The Early Period Of The Calculus Of Variations
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The Early Period Of The Calculus Of Variations
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Author : Paolo Freguglia
language : en
Publisher: Birkhäuser
Release Date : 2018-05-31
The Early Period Of The Calculus Of Variations written by Paolo Freguglia and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-05-31 with Mathematics categories.
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.
The Early Period Of The Calculus Of Variations
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Author : Paolo Freguglia
language : en
Publisher: Birkhäuser
Release Date : 2016-06-27
The Early Period Of The Calculus Of Variations written by Paolo Freguglia and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-06-27 with Mathematics categories.
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.
A History Of The Calculus Of Variations From The 17th Through The 19th Century
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Author : H. H. Goldstine
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
A History Of The Calculus Of Variations From The 17th Through The 19th Century written by H. H. Goldstine and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
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.
A History Of The Calculus Of Variations In The Eighteenth Century
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Author : Robert Woodhouse
language : en
Publisher: American Mathematical Society
Release Date : 2004-04-13
A History Of The Calculus Of Variations In The Eighteenth Century written by Robert Woodhouse and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-04-13 with Mathematics categories.
Shortly after the invention of differential and integral calculus, the calculus of variations was developed. The new calculus looks for functions that minimize or maximize some quantity, such as the brachistochrone problem, which was solved by Johann Bernoulli, Leibniz, Newton, Jacob Bernoulli and l'Hopital and is sometimes considered as the starting point of the calculus of variations. In Woodhouse's book, first published in 1810, he has interwoven the historical progress with the scientific development of the subject. The reader will have the opportunity to see how calculus, during its first one hundred years, developed by seemingly tiny increments to become the highly polished subject that we know today. Here, Woodhouse's interweaving of history and science gives his special point of view on the mathematics. As he states in his preface: ""Indeed the authors who write near the beginnings of science are, in general, the most instructive; they take the reader more along with them, show him the real difficulties and, which is the main point, teach him the subject, the way they themselves learned it.
Calculus Of Variations
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Author : Gilbert Ames Bliss
language : en
Publisher: American Mathematical Soc.
Release Date : 1925-12-31
Calculus Of Variations written by Gilbert Ames Bliss and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1925-12-31 with Calculus of variations categories.
The development of the calculus of variations has, from the beginning, been interlaced with that of the differential and integral calculus. Without any knowledge of the calculus, one can readily understand at least the geometrical or mechanical statements of many of the problems of the calculus of variations and the character of their solutions. The discovery and justification of the results in this book, apart from their simple statements, do require, however, acquaintance with the principles of the calculus, and it is assumed that the reader has such an acquaintance. Calculus of Variations begins by studying special problems rather than the general theory. The first chapter of the book describes the historical setting out of which the theory of the calculus of variations grew and the character of some of the simpler problems. The next three chapters are devoted to the development, in detail, of the then known results for three special problems (shortest distances, brachistochrone, and surfaces of revolution of minimum area) which illustrate in excellent fashion the essential characteristics of the general theory contained in Chapter V with which the book concludes.
A History Of The Progress Of The Calculus Of Variations During The Nineteenth Century
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Author : Isaac Todhunter
language : en
Publisher:
Release Date : 1861
A History Of The Progress Of The Calculus Of Variations During The Nineteenth Century written by Isaac Todhunter and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1861 with Calculus of variations categories.
Introduction To The Calculus Of Variations
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Author : Hans Sagan
language : en
Publisher: Courier Corporation
Release Date : 2012-04-26
Introduction To The Calculus Of Variations written by Hans Sagan and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-04-26 with Mathematics categories.
Provides a thorough understanding of calculus of variations and prepares readers for the study of modern optimal control theory. Selected variational problems and over 400 exercises. Bibliography. 1969 edition.
The Calculus Of Variations
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Author : N.I. Akhiezer
language : en
Publisher: CRC Press
Release Date : 1988-01-01
The Calculus Of Variations written by N.I. Akhiezer and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988-01-01 with Mathematics categories.
An authoritative text on the calculus of variations for first-year graduate students. From a study of the simplest problem it goes on to cover Lagrangian derivatives, Jacobi’s condition, and field theory. Devotes considerable attention to direct methods and the Sturm-Liouville problem in a finite interval. Contains numerous interesting and challenging exercises plus five appendices on important results, generalizations, and applications of the material,
A First Course In The Calculus Of Variations
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Author : Mark Kot
language : en
Publisher: American Mathematical Society
Release Date : 2014-10-06
A First Course In The Calculus Of Variations written by Mark Kot and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-10-06 with Mathematics categories.
This book is intended for a first course in the calculus of variations, at the senior or beginning graduate level. The reader will learn methods for finding functions that maximize or minimize integrals. The text lays out important necessary and sufficient conditions for extrema in historical order, and it illustrates these conditions with numerous worked-out examples from mechanics, optics, geometry, and other fields. The exposition starts with simple integrals containing a single independent variable, a single dependent variable, and a single derivative, subject to weak variations, but steadily moves on to more advanced topics, including multivariate problems, constrained extrema, homogeneous problems, problems with variable endpoints, broken extremals, strong variations, and sufficiency conditions. Numerous line drawings clarify the mathematics. Each chapter ends with recommended readings that introduce the student to the relevant scientific literature and with exercises that consolidate understanding.
A History Of The Calculus Of Variations
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Author : I. Todhunter
language : en
Publisher: BoD – Books on Demand
Release Date : 2022-06-12
A History Of The Calculus Of Variations written by I. Todhunter and has been published by BoD – Books on Demand this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-06-12 with Fiction categories.
Reprint of the original, first published in 1861.