1 CHAPTER 2 CONIC SECTIONS 2.1 Introduction a Particle Moving Under the Influence of an Inverse Square Force Moves in an Orbit T
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How Should the College Teach Analytic Geometry?*
1906.] THE TEACHING OF ANALYTIC GEOMETRY. 493 HOW SHOULD THE COLLEGE TEACH ANALYTIC GEOMETRY?* BY PROFESSOR HENRY S. WHITE. IN most American colleges, analytic geometry is an elective study. This fact, and the underlying causes of this fact, explain the lack of uniformity in content or method of teaching this subject in different institutions. Of many widely diver gent types of courses offered, a few are likely to survive, each because it is best fitted for some one purpose. It is here my purpose to advocate for the college of liberal arts a course that shall draw more largely from projective geometry than do most of our recent college text-books. This involves a restriction of the tendency, now quite prevalent, to devote much attention at the outset to miscellaneous graphs of purely statistical nature or of physical significance. As to the subject matter used in a first semester, one may formulate the question : Shall we teach in one semester a few facts about a wide variety of curves, or a wide variety of propositions about conies ? Of these alternatives the latter is to be preferred, for the educational value of a subject is found less in its extension than in its intension ; less in the multiplicity of its parts than in their unification through a few fundamental or climactic princi ples. Some teachers advocate the study of a wider variety of curves, either for the sake of correlation wTith physical sci ences, or in order to emphasize what they term the analytic method. As to the first, there is a very evident danger of dis sipating the student's energy ; while to the second it may be replied that there is no one general method which can be taught, but many particular and ingenious methods for special problems, whose resemblances may be understood only when the problems have been mastered. -
Greece Part 3
Greece Chapters 6 and 7: Archimedes and Apollonius SOME ANCIENT GREEK DISTINCTIONS Arithmetic Versus Logistic • Arithmetic referred to what we now call number theory –the study of properties of whole numbers, divisibility, primality, and such characteristics as perfect, amicable, abundant, and so forth. This use of the word lives on in the term higher arithmetic. • Logistic referred to what we now call arithmetic, that is, computation with whole numbers. Number Versus Magnitude • Numbers are discrete, cannot be broken down indefinitely because you eventually came to a “1.” In this sense, any two numbers are commensurable because they could both be measured with a 1, if nothing bigger worked. • Magnitudes are continuous, and can be broken down indefinitely. You can always bisect a line segment, for example. Thus two magnitudes didn’t necessarily have to be commensurable (although of course they could be.) Analysis Versus Synthesis • Synthesis refers to putting parts together to obtain a whole. • It is also used to describe the process of reasoning from the general to the particular, as in putting together axioms and theorems to prove a particular proposition. • Proofs of the kind Euclid wrote are referred to as synthetic. Analysis Versus Synthesis • Analysis refers to taking things apart to see how they work, or so you can understand them. • It is also used to describe reasoning from the particular to the general, as in studying a particular problem to come up with a solution. • This is one general meaning of analysis: a way of solving problems, of finding the answers. Analysis Versus Synthesis • A second meaning for analysis is specific to logic and theorem proving: beginning with what you wish to prove, and reasoning from that point in hopes you can arrive at the hypotheses, and then reversing the logical steps. -
On the Constructibility of the Axes of an Ellipsoid
