DOCSLIB.ORG

Solving Differential Equations on Manifolds

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Solving Differential Equations on Manifolds Solving Differential Equations on Manifolds Ernst Hairer Universit´ede Gen`eve June 2011 Section de math´ematiques 2-4 rue du Li`evre,CP 64 CH-1211 Gen`eve 4 Acknowledgement. These notes have been distributed during the lecture “Equations´ diff´erentielles sur des sous-vari´et´es”(2 hours per week) given in the spring term 2011. At several places, the text and figures are taken from one of the monographs by the author. Contents I Introduction by Examples . 1 I.1 Differential equation on a sphere – the rigid body . 1 I.2 Problems in control theory . 3 I.3 Constrained mechanical systems . 4 I.4 Exercises . 6 II Submanifolds of Rn ................................ 7 II.1 Definition and characterization of submanifolds . 7 II.2 Tangent space . 9 II.3 Differential equations on submanifolds . 11 II.4 Differential equations on Lie groups . 14 II.5 Exercises . 16 III Integrators on Manifolds . 19 III.1 Projection methods . 19 III.2 Numerical methods based on local coordinates . 21 III.3 Derivative of the exponential and its inverse . 23 III.4 Methods based on the Magnus series expansion . 24 III.5 Convergence of methods on submanifolds . 26 III.6 Exercises . 27 IV Differential-Algebraic Equations . 29 IV.1 Linear equations with constant coefficients . 29 IV.2 Differentiation index . 30 IV.3 Control problems . 32 IV.4 Mechanical systems . 34 IV.5 Exercises . 36 V Numerical Methods for DAEs . 39 V.1 Runge–Kutta and multistep methods . 39 V.2 Index 1 problems . 42 V.3 Index 2 problems . 45 V.4 Constrained mechanical systems . 47 V.5 Shake and Rattle . 50 V.6 Exercises . 51 3 Recommended Literature There are many monographs treating manifolds and submanifolds. Many of them can be found under the numbers 53 and 57 in the mathematics library. Books specially devoted to the numerical treatment of differential equations on manifolds (differential-algebraic equa- tions) are listed under number 65 in the library. We give here an incomplete list for further reading: the numbers in brackets (e.g. [MA 65/403]) allow one to find the book without computer search. R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition, Applied Mathematical Sciences 75, Springer-Verlag, 1988. [MA 57/266] V. Arnold, Equations Diff´erentielles Ordinaires, Editions Mir (traduction fran¸caise),Moscou, 1974. [MA 34/102] U.M. Ascher and L.R. Petzold, Computer Methods for Ordinary Differential Equations and Differential- Algebraic Equations, Society for Industrial and Applied Mathematics (SIAM), Philadel- phia, PA, 1998. K.E. Brenan, S.L. Campbell and L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Revised and corrected reprint of the 1989 original, Classics in Applied Mathematics 14, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. [MA 65/294] E. Eich-Soellner and C. F¨uhrer, Numerical methods in multibody dynamics, European Consortium for Mathematics in Industry. B.G. Teubner, Stuttgart, 1998. E. Griepentrog and R. M¨arz, Differential-Algebraic Equations and Their Numerical Treatment, Teubner-Texte zur Mathematik 88, Teubner Verlagsgesellschaft, Leipzig, 1986. [MA 65/256] E. Hairer, C. Lubich and M. Roche, The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Mathematics 1409, Springer Berlin, 1989. [MA 00.04/3 1409] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Al- gorithms for Ordinary Differential Equations, 2nd edition, Springer Series in Compu- tational Mathematics 31, Springer Berlin, 2006. [MA 65/448] E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Prob- lems, 2nd edition, Springer Series in Computational Mathematics 8, Springer Berlin, 1993. [MA 65/245] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Agebraic Problems, 2nd edition, Springer Series in Computational Mathematics 14, Springer Berlin, 1996. [MA 65/245] E. Hairer and G. Wanner, Analysis by Its History, Undergraduate Texts in Mathematics, Springer New York, 1995. [MA 27/256] P. Kunkel and