlag, Leipzig, 1976, 1977, and 1978. The present English edition was translated from a completely rewritten manuscript which is significantly longer than the original version in the Teubner-Texte series. The material is organized in the following way: Part I: Fixed Point Theorems. Part II: Monotone Operators. Part III: Variational Methods and Optimization. Parts IV jV: Applications to Mathematical Physics. The exposition is guided by the following considerations: (a) What are the supporting basic ideas and what intrinsic interrelations exist between them? (/3) In what relation do the basic ideas stand to the known propositions of classical analysis and linear functional analysis? ( y) What typical applications are there? Vll Preface viii Special emphasis is placed on motivation.
Book Synopsis Joseph Liouville 1809–1882 by : Jesper Lützen
Download or read book Joseph Liouville 1809–1882 written by Jesper Lützen and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 893 pages. Available in PDF, EPUB and Kindle. Book excerpt: This scientific biography of the mathematician Joseph Liouville is divided into two parts. The first part is a chronological account of Liouville's career including a description of the institutions he worked in, his relations with his teachers, colleagues and students, and the historical context of his works. It portrays the French scientific community in a period when Germany and England had surpassed France as the leading nations in mathematics and physics. The second part of the book gives a detailed analysis of Liouville's major contributions to mathematics and mechanics. The gradual development of Liouville's ideas, as reflected in his publications and notebooks, are related to the works of his predecessors and his contemporaries as well as to later developments in the field. On the basis of Liouville's unpublished notes the book reconstructs Liouville's hitherto unknown theories of stability of rotating masses of fluid, potential theory, Galois theory and electrodynamics. It also incorporates valuable added information from Liouville's notes regarding his works on differentiation of arbitrary order, integration in finite terms, Sturm-Liouville theory, transcendental numbers, doubly periodic functions, geometry and mechanics.
Book Synopsis Thomas Harriot's Artis Analyticae Praxis by : Muriel Seltman
Download or read book Thomas Harriot's Artis Analyticae Praxis written by Muriel Seltman and published by Springer Science & Business Media. This book was released on 2007-05-09 with total page 303 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first English translation of Thomas Harriot’s seminal Artis Analyticae Praxis, first published in Latin in 1631. It has recently become clear that Harriot's editor substantially rearranged the work, and omitted sections beyond his comprehension. Commentary included with this translation relates to corresponding pages in the manuscript papers, enabling exploration of Harriot's novel and advanced mathematics. This publication provides the basis for a reassessment of the development of algebra.