ON THE CONSTRUCTIBILITY OF THE AXES OF AN ELLIPSOID. THE CONSTRUCTION OF CHASLES IN PRACTICE AKOS´ G.HORVATH´ AND ISTVAN´ PROK Dedicated to the memory of the teaching of Descriptive Geometry in BME Faculty of Mechanical Engineering Abstract. In this paper we discuss Chasles’s construction on ellipsoid to draw the semi-axes from a complete system of conjugate diameters. We prove that there is such situation when the construction is not planar (the needed points cannot be constructed with compasses and ruler) and give some others in which the construction is planar. 1. Introduction Who interested in conics know the construction of Rytz, namely a construction which determines the axes of an ellipse from a pair of conjugate diameters. Contrary to this very few mathematicians know that in the first half of the nineteens century Chasles (see in [1]) gave a construction in space to the analogous problem. The only reference which we found in English is also in an old book written by Salmon (see in [6]) on the analytic geometry of the three-dimensional space. In [2] the author extracted this analytic method to prove some results on the quadric of n-dimensional space. However, in practice this construction cannot be done using compasses and ruler only as we will see in present paper (Statement 4). On the other hand those steps of the construction which are planar constructions we can draw by descriptive geometry (see the figures in this article). 2. Basic properties The proof of the statements of this section can be found in [2]. -
11.-HYPERBOLA-THEORY.Pdf
12. HYPERBOLA 1. INTRODUCTION A hyperbola is the locus of a point which moves in the plane in such a way that Z the ratio of its distance from a fixed point in the same plane to its distance X’ P from a fixed line is always constant which is always greater than unity. M The fixed point is called the focus, the fixed line is called the directrix. The constant ratio is generally denoted by e and is known as the eccentricity of the Directrix hyperbola. A hyperbola can also be defined as the locus of a point such that S (focus) the absolute value of the difference of the distances from the two fixed points Z’ (foci) is constant. If S is the focus, ZZ′ is the directrix and P is any point on the hyperbola as show in figure. Figure 12.1 SP Then by definition, we have = e (e > 1). PM Note: The general equation of a conic can be taken as ax22+ 2hxy + by + 2gx + 2fy += c 0 This equation represents a hyperbola if it is non-degenerate (i.e. eq. cannot be written into two linear factors) ahg ∆ ≠ 0, h2 > ab. Where ∆=hb f gfc MASTERJEE CONCEPTS 1. The general equation ax22+ 2hxy + by + 2gx + 2fy += c 0 can be written in matrix form as ahgx ah x x y + 2gx + 2fy += c 0 and xy1hb f y = 0 hb y gfc1 Degeneracy condition depends on the determinant of the 3x3 matrix and the type of conic depends on the determinant of the 2x2 matrix. -
Automated Study of Isoptic Curves: a New Approach with Geogebra∗
Automated study of isoptic curves: a new approach with GeoGebra∗ Thierry Dana-Picard1 and Zoltan Kov´acs2 1 Jerusalem College of Technology, Israel [email protected] 2 Private University College of Education of the Diocese of Linz, Austria [email protected] March 7, 2018 Let C be a plane curve. For a given angle θ such that 0 ≤ θ ≤ 180◦, a θ-isoptic curve of C is the geometric locus of points in the plane through which pass a pair of tangents making an angle of θ.If θ = 90◦, the isoptic curve is called the orthoptic curve of C. For example, the orthoptic curves of conic sections are respectively the directrix of a parabola, the director circle of an ellipse, and the director circle of a hyperbola (in this case, its existence depends on the eccentricity of the hyperbola). Orthoptics and θ-isoptics can be studied for other curves than conics, in particular for closed smooth strictly convex curves,as in [1]. An example of the study of isoptics of a non-convex non-smooth curve, namely the isoptics of an astroid are studied in [2] and of Fermat curves in [3]. The Isoptics of an astroid present special properties: • There exist points through which pass 3 tangents to C, and two of them are perpendicular. • The isoptic curve are actually union of disjoint arcs. In Figure 1, the loops correspond to θ = 120◦ and the arcs connecting them correspond to θ = 60◦. Such a situation has been encountered already for isoptics of hyperbolas. These works combine geometrical experimentation with a Dynamical Geometry System (DGS) GeoGebra and algebraic computations with a Computer Algebra System (CAS). -