V. Mehrmann, Differential-Algebraic Equations. Analysis and Numerical Solution, EMS Textbooks in Mathematics. European Mathematical Society (EMS), Z¨urich, 2006. [MA 34/325] S. Lang, Introduction to Differentiable Manifolds, 2nd edition, Universitext, Springer New York, 2002. [MA 57/15] J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer New York, 2003. [MA 53/302] Chapter I Introduction by Examples Systems of ordinary differential equations in the Euclidean space Rn are given by y˙ = f(y), (0.1) where f : U → Rn with an open set U ⊂ Rn. If f is sufficiently smooth and an initial value n y(0) = y0 is prescribed, it is known that the problem has a unique solution y :(−α, α) → R for some α > 0. This solution can be extended until it approaches the border of U. In the present lecture we are interested in differential equations, where the solution is known to evolve on a submanifold of Rn, and the vector field f(y) is often only defined on this submanifold. We start with presenting a few typical examples, which serve as motivation for the topic. Later, we shall give a precise definition of differential equations on submanifolds, we shall discuss their numerical treatment, and we shall analyze them rigorously. I.1 Differential equation on a sphere – the rigid body Let I1,I2,I3 be the principal moments of inertia of a rigid body. The angular momentum T vector y = (y1, y2, y3) then satisfies Euler’s equations of motion −1 −1 y˙1 = (I3 − I2 ) y3 y2 y˙1 0 −y3 y2 y1/I1 −1 −1 y˙2 = (I − I ) y1 y3 or y˙2 = y3 0 −y1 y2/I2 . (1.1) 1 3 −1 −1 y˙3 = (I2 − I1 ) y2 y1 y˙3 −y2 y1 0 y3/I3 Fig. I.1: Euler’s equation of motion for I1 = 1.6, I2 = 1, I3 = 2/3; left picture: vector field on the sphere; right picture: some solution curves. 2 Introduction by Examples This differential equation has the property that the function 1 C(y) = y2 + y2 + y2 (1.2) 2 1 2 3 is exactly preserved along solutions, a property that can be checked by differentiation: d dt C y(t) = ... = 0. As a consequence, the solution remains forever on the sphere with radius that is determined by the initial values. The left picture of Figure I.1 shows the vector f(y) attached to selected points y of the unit sphere. To study further properties of the solution we write the differential equation as 0 −y y 3 2 2 2 2 1 y1 y2 y3 y˙ = B(y)∇H(y) with B(y) = y3 0 −y1 ,H(y) = + + . 2 I1 I2 I3 −y2 y1 0 The function H(y) is called Hamiltonian of the system, whereas C(y) of (1.2) is called d Casimir function. Exploiting the skew-symmetry of the matrix B(y), we obtain dt H(y(t)) = ∇H(y(t))TB(y)∇H(y(t)) = 0, which implies the preservation of the Hamiltonian H(y) along solutions of (1.1). Consequently, solutions lie on the intersection of a sphere C(y) = Const with an ellipsoid H(y) = Const, and give rise to the closed curves of the right picture in Figure I.1. Solutions are therefore typically periodic. Numerical solutions are displayed in Figure I.2. The top picture shows the numerical result, when the explicit Euler method yn+1 = yn + hf(yn) is applied with step size h = 0.025 and with the initial value y0 = (cos(0.9), 0, sin(0.9)). The numerical solution drifts away from the manifold. The bottom left picture shows the result of the trapezoidal rule h yn+1 = yn + 2 (f(yn+1) + f(yn)) with h = 1, where the numerical solution is orthogonally Fig. I.2: Top picture: integration with explicit Euler; bottom left picture: trapezoidal rule with projection onto the sphere; bottom right picture: implicit midpoint rule. Introduction by Examples 3 projected onto the sphere after every step. The bottom right picture considers the implicit 1 mid-point rule yn+1 = yn + hf( 2 (yn+1 + yn)) with h = 1. Even without any projection, the solution agrees extremely well with the exact solution. All these behaviours will be explained in later chapters. I.2 Problems in control theory In control theory one often encounters problems of the form y˙ = f(y, u) (2.1) 0 = g(y), where u(t) is a control function that permits to steer the motion y(t) of a mechanical system. Differentiating the algebraic equation