Elliptic Integrals and Integration by Substitution Janet Heine Barnett Colorado State University-Pueblo, [email protected]
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
MA-302 Advanced Calculus 8
Prof. D. P.Patil, Department of Mathematics, Indian Institute of Science, Bangalore Aug-Dec 2002 MA-302 Advanced Calculus 8. Inverse function theorem Carl Gustav Jacob Jacobi† (1804-1851) The exercises 8.1 and 8.2 are only to get practice, their solutions need not be submitted. 8.1. Determine at which points a of the domain of definition G the following maps F have differentiable inverse and give the biggest possible open subset on which F define a diffeomorphism. a). F : R2 → R2 with F(x,y) := (x2 − y2, 2xy) . (In the complex case this is the function z → z2.) b). F : R2 → R2 with F(x,y) := (sin x cosh y,cos x sinh y). (In the complex case this is the function z → sin z.) c). F : R2 → R2 with F(x,y) := (x+ y,x2 + y2) . d). F : R3 → R3 with F(r,ϕ,h) := (r cos ϕ,rsin ϕ,h). (cylindrical coordinates) × 2 2 3 3 e). F : (R+) → R with F(x,y) := (x /y , y /x) . xy f). F : R2 → R2 with F(x,y) := (x2 + y2 ,e ) . 8.2. Show that the following maps F are locally invertible at the given point a, and find the Taylor-expansion F(a) of the inverse map at the point upto order 2. a). F : R3 → R3 with F(x,y,z) := − (x+y+z), xy+xz+yz,−xyz at a =(0, 1, 2) resp. a =(−1, 2, 1). ( Hint : The problem consists in approximating the three zeros of the monic polynomial of degree 3 whose coefficients are close to those of X(X−1)(X−2) resp. -
Carl Gustav Jacob Jacobi
CARL GUSTAV JACOB JACOBI Along with Norwegian Niels Abel, German mathematician Carl Gustav Jacob Jacobi (December 10, 1804 – February 18, 1851) was one of the founders of the theory of elliptic functions, which he described in his 1829 work Fundamenta Nova Theoriae Functionum Ellipticarum (New Foundations of the Theory of Elliptic Functions). His achievements in this area drew praise from Joseph-Louis Lagrange, who had spent some 40 years studying elliptic integrals. Jacobi also investigated number theory, mathematical analysis, geometry and differential equations. His work with determinants, in particular the Hamilton-Jacobi theory, a technique of solving a system of partial differential equations by transforming coordinates, is important in the presentation of dynamics and quantum mechanics. V.I. Arnold wrote of the Hamilton-Jacobi method, “… this is the most powerful method known for exact integration.” Born in Potsdam, Jacobi was the son of a prosperous Jewish banker. His older brother Moritz Hermann Jacobi was a well-known physicist and engineer. The latter worked as a leading researcher at the Academy of Sciences in St. Petersburg. During their lifetimes Moritz was the better known of the two for his work on the practical applications of electricity and especially for his discovery in 1838 of galvanoplastics. Also called electrotyping, it is a process something like electroplating for making duplicate plates of relief, or letterpress, printing. Carl was constantly mistaken for his brother or even worse congratulated for having such a distinguished and accomplished brother. To one such compliment he responded with annoyance, “I am not his brother, he is mine.” Carl Jacobi demonstrated great talent for both languages and mathematics from an early age. -
Mathematical Genealogy of the Union College Department of Mathematics
Gemma (Jemme Reinerszoon) Frisius Mathematical Genealogy of the Union College Department of Mathematics Université Catholique de Louvain 1529, 1536 The Mathematics Genealogy Project is a service of North Dakota State University and the American Mathematical Society. Johannes (Jan van Ostaeyen) Stadius http://www.genealogy.math.ndsu.nodak.edu/ Université Paris IX - Dauphine / Université Catholique de Louvain Justus (Joost Lips) Lipsius Martinus Antonius del Rio Adam Haslmayr Université Catholique de Louvain 1569 Collège de France / Université Catholique de Louvain / Universidad de Salamanca 1572, 1574 Erycius (Henrick van den Putte) Puteanus Jean Baptiste Van Helmont Jacobus Stupaeus Primary Advisor Secondary Advisor Universität zu Köln / Université Catholique de Louvain 1595 Université Catholique de Louvain Erhard Weigel Arnold Geulincx Franciscus de le Boë Sylvius Universität Leipzig 1650 Université Catholique de Louvain / Universiteit Leiden 1646, 1658 Universität Basel 1637 Union College Faculty in Mathematics Otto Mencke Gottfried Wilhelm Leibniz Ehrenfried Walter von Tschirnhaus Key Universität Leipzig 1665, 1666 Universität Altdorf 1666 Universiteit Leiden 1669, 1674 Johann Christoph Wichmannshausen Jacob Bernoulli Christian M. von Wolff Universität Leipzig 1685 Universität Basel 1684 Universität Leipzig 1704 Christian August Hausen Johann Bernoulli Martin Knutzen Marcus Herz Martin-Luther-Universität Halle-Wittenberg 1713 Universität Basel 1694 Leonhard Euler Abraham Gotthelf Kästner Franz Josef Ritter von Gerstner Immanuel Kant -
Mathematical Genealogy of the Wellesley College Department Of
