Geometry and Algebra: the Future Flight Equation
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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. -
LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
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. -
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. -
Schaum's Outline of Linear Algebra (4Th Edition)
SCHAUM’S SCHAUM’S outlines outlines Linear Algebra Fourth Edition Seymour Lipschutz, Ph.D. Temple University Marc Lars Lipson, Ph.D. University of Virginia Schaum’s Outline Series New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2009, 2001, 1991, 1968 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior writ- ten permission of the publisher. ISBN: 978-0-07-154353-8 MHID: 0-07-154353-8 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-154352-1, MHID: 0-07-154352-X. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at [email protected]. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. -
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. -
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. -
Problems in Abstract Algebra
STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth 10.1090/stml/082 STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 00A07, 12-01, 13-01, 15-01, 20-01. For additional information and updates on this book, visit www.ams.org/bookpages/stml-82 Library of Congress Cataloging-in-Publication Data Names: Wadsworth, Adrian R., 1947– Title: Problems in abstract algebra / A. R. Wadsworth. Description: Providence, Rhode Island: American Mathematical Society, [2017] | Series: Student mathematical library; volume 82 | Includes bibliographical references and index. Identifiers: LCCN 2016057500 | ISBN 9781470435837 (alk. paper) Subjects: LCSH: Algebra, Abstract – Textbooks. | AMS: General – General and miscellaneous specific topics – Problem books. msc | Field theory and polyno- mials – Instructional exposition (textbooks, tutorial papers, etc.). msc | Com- mutative algebra – Instructional exposition (textbooks, tutorial papers, etc.). msc | Linear and multilinear algebra; matrix theory – Instructional exposition (textbooks, tutorial papers, etc.). msc | Group theory and generalizations – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA162 .W33 2017 | DDC 512/.02–dc23 LC record available at https://lccn.loc.gov/2016057500 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
From Arithmetic to Algebra
From arithmetic to algebra Slightly edited version of a presentation at the University of Oregon, Eugene, OR February 20, 2009 H. Wu Why can’t our students achieve introductory algebra? This presentation specifically addresses only introductory alge- bra, which refers roughly to what is called Algebra I in the usual curriculum. Its main focus is on all students’ access to the truly basic part of algebra that an average citizen needs in the high- tech age. The content of the traditional Algebra II course is on the whole more technical and is designed for future STEM students. In place of Algebra II, future non-STEM would benefit more from a mathematics-culture course devoted, for example, to an understanding of probability and data, recently solved famous problems in mathematics, and history of mathematics. At least three reasons for students’ failure: (A) Arithmetic is about computation of specific numbers. Algebra is about what is true in general for all numbers, all whole numbers, all integers, etc. Going from the specific to the general is a giant conceptual leap. Students are not prepared by our curriculum for this leap. (B) They don’t get the foundational skills needed for algebra. (C) They are taught incorrect mathematics in algebra classes. Garbage in, garbage out. These are not independent statements. They are inter-related. Consider (A) and (B): The K–3 school math curriculum is mainly exploratory, and will be ignored in this presentation for simplicity. Grades 5–7 directly prepare students for algebra. Will focus on these grades. Here, abstract mathematics appears in the form of fractions, geometry, and especially negative fractions. -
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 -
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. -
Choosing Your Math Course
CHOOSING YOUR MATH COURSE When choosing your first math class, it’s important to consider your current skills, the topics covered, and course expectations. It’s also helpful to think about how each course aligns with both your interests and your career and transfer goals. If you have questions, talk with your advisor. The courses on page 1 are developmental-level and the courses on page 2 are college-level. • Developmental math courses (Foundations of Math and Math & Algebra for College) offer the chance to develop concepts and skills you may have forgotten or never had the chance to learn. They focus less on lecture and more on practicing concepts individually and in groups. Your homework will emphasize developing and practicing new skills. • College-level math courses build on foundational skills taught in developmental math courses. Homework, quizzes, and exams will often include word problems that require multiple steps and incorporate a variety of math skills. You should expect three to four exams per semester, which cover a variety of concepts, and shorter weekly quizzes which focus on one or two concepts. You may be asked to complete a final project and submit a paper using course concepts, research, and writing skills. Course Your Skills Readiness Topics Covered & Course Expectations Foundations of 1. All multiplication facts (through tens, preferably twelves) should be memorized. In Foundations of Mathematics, you will: Mathematics • learn to use frections, decimals, percentages, whole numbers, & 2. Whole number addition, subtraction, multiplication, division (without a calculator): integers to solve problems • interpret information that is communicated in a graph, chart & table Math & Algebra 1.