g(y(t)) = 0 with respect to time yields g0(y)f(y, u) = 0. Under suitable regularity assumptions this relation permits us to express u as a function of y (using the implicit function theorem). Inserting u = G(y) into (2.1) gives a differential equation for y on the manifold M = {y ; g(y) = 0}. θ2 θ1 θ4 θ3 Fig. I.3: Sketch of an articulated robot arm. Example 2.1 (articulated robot arm). Consider n > 2 segments of fixed length 1 that are connected with joints as illustrated in Figure I.3. We assume that the starting point of the first segment is fixed at the origin. Denoting by θj the angle of the jth segment with respect to the horizontal axis, the endpoint of the last segment is given by g(θ), where for θ = (θ1, . , θn) we have cos θ + cos θ + ... + cos θ ! g(θ) = 1 2 n . (2.2) sin θ1 + sin θ2 + ... + sin θn The problem consists in finding the motion θ(t) of the articulated robot arm such that the endpoint of the last segment follows a given parametrized curve γ(t) in the plane and kθ˙(t)k → min subject to g(θ(t)) = γ(t).
Load more

Recommended publications
  • Chapter 11. Three Dimensional Analytic Geometry and Vectors
    Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1.
    [Show full text]
  • A Quick Algebra Review
    A Quick Algebra Review 1. Simplifying Expressions 2. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Exponents 9. Quadratics 10. Rationals 11. Radicals Simplifying Expressions An expression is a mathematical “phrase.” Expressions contain numbers and variables, but not an equal sign. An equation has an “equal” sign. For example: Expression: Equation: 5 + 3 5 + 3 = 8 x + 3 x + 3 = 8 (x + 4)(x – 2) (x + 4)(x – 2) = 10 x² + 5x + 6 x² + 5x + 6 = 0 x – 8 x – 8 > 3 When we simplify an expression, we work until there are as few terms as possible. This process makes the expression easier to use, (that’s why it’s called “simplify”). The first thing we want to do when simplifying an expression is to combine like terms. For example: There are many terms to look at! Let’s start with x². There Simplify: are no other terms with x² in them, so we move on. 10x x² + 10x – 6 – 5x + 4 and 5x are like terms, so we add their coefficients = x² + 5x – 6 + 4 together. 10 + (-5) = 5, so we write 5x. -6 and 4 are also = x² + 5x – 2 like terms, so we can combine them to get -2. Isn’t the simplified expression much nicer? Now you try: x² + 5x + 3x² + x³ - 5 + 3 [You should get x³ + 4x² + 5x – 2] Order of Operations PEMDAS – Please Excuse My Dear Aunt Sally, remember that from Algebra class? It tells the order in which we can complete operations when solving an equation.
    [Show full text]
  • Writing the Equation of a Line
    Name______________________________ Writing the Equation of a Line When you find the equation of a line it will be because you are trying to draw scientific information from it. In math, you write equations like y = 5x + 2 This equation is useless to us. You will never graph y vs. x. You will be graphing actual data like velocity vs. time. Your equation should therefore be written as v = (5 m/s2) t + 2 m/s. See the difference? You need to use proper variables and units in order to compare it to theory or make actual conclusions about physical principles. The second equation tells me that when the data collection began (t = 0), the velocity of the object was 2 m/s. It also tells me that the velocity was changing at 5 m/s every second (m/s/s = m/s2). Let’s practice this a little, shall we? Force vs. mass F (N) y = 6.4x + 0.3 m (kg) You’ve just done a lab to see how much force was necessary to get a mass moving along a rough surface. Excel spat out the graph above. You labeled each axis with the variable and units (well done!). You titled the graph starting with the variable on the y-axis (nice job!). Now we turn to the equation. First we replace x and y with the variables we actually graphed: F = 6.4m + 0.3 Then we add units to our slope and intercept. The slope is found by dividing the rise by the run so the units will be the units from the y-axis divided by the units from the x-axis.
