Comment on Clebsch's 1857 and 1859 Papers
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Variational Formulation of Fluid and Geophysical Fluid Dynamics
Advances in Geophysical and Environmental Mechanics and Mathematics Gualtiero Badin Fulvio Crisciani Variational Formulation of Fluid and Geophysical Fluid Dynamics Mechanics, Symmetries and Conservation Laws Advances in Geophysical and Environmental Mechanics and Mathematics Series editor Holger Steeb, Institute of Applied Mechanics (CE), University of Stuttgart, Stuttgart, Germany More information about this series at http://www.springer.com/series/7540 Gualtiero Badin • Fulvio Crisciani Variational Formulation of Fluid and Geophysical Fluid Dynamics Mechanics, Symmetries and Conservation Laws 123 Gualtiero Badin Fulvio Crisciani Universität Hamburg University of Trieste Hamburg Trieste Germany Italy ISSN 1866-8348 ISSN 1866-8356 (electronic) Advances in Geophysical and Environmental Mechanics and Mathematics ISBN 978-3-319-59694-5 ISBN 978-3-319-59695-2 (eBook) DOI 10.1007/978-3-319-59695-2 Library of Congress Control Number: 2017949166 © Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. -
Differential Calculus and by Era Integral Calculus, Which Are Related by in Early Cultures in Classical Antiquity the Fundamental Theorem of Calculus
History of calculus - Wikipedia, the free encyclopedia 1/1/10 5:02 PM History of calculus From Wikipedia, the free encyclopedia History of science This is a sub-article to Calculus and History of mathematics. History of Calculus is part of the history of mathematics focused on limits, functions, derivatives, integrals, and infinite series. The subject, known Background historically as infinitesimal calculus, Theories/sociology constitutes a major part of modern Historiography mathematics education. It has two major Pseudoscience branches, differential calculus and By era integral calculus, which are related by In early cultures in Classical Antiquity the fundamental theorem of calculus. In the Middle Ages Calculus is the study of change, in the In the Renaissance same way that geometry is the study of Scientific Revolution shape and algebra is the study of By topic operations and their application to Natural sciences solving equations. A course in calculus Astronomy is a gateway to other, more advanced Biology courses in mathematics devoted to the Botany study of functions and limits, broadly Chemistry Ecology called mathematical analysis. Calculus Geography has widespread applications in science, Geology economics, and engineering and can Paleontology solve many problems for which algebra Physics alone is insufficient. Mathematics Algebra Calculus Combinatorics Contents Geometry Logic Statistics 1 Development of calculus Trigonometry 1.1 Integral calculus Social sciences 1.2 Differential calculus Anthropology 1.3 Mathematical analysis -
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. -
Analyzing Applied Calculus Student Understanding of Definite Integrals in Real-Life Applications
Graduate Theses, Dissertations, and Problem Reports 2021 ANALYZING APPLIED CALCULUS STUDENT UNDERSTANDING OF DEFINITE INTEGRALS IN REAL-LIFE APPLICATIONS CODY HOOD [email protected] Follow this and additional works at: https://researchrepository.wvu.edu/etd Part of the Mathematics Commons, and the Science and Mathematics Education Commons Recommended Citation HOOD, CODY, "ANALYZING APPLIED CALCULUS STUDENT UNDERSTANDING OF DEFINITE INTEGRALS IN REAL-LIFE APPLICATIONS" (2021). Graduate Theses, Dissertations, and Problem Reports. 8260. https://researchrepository.wvu.edu/etd/8260 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Dissertation has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. Graduate Theses, Dissertations, and Problem Reports 2021 ANALYZING APPLIED CALCULUS STUDENT UNDERSTANDING OF DEFINITE INTEGRALS IN REAL-LIFE APPLICATIONS CODY HOOD Follow this and additional works at: https://researchrepository.wvu.edu/etd Part of the Mathematics Commons, and the Science and Mathematics Education Commons ANALYZING APPLIED CALCULUS STUDENT UNDERSTANDING OF DEFINITE INTEGRALS IN REAL-LIFE APPLICATIONS Cody Hood Dissertation submitted to the Eberly College of Arts and Sciences at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Vicki Sealey, Ph.D., Chair Harvey Diamond, Ph.D. -