Nilos Kabasilas Mathematical Genealogy of the Wellesley College Department of Mathematics Elissaeus Judaeus Demetrios Kydones The Mathematics Genealogy Project is a service of North Dakota State University and the American Mathematical Society. http://www.genealogy.math.ndsu.nodak.edu/ Georgios Plethon Gemistos Manuel Chrysoloras 1380, 1393 Basilios Bessarion 1436 Mystras Johannes Argyropoulos Guarino da Verona 1444 Università di Padova 1408 Cristoforo Landino Marsilio Ficino Vittorino da Feltre 1462 Università di Firenze 1416 Università di Padova Angelo Poliziano Theodoros Gazes Ognibene (Omnibonus Leonicenus) Bonisoli da Lonigo 1477 Università di Firenze 1433 Constantinople / Università di Mantova Università di Mantova Leo Outers Moses Perez Scipione Fortiguerra Demetrios Chalcocondyles Jacob ben Jehiel Loans Thomas à Kempis Rudolf Agricola Alessandro Sermoneta Gaetano da Thiene Heinrich von Langenstein 1485 Université Catholique de Louvain 1493 Università di Firenze 1452 Mystras / Accademia Romana 1478 Università degli Studi di Ferrara 1363, 1375 Université de Paris Maarten (Martinus Dorpius) van Dorp Girolamo (Hieronymus Aleander) Aleandro François Dubois Jean Tagault Janus Lascaris Matthaeus Adrianus Pelope Johann (Johannes Kapnion) Reuchlin Jan Standonck Alexander Hegius Pietro Roccabonella Nicoletto Vernia Johannes von Gmunden 1504, 1515 Université Catholique de Louvain 1499, 1508 Università di Padova 1516 Université de Paris 1472 Università di Padova 1477, 1481 Universität Basel / Université de Poitiers 1474, 1490 Collège Sainte-Barbe -
Hamiltonian System and Dissipative System
Hamiltonian system and Dissipative system: Motivating example: (energy and DE) y00 + qy = 0 system: y0 = v, v0 = −qy y00 + py0 + qy = 0 system: y0 = v, v0 = −qy − pv 1 1 Energy function: E(y, v) = v2 + qy2 2 2 1 where v2 is the kinetic energy (note that m = 1) 2 1 and qy2 is the potential energy (example: q = g) 2 1 y00 + qy = 0 d d 1 1 E(y(t), v(t)) = v2(t) + qy2(t) = v(t)v0(t) + qy(t)y0(t) dt dt 2 2 = v(t)(−qy(t)) + qy(t) · v(t) = 0 (energy is conserved) y00 + py0 + qy = 0 d d 1 1 E(y(t), v(t)) = v2(t) + qy2(t) = v(t)v0(t) + qy(t)y0(t) dt dt 2 2 = v(t)(−qy(t) − pv(t)) + qy(t) · v(t) = −p[v(t)]2 ≤ 0 (energy is dissipated) 2 Definition: dx dy = f(x, y), = g(x, y). dt dt If there is a function H(x, y) such that for each solution orbit d (x(t), y(t)), we have H(x(t), y(t)) = 0, then the system is a dt Hamiltonian system, and H(x, y) is called conserved quantity. (or energy function, Hamiltonian) If there is a function H(x, y) such that for each solution orbit d (x(t), y(t)), we have H(x(t), y(t)) ≤ 0, then the system is a dt dissipative system, and H(x, y) is called Lyapunov function. (or energy function) 3 Example: If a satellite is circling around the earth, it is a Hamil- tonian system; but if it drops to the earth, it is a dissipative system. -
From Abraham De Moivre to Johann Carl Friedrich Gauss
International Journal of Engineering Science Invention (IJESI) ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726 www.ijesi.org ||Volume 7 Issue 6 Ver V || June 2018 || PP 28-34 A Brief Historical Overview Of the Gaussian Curve: From Abraham De Moivre to Johann Carl Friedrich Gauss Edel Alexandre Silva Pontes1 1Department of Mathematics, Federal Institute of Alagoas, Brazil Abstract : If there were only one law of probability to be known, this would be the Gaussian distribution. Faced with this uneasiness, this article intends to discuss about this distribution associated with its graph called the Gaussian curve. Due to the scarcity of texts in the area and the great demand of students and researchers for more information about this distribution, this article aimed to present a material on the history of the Gaussian curve and its relations. In the eighteenth and nineteenth centuries, there were several mathematicians who developed research on the curve, including Abraham de Moivre, Pierre Simon Laplace, Adrien-Marie Legendre, Francis Galton and Johann Carl Friedrich Gauss. Some researchers refer to the Gaussian curve as the "curve of nature itself" because of its versatility and inherent nature in almost everything we find. Virtually all probability distributions were somehow part or originated from the Gaussian distribution. We believe that the work described, the study of the Gaussian curve, its history and applications, is a valuable contribution to the students and researchers of the different areas of science, due to the lack of more detailed research on the subject. Keywords - History of Mathematics, Distribution of Probabilities, Gaussian Curve. ----------------------------------------------------------------------------------------------------------------------------- --------- Date of Submission: 09-06-2018 Date of acceptance: 25-06-2018 ----------------------------------------------------------------------------------------------------------------------------- ---------- I. -
Carl Friedrich Gauss Seminarski Rad