HYPERBOLA HYPERBOLA a Conic Is Said to Be a Hyperbola If Its Eccentricity Is Greater Than One
HYPERBOLA HYPERBOLA A Conic is said to be a hyperbola if its eccentricity is greater than one. If S ax22 hxy by 2 2 gx 2 fy c 0 represents a hyperbola then h2 ab 0 and 0 2 2 If S ax2 hxy by 2 gx 2 fy c 0 represents a rectangular hyperbola then h2 ab 0 , 0 and a b 0 x2 y 2 Hyperbola: Equation of the hyperbola in standard form is 1. a2 b2 Hyperbola is symmetric about both the axes. Hyperbola does not passing through the origin and does not meet the Y - axis. Hyperbola curve does not exist between the lines x a and x a . x2 y 2 xx yy x2 y 2 x x y y Notation: S 1 , S1 1 1, S1 1 1, S1 2 1 2 1. a2 b 2 1 a2 b 2 11 a2 b 2 12 a2 b 2 Rectangular hyperbola (or) Equilateral hyperbola : In a hyperbola if the length of the transverse axis (2a) is equal to the length of the conjugate axis (2b) , then the hyperbola is called a rectangular hyperbola or an equilateral hyperbola. The eccentricity of a rectangular hyperbola is 2 . Conjugate hyperbola : The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called the conjugate hyperbola of the given hyperbola. 2 2 1 x y The equation of the hyperbola conjugate to S 0 is S 1 0 . a2 b 2 If e1 and e2 be the eccentricities of the hyperbola and its conjugate hyperbola then 1 1 2 2 1. -
Attoclock Ultrafast Timing of Inversive Chord-Contact Dynamics
Attoclock Ultrafast Timing of Inversive Chord-Contact Dynamics - A Spectroscopic FFT-Glimpse onto Harmonic Analysis of Gravitational Waves - Walter J. Schempp Lehrstuhl f¨urMathematik I Universit¨atSiegen 57068 Siegen, Germany [email protected] According to Bohr's model of the hydrogen atom, it takes about 150 as for an electron in the ground state to orbit around the atomic core. Therefore, the attosecond (1 as = 10−18s) is the typical time scale for Bloch wave packet dynamics operating at the interface of quantum field theory and relativity theory. Attoclock spectroscopy is mathematically dominated by the spectral dual pair of spin groups with the commuting action of the stabilized symplectic spinor quantization (Mp(2; R); PSO(1; 3; R)) In the context of the Mp(2; R) × PSO(1; 3; R) module structure of the complex Schwartz space 2 2 ∼ S(R ⊕ R ) = S(C ⊕ C) of quantum field theory, the metaplectic Lie group Mp(2; R) is the ∼ two-sheeted covering group Sp(2f ; R) of the symplectic group Sp(2; R) = SL(2; R), where the ∼ special linear group SL(2; R) is a simple Lie group, and the two-cycle rotation group SO(2f ; R) = U(1e ; C) is a maximal compact subgroup of Mp(2; R). The two-fold hyperbolically, respectively ∼ elliptically ruled conformal Lie group SU(1; 1; C) is conjugate to SL(2; R) = Spin(1; 2; R) in the ∼ ∼ complexification SL(2; C) = Spin(1; 3; R); the projective Lorentz-M¨obiusgroup PSO(1; 3; R) = ∼ PSL(2; C) = SL(2; C)=(Z2:id) represents a semi-simple Lie group which is isomorphic to the "+ four-dimensional proper orthochroneous Lorentz group L4 of relativity theory. -
Department of Mathematics Scheme of Studies B.Sc