Representation of Divergence-Free Vector Fields
CENTRO DE MATEMATICA´ E APLICAC¸ OES˜ FUNDAMENTAIS PrePrint Pre-2009-017 http://cmaf.ptmat.fc.ul.pt/preprints.html REPRESENTATION OF DIVERGENCE-FREE VECTOR FIELDS CRISTIAN BARBAROSIE CMAF, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal web page: http://cmaf.ptmat.fc.ul.pt/~barbaros/ e-mail: [email protected] Abstract. This paper focuses on a representation result for divergence-free vector fields. Known results are recalled, namely the representation of divergence-free vector fields as curls in two and three dimensions. The representation proposed in the present paper expresses the vector field as exterior product of gradients and stands valid in arbitrary dimension. Links to computer graphics and partial differential equations are discussed. 1. Introduction. The aim of this paper is to study properties of vector fields having zero divergence, with particular emphasis on the representation of such vector fields in terms of a potential (which may be vector-valued). More precisely, a divergence-free vector field in Rn is expressed as exterior product of n − 1 gradients. The main result is stated in Theorem 7.3. This representation appears sometimes in textbooks on mechanics (especially fluid mechanics) and electromagnetism is a somewhat vage formulation; see the bibliographical comments at the end of Section 5. One motivation for seeking this type of representation result comes from computer graphics, see Section 4. Another motivation is related to elliptic partial differential equations, see Section 8. The outline of the paper is as follows. Section 2 presents results on two-dimensional vector fields, some of which are well-known from graduate-level calculus. -
The Fluid Dynamics of Spin - a Fisher Information Perspective and Comoving Scalars
Chaotic Modeling and Simulation (CMSIM) 1: 17{30, 2020 The Fluid Dynamics of Spin - a Fisher Information Perspective and Comoving Scalars Asher Yahalom Ariel University, Ariel 40700, Israel (E-mail: [email protected]) Abstract. Two state systems have wide applicability in quantum modelling of var- ious systems. Here I return to the original two state system, that is Pauli's electron with a spin. And show how this system can be interpreted as a vortical fluid. The similarities and difference between spin flows and classical ideal flows are elucidated. It will be shown how the internal energy of the spin fluid can partially be interpreted in terms of Fisher Information. Keywords: Spin, Fluid dynamics. 1 Introduction The Copenhagen interpretation of quantum mechanics is probably the most prevalent approach. This approach defies any ontology quantum theory and declares it to be completely epistemological in accordance to the Kantian [1] conception of reality. However, in addition to this approach we see the devel- opment of another school that believed in the realism of the wave function. This approach that was championed by Einstein and Bohm [2{4] led to other interpretations of quantum mechanics among them the fluid interpretation due to do Madelung [5,6] which interpreted the modulus square as the fluid density and the phase as a potential of a velocity field. However, this model was limited to spin less electrons and could not take into account a complete set of electron attributes even for non relativistic electrons. A spin dependent non relativistic quantum equation was first introduced by Wolfgang Pauli in 1927 [7]. -
Connes on the Role of Hyperreals in Mathematics
Found Sci DOI 10.1007/s10699-012-9316-5 Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics Vladimir Kanovei · Mikhail G. Katz · Thomas Mormann © Springer Science+Business Media Dordrecht 2012 Abstract We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in func- tional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being “vir- tual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnera- ble to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace − (featured on the front cover of Connes’ magnum opus) V. -
The Pressure Boundary Condition and the Pressure As Lagrangian for Water Waves