    [Show full text]
  • The Geometry of the Phase Diffusion Equation
    J. Nonlinear Sci. Vol. 10: pp. 223–274 (2000) DOI: 10.1007/s003329910010 © 2000 Springer-Verlag New York Inc. The Geometry of the Phase Diffusion Equation N. M. Ercolani,1 R. Indik,1 A. C. Newell,1,2 and T. Passot3 1 Department of Mathematics, University of Arizona, Tucson, AZ 85719, USA 2 Mathematical Institute, University of Warwick, Coventry CV4 7AL, UK 3 CNRS UMR 6529, Observatoire de la Cˆote d’Azur, 06304 Nice Cedex 4, France Received on October 30, 1998; final revision received July 6, 1999 Communicated by Robert Kohn E E Summary. The Cross-Newell phase diffusion equation, (|k|)2T =∇(B(|k|) kE), kE =∇2, and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order “self-dual” equation. Moreover, the self-dual solutions give upper bounds for the free energy which imply the existence of weak limits for the asymptotic minimizers. In certain cases, some recent results of Jin and Kohn [28] combined with these upper bounds enable us to demonstrate that the energy of the asymptotic minimizers converges to that of the self-dual solutions in a viscosity limit. 1. Introduction The mathematical models discussed in this paper are motivated by physical systems, far from equilibrium, which spontaneously form patterns.
    [Show full text]
  • Numerical Algebraic Geometry and Algebraic Kinematics
    Numerical Algebraic Geometry and Algebraic Kinematics Charles W. Wampler∗ Andrew J. Sommese† January 14, 2011 Abstract In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work. Contents 1 Introduction 4 2 Notation 5 I Fundamentals of Algebraic Kinematics 6 3 Some Motivating Examples 6 3.1 Serial-LinkRobots ............................... ..... 6 3.1.1 Planar3Rrobot ................................. 6 3.1.2 Spatial6Rrobot ................................ 11 3.2 Four-BarLinkages ................................ 13 3.3 PlatformRobots .................................. 18 ∗General Motors Research and Development, Mail Code 480-106-359, 30500 Mound Road, Warren, MI 48090- 9055, U.S.A. Email: [email protected] URL: www.nd.edu/˜cwample1. This material is based upon work supported by the National Science Foundation under Grant DMS-0712910 and by General Motors Research and Development. †Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618, U.S.A. Email: [email protected] URL: www.nd.edu/˜sommese. This material is based upon work supported by the National Science Foundation under Grant DMS-0712910 and the Duncan Chair of the University of Notre Dame.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Chapter 1: Analytic Geometry
    1 Analytic Geometry Much of the mathematics in this chapter will be review for you. However, the examples will be oriented toward applications and so will take some thought. In the (x,y) coordinate system we normally write the x-axis horizontally, with positive numbers to the right of the origin, and the y-axis vertically, with positive numbers above the origin. That is, unless stated otherwise, we take “rightward” to be the positive x- direction and “upward” to be the positive y-direction. In a purely mathematical situation, we normally choose the same scale for the x- and y-axes. For example, the line joining the origin to the point (a,a) makes an angle of 45◦ with the x-axis (and also with the y-axis). In applications, often letters other than x and y are used, and often different scales are chosen in the horizontal and vertical directions. For example, suppose you drop something from a window, and you want to study how its height above the ground changes from second to second. It is natural to let the letter t denote the time (the number of seconds since the object was released) and to let the letter h denote the height. For each t (say, at one-second intervals) you have a corresponding height h. This information can be tabulated, and then plotted on the (t, h) coordinate plane, as shown in figure 1.0.1. We use the word “quadrant” for each of the four regions into which the plane is divided by the axes: the first quadrant is where points have both coordinates positive, or the “northeast” portion of the plot, and the second, third, and fourth quadrants are counted off counterclockwise, so the second quadrant is the northwest, the third is the southwest, and the fourth is the southeast.