BHUPAL NOBLES` UNIVERSITY, UDAIPUR FACULTY OF SCIENCE Department Of Mathematics Scheme of Studies B.Sc. I Year (Annual Scheme) S. No. PAPER NOMENCLATURE COURSE UNIVERSITY INTERNAL MAX. CODE EXAM ASSESSMENT MARKS 1. Paper I Algebra MAT-111 53 22 75 2. Paper II Calculus MAT- 112 53 22 75 3. Paper III Geometry MAT- 113 53 22 75 The marks distribution of internal assessment- 1. Mid Term Examination – 15 marks 2. Attendance – 7 marks B.Sc. II Year (Annual Scheme) S. No. PAPER NOMENCLATURE COURSE UNIVERSITY INTERNAL MAX. CODE EXAM ASSESSMENT MARKS 1. Paper I Advanced Calculus MAT-221 53 22 75 2. Paper II Differential Equation MAT- 222 53 22 75 3. Paper III Mechanics MAT- 223 53 22 75 The marks distribution of internal assessment- 1. Mid Term Examination – 15 marks 2. Attendance – 7 marks B.Sc. III Year (Annual Scheme) S. PAPER NOMENCLATURE COURSE CODE UNIVERSITY INTERNAL MAX. No. EXAM ASSESSMENT MARKS 1. Paper I Real Analysis MAT-331 53 22 75 2. Paper II Advanced Algebra MAT- 332 53 22 75 3. Paper Numerical Analysis MAT-333 (A) 53 22 75 III Mathematical MAT-333 (B) Quantitative Techniques Mathematical MAT-333 (C) Statistics The marks distribution of internal assessment- 1. Mid Term Examination – 15 marks 2. Attendance – 7 marks BHUPAL NOBLES’ UNIVERSITY, UDAIPUR Department of Mathematics Syllabus 2017 -2018 (COMMON FOR THE FACULTIES OF ARTS & SCIENCE) B.A. / B. Sc. FIRST YEAR EXAMINATIONS 2017-2020 MATHEMATICS Theory Papers Name Papers Code Papers Maximum Papers hours/ Marks BA/ week B.Sc. Paper I ALGEBRA MAT-111 3 75 Paper II CALCULUS MAT-112 3 75 Paper III GEOMETRY MAT-113 3 75 Total Marks 225 NOTE: 1. -
Syllabus for Bachelor of Science I-Year (Semester-I) Session 2018
Syllabus for Bachelor of Science I-Year (Semester-I) Session 2018-2019 Code Description Pd/W Exam CIA ESE Total BSMT111 Algebra 3 3 hrs 20 80 100 BSMT112 Differential Calculus 3 3 hrs 20 80 100 BSMT113 Co-ordinate Geometry of Two and 3 3 hrs 20 80 100 Three Dimensions Total 60 240 300 BSMT111: ALGEBRA Matrix: The characteristic equation of matrix: Eigen Values and Eigen Unit-I Vectors, Diagonalization of matrix, Cayley-Hamilton theorem(Statement 9 and proof),and its use in finding the inverse of a Matrix. Theory of equation: Relation between roots and coefficient of the equation Unit-II Symmetric function of roots. Solution of cubic equation by Cordon’s 9 method and Biquadratic equitation by Ferrari’s method. Infinite series: Convergent series, convergence of geometric series, And Unit-III necessary condition for the convergent series, comparison tests: Cauchy 9 root test. D’Alembert’s Ratio test, Logarithmic test, Raabe’s test, De’ Morgan and Unit-IV Bertrand’s test, Cauchy’s condensation test, Leibnitz’s test of alternative 9 series, Absolute convergent. Unit-V Inequalities and Continued fractions. 9 Suggested Readings: 1. Bhargav and Agarwal:Algebra, Jaipur publishing House, Jaipur. 2. Vashistha and Vashistha : Modern Algebra, Krishna Prakashan, Meerut. 3. Gokhroo, Saini and Tak: Algebra,Navkar prakashan, Ajmer.. 4. M-Ray and H. S. Sharma: A Text book of Higher Algebra, New Delhi, BSMT112: Differential Calculus Unit-I Polar Co-ordinates, Angle between radius vector and the tangents. Angle 9 between curves in polar form, length of polar subs tangent and subnormal, pedal equation of a curve, derivatives of an arc. -
A Survey of the Development of Geometry up to 1870
A Survey of the Development of Geometry up to 1870∗ Eldar Straume Department of mathematical sciences Norwegian University of Science and Technology (NTNU) N-9471 Trondheim, Norway September 4, 2014 Abstract This is an expository treatise on the development of the classical ge- ometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of non-Euclidean geometry, about 40 years earlier, had just been demonstrated to be a ”true” geometry on the same footing as Euclidean geometry. These were radically new ideas, but henceforth the importance of the topic became gradually realized. As a consequence, the conventional attitude to the basic geometric questions, including the possible geometric structure of the physical space, was challenged, and foundational problems became an important issue during the following decades. Such a basic understanding of the status of geometry around 1870 enables one to study the geometric works of Sophus Lie and Felix Klein at the beginning of their career in the appropriate historical perspective. arXiv:1409.1140v1 [math.HO] 3 Sep 2014 Contents 1 Euclideangeometry,thesourceofallgeometries 3 1.1 Earlygeometryandtheroleoftherealnumbers . 