The Birth of Calculus: Towards a More Leibnizian View
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CALCULUS of VARIATIONS and TENSOR CALCULUS
CALCULUS OF VARIATIONS and TENSOR CALCULUS U. H. Gerlach September 22, 2019 Beta Edition 2 Contents 1 FUNDAMENTAL IDEAS 5 1.1 Multivariable Calculus as a Prelude to the Calculus of Variations. 5 1.2 Some Typical Problems in the Calculus of Variations. ...... 6 1.3 Methods for Solving Problems in Calculus of Variations. ....... 10 1.3.1 MethodofFiniteDifferences. 10 1.4 TheMethodofVariations. 13 1.4.1 Variants and Variations . 14 1.4.2 The Euler-Lagrange Equation . 17 1.4.3 Variational Derivative . 20 1.4.4 Euler’s Differential Equation . 21 1.5 SolvedExample.............................. 24 1.6 Integration of Euler’s Differential Equation. ...... 25 2 GENERALIZATIONS 33 2.1 Functional with Several Unknown Functions . 33 2.2 Extremum Problem with Side Conditions. 38 2.2.1 HeuristicSolution. 40 2.2.2 Solution via Constraint Manifold . 42 2.2.3 Variational Problems with Finite Constraints . 54 2.3 Variable End Point Problem . 55 2.3.1 Extremum Principle at a Moment of Time Symmetry . 57 2.4 Generic Variable Endpoint Problem . 60 2.4.1 General Variations in the Functional . 62 2.4.2 Transversality Conditions . 64 2.4.3 Junction Conditions . 66 2.5 ManyDegreesofFreedom . 68 2.6 Parametrization Invariant Problem . 70 2.6.1 Parametrization Invariance via Homogeneous Function .... 71 2.7 Variational Principle for a Geodesic . 72 2.8 EquationofGeodesicMotion . 76 2.9 Geodesics: TheirParametrization.. 77 3 4 CONTENTS 2.9.1 Parametrization Invariance. 77 2.9.2 Parametrization in Terms of Curve Length . 78 2.10 Physical Significance of the Equation for a Geodesic . ....... 80 2.10.1 Freefloatframe ........................ -
Notes on Calculus II Integral Calculus Miguel A. Lerma
Notes on Calculus II Integral Calculus Miguel A. Lerma November 22, 2002 Contents Introduction 5 Chapter 1. Integrals 6 1.1. Areas and Distances. The Definite Integral 6 1.2. The Evaluation Theorem 11 1.3. The Fundamental Theorem of Calculus 14 1.4. The Substitution Rule 16 1.5. Integration by Parts 21 1.6. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. Partial Fractions 32 1.8. Integration using Tables and CAS 39 1.9. Numerical Integration 41 1.10. Improper Integrals 46 Chapter 2. Applications of Integration 50 2.1. More about Areas 50 2.2. Volumes 52 2.3. Arc Length, Parametric Curves 57 2.4. Average Value of a Function (Mean Value Theorem) 61 2.5. Applications to Physics and Engineering 63 2.6. Probability 69 Chapter 3. Differential Equations 74 3.1. Differential Equations and Separable Equations 74 3.2. Directional Fields and Euler’s Method 78 3.3. Exponential Growth and Decay 80 Chapter 4. Infinite Sequences and Series 83 4.1. Sequences 83 4.2. Series 88 4.3. The Integral and Comparison Tests 92 4.4. Other Convergence Tests 96 4.5. Power Series 98 4.6. Representation of Functions as Power Series 100 4.7. Taylor and MacLaurin Series 103 3 CONTENTS 4 4.8. Applications of Taylor Polynomials 109 Appendix A. Hyperbolic Functions 113 A.1. Hyperbolic Functions 113 Appendix B. Various Formulas 118 B.1. Summation Formulas 118 Appendix C. Table of Integrals 119 Introduction These notes are intended to be a summary of the main ideas in course MATH 214-2: Integral Calculus. -
A Brief Tour of Vector Calculus
A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0. -
Differentiation Rules (Differential Calculus)
Differentiation Rules (Differential Calculus) 1. Notation The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by D f . Note that f (x) and (D f )(x) are the values of these functions at x. 2. Alternate Notations for (D f )(x) d d f (x) d f 0 (1) For functions f in one variable, x, alternate notations are: Dx f (x), dx f (x), dx , dx (x), f (x), f (x). The “(x)” part might be dropped although technically this changes the meaning: f is the name of a function, dy 0 whereas f (x) is the value of it at x. If y = f (x), then Dxy, dx , y , etc. can be used. If the variable t represents time then Dt f can be written f˙. The differential, “d f ”, and the change in f ,“D f ”, are related to the derivative but have special meanings and are never used to indicate ordinary differentiation. dy 0 Historical note: Newton used y,˙ while Leibniz used dx . About a century later Lagrange introduced y and Arbogast introduced the operator notation D. 3. Domains The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below. -
Newton.Indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag
omslag Newton.indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag. 1 e Dutch Republic proved ‘A new light on several to be extremely receptive to major gures involved in the groundbreaking ideas of Newton Isaac Newton (–). the reception of Newton’s Dutch scholars such as Willem work.’ and the Netherlands Jacob ’s Gravesande and Petrus Prof. Bert Theunissen, Newton the Netherlands and van Musschenbroek played a Utrecht University crucial role in the adaption and How Isaac Newton was Fashioned dissemination of Newton’s work, ‘is book provides an in the Dutch Republic not only in the Netherlands important contribution to but also in the rest of Europe. EDITED BY ERIC JORINK In the course of the eighteenth the study of the European AND AD MAAS century, Newton’s ideas (in Enlightenment with new dierent guises and interpre- insights in the circulation tations) became a veritable hype in Dutch society. In Newton of knowledge.’ and the Netherlands Newton’s Prof. Frans van Lunteren, sudden success is analyzed in Leiden University great depth and put into a new perspective. Ad Maas is curator at the Museum Boerhaave, Leiden, the Netherlands. Eric Jorink is researcher at the Huygens Institute for Netherlands History (Royal Dutch Academy of Arts and Sciences). / www.lup.nl LUP Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 1 Newton and the Netherlands Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 2 Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. -
Introduction to the Modern Calculus of Variations
MA4G6 Lecture Notes Introduction to the Modern Calculus of Variations Filip Rindler Spring Term 2015 Filip Rindler Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom [email protected] http://www.warwick.ac.uk/filiprindler Copyright ©2015 Filip Rindler. Version 1.1. Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. This is not an encyclopedic work; many important results are omitted and sometimes I only present a special case of a more general theorem. I have, however, tried to strike a balance between a pure introduction and a text that can be used for later revision of forgotten material. The presentation is based around a few principles: • The presentation is quite “modern” in that I use several techniques which are perhaps not usually found in an introductory text or that have only recently been developed. • For most results, I try to use “reasonable” assumptions, not necessarily minimal ones. • When presented with a choice of how to prove a result, I have usually preferred the (in my opinion) most conceptually clear approach over more “elementary” ones. For example, I use Young measures in many instances, even though this comes at the expense of a higher initial burden of abstract theory. • Wherever possible, I first present an abstract result for general functionals defined on Banach spaces to illustrate the general structure of a certain result. -
The Newton-Leibniz Controversy Over the Invention of the Calculus
The Newton-Leibniz controversy over the invention of the calculus S.Subramanya Sastry 1 Introduction Perhaps one the most infamous controversies in the history of science is the one between Newton and Leibniz over the invention of the infinitesimal calculus. During the 17th century, debates between philosophers over priority issues were dime-a-dozen. Inspite of the fact that priority disputes between scientists were ¡ common, many contemporaries of Newton and Leibniz found the quarrel between these two shocking. Probably, what set this particular case apart from the rest was the stature of the men involved, the significance of the work that was in contention, the length of time through which the controversy extended, and the sheer intensity of the dispute. Newton and Leibniz were at war in the later parts of their lives over a number of issues. Though the dispute was sparked off by the issue of priority over the invention of the calculus, the matter was made worse by the fact that they did not see eye to eye on the matter of the natural philosophy of the world. Newton’s action-at-a-distance theory of gravitation was viewed as a reversion to the times of occultism by Leibniz and many other mechanical philosophers of this era. This intermingling of philosophical issues with the priority issues over the invention of the calculus worsened the nature of the dispute. One of the reasons why the dispute assumed such alarming proportions and why both Newton and Leibniz were anxious to be considered the inventors of the calculus was because of the prevailing 17th century conventions about priority and attitude towards plagiarism. -
The Infinite and Contradiction: a History of Mathematical Physics By