    [Show full text]
  • The Evolution of Equation-Solving: Linear, Quadratic, and Cubic
    California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2006 The evolution of equation-solving: Linear, quadratic, and cubic Annabelle Louise Porter Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Porter, Annabelle Louise, "The evolution of equation-solving: Linear, quadratic, and cubic" (2006). Theses Digitization Project. 3069. https://scholarworks.lib.csusb.edu/etd-project/3069 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. THE EVOLUTION OF EQUATION-SOLVING LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degre Master of Arts in Teaching: Mathematics by Annabelle Louise Porter June 2006 THE EVOLUTION OF EQUATION-SOLVING: LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino by Annabelle Louise Porter June 2006 Approved by: Shawnee McMurran, Committee Chair Date Laura Wallace, Committee Member , (Committee Member Peter Williams, Chair Davida Fischman Department of Mathematics MAT Coordinator Department of Mathematics ABSTRACT Algebra and algebraic thinking have been cornerstones of problem solving in many different cultures over time. Since ancient times, algebra has been used and developed in cultures around the world, and has undergone quite a bit of transformation. This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work.
    [Show full text]
  • Geometry Formula Sheet ─ Page 1
    KEYSTONE RefEFERENCE GEOMETRY FORMULA SHEET ─ PAGE 1 Formulas that you may need to solve questions on this exam are found below. You may use calculator π or the number 3.14. Properties of Circles Right Triangle Formulas Angle measure is represented by x. Arc measure is represented Pythagorean Theorem: by m and n. Lengths are given by a, b, c, and d. c If a right triangle has legs with a measures a and b and hypotenuse with measure c, then... n° Inscribed Angle b 2 + 2 = 2 x° 1 a b c x = n 2 Trigonometric Ratios: opposite sin θ = x° Tangent-Chord hypotenuse n° = 1 x n hypotenuse adjacent 2 opposite cos θ = hypotenuse θ adjacent opposite tan θ = 2 Chords adjacent d a · = · m°x° n° a b c d c 1 b x = (m + n) 2 Coordinate Geometry Properties a x° Tangent-Secant = – 2 + – 2 b Distance Formula: d (x2 x 1) (y2 y 1) n° 2 = + a b (b c) m° x + x y + y 1 1 2 , 1 2 x = (m − n) Midpoint: c 2 2 2 y − y = 2 1 Slope: m − x 2 x 1 a 2 Secants Point-Slope Formula: (y − y ) = m(x − x ) b + = + 1 1 b (a b) d (c d ) m° n° x° 1 c d x = (m − n) Slope Intercept Formula: y = mx + b 2 Standard Equation of a Line: Ax + By = C a 2 Tangents a = b m° n° x° 1 x = (m − n) b 2 Copyright © 2011 by the Pennsylvania Department of Education. The materials contained in this publication may be duplicated by Pennsylvania educators for local classroom use.
    [Show full text]
  • Analytic Geometry
    INTRODUCTION TO ANALYTIC GEOMETRY BY PEECEY R SMITH, PH.D. N PROFESSOR OF MATHEMATICS IN THE SHEFFIELD SCIENTIFIC SCHOOL YALE UNIVERSITY AND AKTHUB SULLIVAN GALE, PH.D. ASSISTANT PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF ROCHESTER GINN & COMPANY BOSTON NEW YORK CHICAGO LONDON COPYRIGHT, 1904, 1905, BY ARTHUR SULLIVAN GALE ALL BIGHTS RESERVED 65.8 GINN & COMPANY PRO- PRIETORS BOSTON U.S.A. PEE FACE In preparing this volume the authors have endeavored to write a drill book for beginners which presents, in a manner conform- ing with modern ideas, the fundamental concepts of the subject. The subject-matter is slightly more than the minimum required for the calculus, but only as much more as is necessary to permit of some choice on the part of the teacher. It is believed that the text is complete for students finishing their study of mathematics with a course in Analytic Geometry. The authors have intentionally avoided giving the book the form of a treatise on conic sections. Conic sections naturally appear, but chiefly as illustrative of general analytic methods. Attention is called to the method of treatment. The subject is developed after the Euclidean method of definition and theorem, without, however, adhering to formal presentation. The advan- tage is obvious, for the student is made sure of the exact nature of each acquisition. Again, each method is summarized in a rule stated in consecutive steps. This is a gain in clearness. Many illustrative examples are worked out in the text. Emphasis has everywhere been put upon the analytic side, that is, the student is taught to start from the equation.