4 1.1.1 Geometric algebra, constructivism, and the real numbers 7 1.1.2 Thedownfalloftheancientgeometry . 8 ∗This monograph was written up in 2008-2009, as a preparation to the further study of the early geometrical works of Sophus Lie and Felix Klein at the beginning of their career around 1870. The author apologizes for possible historiographic shortcomings, errors, and perhaps lack of updated information on certain topics from the history of mathematics. -
The Project Gutenberg Ebook #43006: Space–Time–Matter
The Project Gutenberg EBook of Space--Time--Matter, by Hermann Weyl This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Space--Time--Matter Author: Hermann Weyl Translator: Henry L. Brose Release Date: June 21, 2013 [EBook #43006] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK SPACE--TIME--MATTER *** Produced by Andrew D. Hwang, using scanned images and OCR text generously provided by the University of Toronto Gerstein Library through the Internet Archive. transcriber’s note The camera-quality files for this public-domain ebook may be downloaded gratis at www.gutenberg.org/ebooks/43006. This ebook was produced using scanned images and OCR text generously provided by the University of Toronto Gerstein Library through the Internet Archive. Typographical corrections, uniformization of punctuation, and minor presentational changes have been effected without comment. This PDF file is optimized for screen viewing, but may be recompiled for printing. Please consult the preamble of the LATEX source file for instructions and other particulars. SPACE—TIME—MATTER BY HERMANN WEYL TRANSLATED FROM THE GERMAN BY HENRY L. BROSE WITH FIFTEEN DIAGRAMS METHUEN & CO. LTD. 36 ESSEX STREET W.C. LONDON First Published in 1922 FROM THE AUTHOR’S PREFACE TO THE FIRST EDITION Einstein’s Theory of Relativity has advanced our ideas of the structure of the cosmos a step further. -
Group B Higher Algebra & Trigonometry
UNIVERSITY DEPARTMENT OF MATHEMATICS, DSPMU, RANCHI CBCS PATTERN SYLLABUS Semester Mat/ Sem VC1- Analytical Geometry 2D, Trigonometry Instruction: Ten questions will be set. Candidates will be required to answer Seven Questions. Question no. 1 will be Compulsory consisting of 10 short answer type covering entire syllabus uniformly. Each question will be of 2 marks. Out of remaining 9 questions candidates will be required to answer 6 questions selecting at least one from each group. Each question will be of 10 marks. GROUP-A ANALYTICAL GEOMETRY OF TWO DIMENSIONS Change of rectangular axes. Condition for the general equation of second degree to represent parabola, ellipse, hyperbola and reduction into standard forms. Equations of tangent and normal (Using Calculus). Chord of contact, Pole and Polar. Pair of tangents in reference to general equation of conic. Axes, centre, director circle in reference to general equation of conic. Polar equation of conic. 5 Questions GROUP B HIGHER ALGEBRA & TRIGONOMETRY Statement and proof of binomial theorem for any index, exponential and logarithmic series. 1 Question De Moivre's theorem and its applications. Trigonometric and Exponential functions of complex argument and hyperbolic functions. Summation of Trigonometrical series. Factorisation of sin , cos 3 Questions Books Recommended: 1. Analytical Geometry& Vector Analysis - B. K. Kar, Books& Allied Co., Kolkata 2. Analytical Geometry of two dimension - Askwith 3. Coordinate Geometry- SL Loney. 4. Trigonometry- Das and Mukheriee 5. Trigonometry- Dasgupta UNIVERSITY DEPARTMENT OF MATHEMATICS, DSPMU, RANCHI CBCS PATTERN SYLLABUS Semester Mat/ Sem IVC 2- Differential Calculus and Vector Calculus Instruction: Ten questions will be set. Candidates will be required to answer Seven Questions.