The infinite and contradiction: A history of mathematical physics by dialectical approach Ichiro Ueki January 18, 2021 Abstract The following hypothesis is proposed: \In mathematics, the contradiction involved in the de- velopment of human knowledge is included in the form of the infinite.” To prove this hypothesis, the author tries to find what sorts of the infinite in mathematics were used to represent the con- tradictions involved in some revolutions in mathematical physics, and concludes \the contradiction involved in mathematical description of motion was represented with the infinite within recursive (computable) set level by early Newtonian mechanics; and then the contradiction to describe discon- tinuous phenomena with continuous functions and contradictions about \ether" were represented with the infinite higher than the recursive set level, namely of arithmetical set level in second or- der arithmetic (ordinary mathematics), by mechanics of continuous bodies and field theory; and subsequently the contradiction appeared in macroscopic physics applied to microscopic phenomena were represented with the further higher infinite in third or higher order arithmetic (set-theoretic mathematics), by quantum mechanics". 1 Introduction Contradictions found in set theory from the end of the 19th century to the beginning of the 20th, gave a shock called \a crisis of mathematics" to the world of mathematicians. One of the contradictions was reported by B. Russel: \Let w be the class [set]1 of all classes which are not members of themselves. Then whatever class x may be, 'x is a w' is equivalent to 'x is not an x'. Hence, giving to x the value w, 'w is a w' is equivalent to 'w is not a w'."[52] Russel described the crisis in 1959: I was led to this contradiction by Cantor's proof that there is no greatest cardinal number. -
The British Journal for the History of Science Mechanical Experiments As Moral Exercise in the Education of George
The British Journal for the History of Science http://journals.cambridge.org/BJH Additional services for The British Journal for the History of Science: Email alerts: Click here Subscriptions: Click here Commercial reprints: Click here Terms of use : Click here Mechanical experiments as moral exercise in the education of George III FLORENCE GRANT The British Journal for the History of Science / Volume 48 / Issue 02 / June 2015, pp 195 - 212 DOI: 10.1017/S0007087414000582, Published online: 01 August 2014 Link to this article: http://journals.cambridge.org/abstract_S0007087414000582 How to cite this article: FLORENCE GRANT (2015). Mechanical experiments as moral exercise in the education of George III. The British Journal for the History of Science, 48, pp 195-212 doi:10.1017/ S0007087414000582 Request Permissions : Click here Downloaded from http://journals.cambridge.org/BJH, IP address: 130.132.173.207 on 07 Jul 2015 BJHS 48(2): 195–212, June 2015. © British Society for the History of Science 2014 doi:10.1017/S0007087414000582 First published online 1 August 2014 Mechanical experiments as moral exercise in the education of George III FLORENCE GRANT* Abstract. In 1761, George III commissioned a large group of philosophical instruments from the London instrument-maker George Adams. The purchase sprang from a complex plan of moral education devised for Prince George in the late 1750s by the third Earl of Bute. Bute’s plan applied the philosophy of Frances Hutcheson, who placed ‘the culture of the heart’ at the foundation of moral education. To complement this affective development, Bute also acted on seventeenth-century arguments for the value of experimental philosophy and geometry as exercises that habituated the student to recognizing truth, and to pursuing it through long and difficult chains of reasoning. -
J. Wallis the Arithmetic of Infinitesimals Series: Sources and Studies in the History of Mathematics and Physical Sciences
J. Wallis The Arithmetic of Infinitesimals Series: Sources and Studies in the History of Mathematics and Physical Sciences John Wallis was appointed Savilian Professor of Geometry at Oxford University in 1649. He was then a relative newcomer to mathematics, and largely self-taught, but in his first few years at Oxford he produced his two most significant works: De sectionibus conicis and Arithmetica infinitorum. In both books, Wallis drew on ideas originally developed in France, Italy, and the Netherlands: analytic geometry and the method of indivisibles. He handled them in his own way, and the resulting method of quadrature, based on the summation of indivisible or infinitesimal quantities, was a crucial step towards the development of a fully fledged integral calculus some ten