    [Show full text]
  • Asce Standardized Reference Evapotranspiration Equation
    THE ASCE STANDARDIZED REFERENCE EVAPOTRANSPIRATION EQUATION Task Committee on Standardization of Reference Evapotranspiration Environmental and Water Resources Institute of the American Society of Civil Engineers January, 2005 Final Report ASCE Standardized Reference Evapotranspiration Equation Page i THE ASCE STANDARDIZED REFERENCE EVAPOTRANSPIRATION EQUATION PREPARED BY Task Committee on Standardization of Reference Evapotranspiration of the Environmental and Water Resources Institute TASK COMMITTEE MEMBERS Ivan A. Walter (chair), Richard G. Allen (vice-chair), Ronald Elliott, Daniel Itenfisu, Paul Brown, Marvin E. Jensen, Brent Mecham, Terry A. Howell, Richard Snyder, Simon Eching, Thomas Spofford, Mary Hattendorf, Derrell Martin, Richard H. Cuenca, and James L. Wright PRINCIPAL EDITORS Richard G. Allen, Ivan A. Walter, Ronald Elliott, Terry Howell, Daniel Itenfisu, Marvin Jensen ENDORSEMENTS Irrigation Association, 2004 ASCE-EWRI Task Committee Report, January, 2005 ASCE Standardized Reference Evapotranspiration Equation Page ii ABSTRACT This report describes the standardization of calculation of reference evapotranspiration (ET) as recommended by the Task Committee on Standardization of Reference Evapotranspiration of the Environmental and Water Resources Institute of the American Society of Civil Engineers. The purpose of the standardized reference ET equation and calculation procedures is to bring commonality to the calculation of reference ET and to provide a standardized basis for determining or transferring crop coefficients for agricultural and landscape use. The basis of the standardized reference ET equation is the ASCE Penman-Monteith (ASCE-PM) method of ASCE Manual 70. For the standardization, the ASCE-PM method is applied for two types of reference surfaces representing clipped grass (a short, smooth crop) and alfalfa (a taller, rougher agricultural crop), and the equation is simplified to a reduced form of the ASCE–PM.
    [Show full text]
  • The Roots of Any Polynomial Equation
    The roots of any polynomial equation G.A.Uytdewilligen, Bergen op Zoomstraat 76, 5652 KE Eindhoven. [email protected] Abstract We provide a method for solving the roots of the general polynomial equation n n−1 a ⋅x + a ⋅x + . + a ⋅x + s 0 n n−1 1 (1) To do so, we express x as a powerseries of s, and calculate the first n-1 coefficients. We turn the polynomial equation into a differential equation that has the roots as solutions. Then we express the powerseries’ coefficients in the first n-1 coefficients. Then the variable s is set to a0. A free parameter is added to make the series convergent. © 2004 G.A.Uytdewilligen. All rights reserved. Keywords: Algebraic equation The method The method is based on [1]. Let’s take the first n-1 derivatives of (1) to s. Equate these derivatives to zero. di Then find x ( s ) in terms of x(s) for i from 1 to n-1. Now make a new differential equation dsi n 1 n 2 d − d − m1⋅ x(s) + m2⋅ x(s) + . + m ⋅x(s) + m 0 n−1 n−2 n n+1 ds ds (2) di and fill in our x ( s ) in (2). Multiply by the denominator of the expression. Now we have a dsi polynomial in x(s) of degree higher then n. Using (1) as property, we simplify this polynomial to the degree of n. Set it equal to (1) and solve m1 .. mn+1 in terms of s and a1 .. an Substituting these in (2) gives a differential equation that has the zeros of (1) among its solutions.
    [Show full text]