Water Waves (2019) 1:131–143 https://doi.org/10.1007/s42286-019-00001-0 ORIGINAL ARTICLE The Pressure Boundary Condition and the Pressure as Lagrangian for Water Waves Thomas J. Bridges1 Received: 24 July 2018 / Accepted: 2 January 2019 / Published online: 11 March 2019 © The Author(s) 2019 Abstract The pressure boundary condition for the full Euler equations with a free surface and general vorticity field is formulated in terms of a generalized Bernoulli equa- tion deduced from the Gavrilyuk–Kalisch–Khorsand conservation law. The use of pressure as a Lagrangian density, as in Luke’s variational principle, is reviewed and extension to a full vortical flow is attempted with limited success. However, a new variational principle for time-dependent water waves in terms of the stream function is found. The variational principle generates vortical boundary conditions but with a harmonic stream function. Other aspects of vorticity in variational principles are also discussed. Keywords Oceanography · Lagrangian · Vorticity · Streamfunction · Variational principle 1 Introduction The dynamic boundary condition in the theory of water waves reduces, in the case of inviscid flow, to continuity of the pressure field. In inviscid irrotational flow this condition is replaced by the Bernoulli equation evaluated at the surface. In this paper, it is shown that there is a generalized Bernoulli equation valid for all inviscid flows. Restricting to two space dimensions with a free surface at y = h(x, t), the Bernoulli equation is + + − 1 2 − 1 2 + = ( ), t U x gh 2 U 2 V Pa f t (1.1) where (U, V ) is the velocity field evaluated at the free surface, Pa(x, t) is the imposed external pressure field, and f (t) is an arbitrary function of time. -
Epistemic Criteria for Designing Limit Tasks on a Real Variable Function
ISSN 1980-4415 DOI: http://dx.doi.org/10.1590/1980-4415v35n69a09 Epistemic Criteria for Designing Limit Tasks on a Real Variable Function Criterios Epistémicos para el Diseño de Tareas sobre Límite de una función en una variable Real Daniela Araya Bastias* ORCID iD 0000-0002-3395-3348 Luis R. Pino-Fan** ORCID iD 0000-0003-4060-7408 Iván G. Medrano*** ORCID iD 0000-0003-4910-1235 Walter F. Castro**** ORCID iD 0000-0002-7890-681X Abstract This article aims at presenting the results of a historical-epistemological study conducted to identify criteria for designing tasks that promote the understanding of the limit notion on a real variable function. As a theoretical framework, we used the Onto-Semiotic Approach (OSA) to mathematical knowledge and instruction, to identify the regulatory elements of mathematical practices developed throughout history, and that gave way to the emergence, evolution, and formalization of limit. As a result, we present a proposal of criteria that summarizes fundamental epistemic aspects, which could be considered when designing tasks that allow the promotion of each of the six meanings identified for the limit notion. The criteria presented allow us to highlight not only the mathematical complexity underlying the study of limit on a real variable function but also the richness of meanings that could be developed to help understand this notion. Keywords: Limit. Task design. Onto-Semiotic Approach. Resumen El objetivo de este artículo es presentar los resultados de un estudio de tipo histórico-epistemológico, que se llevó a cabo para identificar criterios a considerar en el diseño de tareas para promover la comprensión de la noción de * Magíster en Matemática por la Universidad de Santiago de Chile (USACH). -
List of Mathematical Symbols by Subject from Wikipedia, the Free Encyclopedia
List of mathematical symbols by subject From Wikipedia, the free encyclopedia This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are included. Many of the characters are standardized, for example in DIN 1302 General mathematical symbols or DIN EN ISO 80000-2 Quantities and units – Part 2: Mathematical signs for science and technology. The following list is largely limited to non-alphanumeric characters. It is divided by areas of mathematics and grouped within sub-regions. Some symbols have a different meaning depending on the context and appear accordingly several times in the list. Further information on the symbols and their meaning can be found in the respective linked articles. Contents 1 Guide 2 Set theory 2.1 Definition symbols 2.2 Set construction 2.3 Set operations 2.4 Set relations 2.5 Number sets 2.6 Cardinality 3 Arithmetic 3.1 Arithmetic