SREDNJA ŠKOLA AMBROZA HARAČIĆA PODRUČNI ODJEL CRES CARL FRIEDRICH GAUSS SEMINARSKI RAD Cres, 2014. SREDNJA ŠKOLA AMBROZA HARAČIĆA PODRUČNI ODJEL CRES CARL FRIEDRICH GAUSS SEMINARSKI RAD Učenice: Marina Kučica Giulia Muškardin Brigita Novosel Razred: 3.g. Mentorica: prof. Melita Chiole Predmet: Matematika Cres, ožujak 2014. II Sadržaj 1. UVOD .................................................................................................................................... 1 2. DJETINJSTVO I ŠKOLOVANJE ......................................................................................... 2 3. PRIVATNI ŽIVOT ................................................................................................................ 5 4. GAUSSOV RAD ................................................................................................................... 6 4.1. Prvi znanstveni rad .......................................................................................................... 6 4.2. Teorija brojeva ................................................................................................................ 8 4.3. Geodezija ......................................................................................................................... 9 4.4. Fizika ............................................................................................................................. 11 4.5. Astronomija ................................................................................................................... 12 4.6. Religija -
2 a Revolutionary Science
2 A Revolutionary Science When the Parisian crowds stormed the Bastille fortress and prison on 14 July 1789, they set in motion a train of events that revolutionized European political culture. To many contem- porary commentators and observers of the French Revolution, it seemed that the growing disenchantment with the absolutist regime of Louis XVI had been fostered in part by a particular kind of philosophy. French philosophes condemning the iniqui- ties of the ancien regime´ drew parallels between the organization of society and the organization of nature. Like many other En- lightenment thinkers, they took it for granted that science, or natural philosophy,could be used as a tool to understand society as well as nature. They argued that the laws of nature showed how unjust and unnatural the government of France really was. It also seemed, to some at least, that the French Revolution pro- vided an opportunity to galvanize science as well as society. The new French Republic was a tabula rasa on which the reform- ers could write what they liked. They could refound society on philosophical principles, making sure this time around that the organization of society really did mirror the organization of nature. Refounding the social and intellectual structures of sci- ence itself was to be part of this process. In many ways, therefore, the storming of the Bastille led to a revolution in science as well. To many in this new generation of radical French natu- ral philosophers, mathematics seemed to provide the key to 22 A Revolutionary Science 23 understanding nature. This was nothing new in itself, of course. -
Newton's Notebook
Newton’s Notebook The Haverford School’s Math & Applied Math Journal Issue I Spring 2017 The Haverford School Newton’s Notebook Spring 2017 “To explain all nature is too difficult a task for any one man or even for any one age. ‘Tis much better to do a little with certainty & leave the rest for others that come after you.” ~Isaac Newton Table of Contents Pure Mathematics: 7 The Golden Ratio.........................................................................................Robert Chen 8 Fermat’s Last Theorem.........................................................................Michael Fairorth 9 Math in Coding............................................................................................Bram Schork 10 The Pythagoreans.........................................................................................Eusha Hasan 12 Transfinite Numbers.................................................................................Caleb Clothier 15 Sphere Equality................................................................................Matthew Baumholtz 16 Interesting Series.......................................................................................Aditya Sardesi 19 Indirect Proofs..............................................................................................Mr. Patrylak Applied Mathematics: 23 Physics in Finance....................................................................................Caleb Clothier 26 The von Bertalanffy Equation..................................................................Will -
The Astronomical Work of Carl Friedrich Gauss