years later. To the modern reader, the Arithmetica Infinitorum reveals much that is of historical and mathematical interest, not least the mid seventeenth-century tension between classical geometry on the one hand, and arithmetic and algebra on the other. Newton was to take up Wallis’s work and transform it into mathematics that has become part of the mainstream, but in Wallis’s text we see what we think of as modern mathematics still 2004, XXXIV, 192 p. struggling to emerge. It is this sense of watching new and significant ideas force their way slowly and sometimes painfully into existence that makes the Arithmetica Infinitorum such a relevant text even now for students and historians of mathematics alike. Printed book Dr J.A. Stedall is a Junior Research Fellow at Queen's University. She has written a number Hardcover of papers exploring the history of algebra, particularly the algebra of the sixteenth and 159,99 € | £139.99 | $199.99 ▶ seventeenth centuries. -
Reposs #19: Newtonianism in the Scandinavian Countries, 1690–1790
RePoSS: Research Publications on Science Studies RePoSS #19: Newtonianism in the Scandinavian Countries, 1690–1790 Helge Kragh August 2012 Centre for Science Studies, University of Aarhus, Denmark Research group: History and philosophy of science Please cite this work as: Helge Kragh (Aug. 2012). Newtonianism in the Scandinavian Countries, 1690–1790. RePoSS: Research Publications on Sci- ence Studies 19. Aarhus: Centre for Science Studies, University of Aarhus. url: http://www.css.au.dk/reposs. Copyright c Helge Kragh, 2012 1 Newtonianism in the Scandinavian Countries, 1690-1790 HELGE KRAGH 1 Introduction In the present context, the Scandinavian countries refer to two national or administrative units, the one being Denmark and the other Sweden. In the period here considered, largely the century from 1690 to 1790, ‘Denmark’ means really Denmark-Norway, for until 1814 Norway was part of the double monarchy ruled by the king and his government in Copenhagen. It should also be kept in mind that parts of what is today Germany, namely Schleswig-Holstein, belonged to the kingdom. However, as far as language and culture were concerned, these parts of southern Denmark were more German than Danish, and they played no important role in the scientific life of the kingdom. Sweden covered a much larger geographical area than it does today. The country had expanded greatly during the seventeenth century, when not only Finland but also parts of the Baltic area and northern Germany came under Swedish rule. About 1720, after the Great Northern War, Sweden lost most of its possessions, but the major part of Finland remained as part of the country until Centre for Science Studies, Department of Physics and Astronomy, Aarhus University, Denmark. -
Cavendish the Experimental Life
Cavendish The Experimental Life Revised Second Edition Max Planck Research Library for the History and Development of Knowledge Series Editors Ian T. Baldwin, Gerd Graßhoff, Jürgen Renn, Dagmar Schäfer, Robert Schlögl, Bernard F. Schutz Edition Open Access Development Team Lindy Divarci, Georg Pflanz, Klaus Thoden, Dirk Wintergrün. The Edition Open Access (EOA) platform was founded to bring together publi- cation initiatives seeking to disseminate the results of scholarly work in a format that combines traditional publications with the digital medium. It currently hosts the open-access publications of the “Max Planck Research Library for the History and Development of Knowledge” (MPRL) and “Edition Open Sources” (EOS). EOA is open to host other open access initiatives similar in conception and spirit, in accordance with the Berlin Declaration on Open Access to Knowledge in the sciences and humanities, which was launched by the Max Planck Society in 2003. By combining the advantages of traditional publications and the digital medium, the platform offers a new way of publishing research and of studying historical topics or current issues in relation to primary materials that are otherwise not easily available. The volumes are available both as printed books and as online open access publications. They are directed at scholars and students of various disciplines, and at a broader public interested in how science shapes our world. Cavendish The Experimental Life Revised Second Edition Christa Jungnickel and Russell McCormmach Studies 7 Studies 7 Communicated by Jed Z. Buchwald Editorial Team: Lindy Divarci, Georg Pflanz, Bendix Düker, Caroline Frank, Beatrice Hermann, Beatrice Hilke Image Processing: Digitization Group of the Max Planck Institute for the History of Science Cover Image: Chemical Laboratory.