operators 3.2 Equality signs 3.3 Comparison 3.4 Divisibility 3.5 Intervals 3.6 Elementary functions 3.7 Complex numbers 3.8 Mathematical constants 4 Calculus 4.1 Sequences and series 4.2 Functions 4.3 Limits 4.4 Asymptotic behaviour 4.5 Differential calculus 4.6 Integral calculus 4.7 Vector calculus 4.8 Topology 4.9 Functional analysis 5 Linear algebra and geometry 5.1 Elementary geometry 5.2 Vectors and matrices 5.3 Vector calculus 5.4 Matrix calculus 5.5 Vector spaces 6 Algebra 6.1 Relations 6.2 Group theory 6.3 Field theory 6.4 Ring theory 7 Combinatorics 8 Stochastics 8.1 Probability theory 8.2 Statistics 9 Logic 9.1 Operators 9.2 Quantifiers 9.3 Deduction symbols 10 See also 11 References 12 External links Guide The following information is provided for each mathematical symbol: Symbol: The symbol as it is represented by LaTeX. -
Pascal and Leibniz: Sines, Circles, and Transmutations
Pascal and Leibniz: Sines, Circles, and Transmutations The title of the last section promised some discussion of sine curves. In Roberval's construction a sine curve was drawn and used to find the area of the cycloidal arch (see Figure 2.13b), but Roberval called this curve the "companion of the cycloid." He did not see this curve as a graph of a sine or cosine, and neither did any of his contemporaries. They did discuss sines and cosines, however, and it is important to understand the conceptual point of view that was then standard. Before I present my last example of curve drawing from Leibniz, I must also return to some of the issues raised in section 2.3. Leibniz's use of the characteristic triangle (Figure 2.3b) was directly inspired by the work of Pascal concerning sines. Trigonometric functions were not defined using ratios, or the unit circle, until the textbooks of Euler were published in 1748 (Euler, 1988). Both Ptolemy and Arabic astronomers made detailed tables of the lengths of chords subtended by circular arcs (Katz, 1993). That is, given two points, A and B, on a circle, to find the length of the line segment AB. Such tables were usually made for a circle of a given (large) radius, and then scaled for use in other settings. It was also found useful by Arabic astronomers to have tables of half chords, and such tables became known in Latin as tables of "sines." 1 Since the perpendicular bisector of any chord passes through the center of the circle, half chords on the unit circle are the same as our sines, but I want to stress that trigonometric quantities were seen as lengths. -
Clebsch Representations in the Theory of Minimum Energy Equilibrium
J. Plasma Physics (1978), vol. 20, part 3, pp. 329-344 329 Printed in Great Britain Clebsch representations in the theory of minimum energy equilibrium solutions in magnetohydrodynamics ByHANNO RUND Department of Mathematics, University of Arizona, Tucson, Arizona 85721 DANIEL R. WELLS AND LAWRENCE CARL HAWKINS Department of Physics, University of Miami, Coral Gables, Florida 33124 (Received 15 February 1977, and in revised form 16 November 1977) The introduction of Clebsch representations allows one to formulate the problem of finding minimum energy solutions for a magneto-fluid as a well-posed problem in the calculus of variations of multiple integrals. When the latter is subjected to integral constraints, the Euler—Lagrange equations of the resulting isoperimetric problem imply that the fluid velocities are collinear with the magnetic field. If, in particular, one constraint is abolished, Alfv6n velocities are obtained. In view of the idealized nature of the model treated here, further investigations of more sophisticated structures by means of Clebsch representations are anticipated. Preliminary results of a similar calculation utilizing a modified two fluid model are discussed. 1. Introduction It is important to find solutions of the variational problem posed by a deter- mination of minimum free energy bounded plasma structures. These solutions correspond to force-free or quasi force-free configurations. The concept of force- free fields has played a major role in both astrophysics and the theory of stable laboratory plasmas. It has proved to be very useful in investigating possibly naturally occurring stable states in magnetically confined high temperature plasmas (Wells & Norwood 1968; Wells 1970, 1976; Taylor 1974).