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector HISTORIA MATHEMATICA 5 (1978), 167-181 THE ASTRONOMICALWORK OF CARL FRIEDRICH GAUSS(17774855) BY ERIC G, FORBES, UNIVERSITY OF EDINBURGH, EDINBURGH EH8 9JY This paper was presented on 3 June 1977 at the Royal Society of Canada's Gauss Symposium at the Ontario Science Centre in Toronto [lj. SUMMARIES Gauss's interest in astronomy dates from his student-days in Gattingen, and was stimulated by his reading of Franz Xavier von Zach's Monatliche Correspondenz... where he first read about Giuseppe Piazzi's discovery of the minor planet Ceres on 1 January 1801. He quickly produced a theory of orbital motion which enabled that faint star-like object to be rediscovered by von Zach and others after it emerged from the rays of the Sun. Von Zach continued to supply him with the observations of contemporary European astronomers from which he was able to improve his theory to such an extent that he could detect the effects of planetary perturbations in distorting the orbit from an elliptical form. To cope with the complexities which these introduced into the calculations of Ceres and more especially the other minor planet Pallas, discovered by Wilhelm Olbers in 1802, Gauss developed a new and more rigorous numerical approach by making use of his mathematical theory of interpolation and his method of least-squares analysis, which was embodied in his famous Theoria motus of 1809. His laborious researches on the theory of Pallas's motion, in whi::h he enlisted the help of several former students, provided the framework of a new mathematical formu- lation of the problem whose solution can now be easily effected thanks to modern computational techniques. -
THE JACOBIAN MATRIX a Thesis Presented to the Department Of
THE JACOBIAN MATRIX A Thesis Presented to the Department of Mathematics The Kansas State Teachers College of Emporia "*' In Partial Fulfillment of the Requirements for the Degree Master of Arts by Sheryl A. -Cole June 1970 aAo.xddv PREFACE The purpose of this paper is to investigate the Jacobian matrix in greater depth than this topic is dealt with in any individual calculus text. The history of Carl Jacobi, the man who discovered the determinant, is reviewed in chapter one. The several illustrations in chapter two demon strate tile mechanics of finding the Jacobian matrix. In chapter three the magnification of area under a transformation is related to ti1e Jacobian. The amount of magnification under coordinate changes is also discussed in this chapter. In chapter four a definition of smooth surface area is arrived at by application of the Jacobian. It is my pleasure to express appreciation to Dr. John Burger for all of his assistance in preparing this paper. An extra special thanks to stan for his help and patience. C.G.J. JACOBI (1804-1851) "Man muss immer umkehren" (Man must always invert.) , ,~' I, "1,1'1 TABLE OF CONTENTS . CHAPTER PAGE I. A HISTORY OF JACOBI • • • • • • • • • •• • 1 II. TIlE JACOBIAN .AND ITS RELATED THEOREMS •• • 8 III. TRANSFORMATIONS OF AREA • • • ••• • • • • 14 IV. SURFACE AREA • • ••• • • • • • • ••• • 23 CONCLUSION •• •• ••• • • • • • •• •• •• • • 28 FOOTNOTES •• •• • • • ••• ••• •• •• • • • 30 BIBLIOGRAPHY • • • • • • • • • • • • ·.. • • • • • 32 CHAPTER I A HISTORY OF JACOBI Carl Gustav Jacob Jacobi was a prominant mathema tician noted chiefly for his pioneering work in the field of elliptic functions. Jacobi was born December 10, 1804, in Potsdam, Prussia. He was the second of four children born to a very prosperous banker. -
Carl Friedrich Gauss Papers
G. Waldo Dunnington, who taught German at Northwestern State University from 1946 until his retirement in 1969, collected these resources over a thirty year period. Dunnington wrote Carl Friedrich Gauss, Titan of Science, the first complete biography on the scientific genius in 1955. Dunnington also wrote an Encyclopedia Britannica article on Gauss. He bequeathed his entire collection to the Cammie Henry Research Center at Northwestern. Titan of Science has recently been republished with additional material by Jeremy Gray and is available at the Mathematical Society of America ISBN: 0883855380 Gauss was appointed director of the University of Göttingen observatory and Professor. Among his other scientific triumphs, Gauss devised a method for the complete determination of the elements of a planet’s orbit from three observations. Gauss and Physicist Wilhelm Weber collaborated in 1833 to produce the electro-magnetic telegraph. They devised an alphabet and could transmit accurate messages of up to eight words a minute. The two men formulated fundamental laws and theories of magnetism. Gauss and his achievements are commemorated in currency, stamps and monuments across Germany. The Research Center holds many examples of these. After his death, a study of Gauss' brain revealed the weight to be 1492 grams with a cerebral area equal to 219,588 square centimeters, a size that could account for his genius Göttingen, the home of Gauss, and site of much of his research. Links to more material on Gauss: Dunnington's Encyclopedia Article Description of Dunnington Collection at the Research Center Gauss-Society, Göttingen Gauss, a Biography Gauß site (German) References for Gauss Nelly Cung's compilation of Gauss material http://www.gausschildren.org This web site gathers together information about the descendants of Carl Friedrich Gauss Contact Information: Cammie G.