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Section 2.6 Cylindrical and Spherical Coordinates
Section 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O,the rotating ray or half line from O with unit tick. A point P in the plane can be uniquely described by its distance to the origin r = dist (P, O) and the angle µ, 0 µ < 2¼ : · Y P(x,y) r θ O X We call (r, µ) the polar coordinate of P. Suppose that P has Cartesian (stan- dard rectangular) coordinate (x, y) .Then the relation between two coordinate systems is displayed through the following conversion formula: x = r cos µ Polar Coord. to Cartesian Coord.: y = r sin µ ½ r = x2 + y2 Cartesian Coord. to Polar Coord.: y tan µ = ( p x 0 µ < ¼ if y > 0, 2¼ µ < ¼ if y 0. · · · Note that function tan µ has period ¼, and the principal value for inverse tangent function is ¼ y ¼ < arctan < . ¡ 2 x 2 1 So the angle should be determined by y arctan , if x > 0 xy 8 arctan + ¼, if x < 0 µ = > ¼ x > > , if x = 0, y > 0 < 2 ¼ , if x = 0, y < 0 > ¡ 2 > > Example 6.1. Fin:>d (a) Cartesian Coord. of P whose Polar Coord. is ¼ 2, , and (b) Polar Coord. of Q whose Cartesian Coord. is ( 1, 1) . 3 ¡ ¡ ³ So´l. (a) ¼ x = 2 cos = 1, 3 ¼ y = 2 sin = p3. 3 (b) r = p1 + 1 = p2 1 ¼ ¼ 5¼ tan µ = ¡ = 1 = µ = or µ = + ¼ = . 1 ) 4 4 4 ¡ 5¼ Since ( 1, 1) is in the third quadrant, we choose µ = so ¡ ¡ 4 5¼ p2, is Polar Coord. -
Designing Patterns with Polar Equations Using Maple
Designing Patterns with Polar Equations using Maple Somasundaram Velummylum Claflin University Department of Mathematics and Computer Science 400 Magnolia St Orangeburg, South Carolina 29115 U.S.A [email protected] The Cartesian coordinates and polar coordinates can be used to identify a point in two dimensions. The polar coordinate system has a fixed point O called the origin or pole and a directed half –line called the polar axis with end point O. A point P on the plane can be described by the polar coordinates ( r, θ ), where r is the radial distance from the origin and θ is the angle made by OP with the polar axis. Several important types of graphs have equations that are simpler in polar form than in rectangular form. For example, the polar equation of a circle having radius a and centered at the origin is simply r = a. In other words , polar coordinate system is useful in describing two dimensional regions that may be difficult to describe using Cartesian coordinates. For example, graphing the circle x 2 + y 2 = a 2 in Cartesian coordinates requires two functions, one for the upper half and one for the lower half. In polar coordinate system, the same circle has the very simple representation r = a. Maple can be used to create patterns in two dimensions using polar coordinate system. When we try to analyze two dimensional patterns mathematically it will be helpful if we can draw a set of patterns in different colors using the help of technology. To investigate and compare some patterns we will go through some examples of cardioids and rose curves that are drawn using Maple. -
Polar Coordinates and Calculus.Wxp
Polar Coordinates and Calculus Recall that in the Polar coordinate system the coordinates ) represent <ß the directed distance from the pole to the point and the directed angle, counterclockwise from the polar axis to the segment from the pole to the point. A polar function would be of the form: ) . < œ 0 To find the slope of the tangent line of a polar graph we will parameterize the equation. ) ( Assume is a differentiable function of ) ) <œ0 0 ¾Bœ<cos))) œ0 cos and Cœ< sin ))) œ0 sin . .C .C .) Recall also that œ .B .B .) 1Þ Find the derivative of < œ # sin ) Remember that horizontal tangent lines occur when and that vertical tangent lines occur when page 1 2 Find the HTL and VTL for cos ) and sketch the graph. Þ < œ # " page 2 Recall our formula for the derivative of a function in polar coordinates. Since Bœ<cos))) œ0 cos and Cœ< sin ))) œ0 sin , we can conclude .C .C .) that œ.B œ .B .) Solutions obtained by setting < œ ! gives equations of tangent lines through the pole. 3Þ Find the equations of the tangent line(s) through the pole if <œ#sin #Þ) page 3 To find the arc length of a polar curve, you have two options. 1) You can use the parametrization of the polar curve: cos))) cos and sin ))) sin Bœ< œ0 Cœ< œ0 then use the arc length formula for parametric curves: " w # w # 'α ÈÐBÐ) ÑÑ ÐCÐ ) ÑÑ . ) or, 2) You can use an alternative formula for arc length in polar form: " <# Ð.< Ñ # . -
CID REBORIDO, Alicia. “Experiencias Del Examen Global En Línea De La UEA De Complementos De Matemáticas, DCBI-A” [Recurso Electrónico]
LÓPEZ BAUTISTA, Ricardo; PULIDO RODRÍGUEZ, Georgina; CID REBORIDO, Alicia. “Experiencias del examen global en línea de la UEA de complementos de matemáticas, DCBI-A” [recurso electrónico]. -- p. 349-368. -- En: Coloquio sobre la Práctica de la Educación Virtual en la UAM-A (1º. : 2012 : UAM Azcapotzalco, Ciudad de México). Memorias del Primer Coloquio sobre la Práctica de la Educación Virtual en la UAM-A. Mesa 3: Estudios de caso, segunda parte / Micheli Thirión, Jordy, coordinador y Armendáriz Torres, Sara, coordinadora. – México: Universidad Autónoma Metropolitana (México), Unidad Azcapotzalco, División de Ciencias Sociales y Humanidades, Coordinación de Difusión y Publicaciones, 2012. 467 páginas. ISBN 978-607-477-830-4 EXPERIENCIAS DEL EXAMEN GLOBAL EN LÍNEA DE LA UEA DE COMPLEMENTOS DE MATEMÁTICAS, DCBI-A Ricardo López Bautista [email protected] Georgina Pulido Rodríguez [email protected] Alicia Cid Reborido [email protected] Resumen Desde 2011 se han aplicado en línea tanto el examen global como el de recuperación de la UEA Complementos de Matemáticas, que pertenece al Tronco General de las licenciaturas de Ingeniería en la UAM-A. En esta comunicación se describen las experiencias, retos y logros de llevar a cabo la aplicación de estos exámenes en línea en forma presencial a todos los sustentantes en tiempo y en forma de acuerdo con la legislación. Se utilizaron aulas virtuales con evaluación continua en los grupos de los autores en 2009, lo que dio lugar a la creación del sistema galoisenlinea, donde se encuentra ahora un Centro de Evaluación en Línea de Matemáticas. Ahí aparecen también autoevaluaciones, una gran variedad de recursos didácticos e información general para los usuarios del sistema, que puede ser un alumno o un académico de la comunidad UAM-A. -
Classical Mechanics Lecture Notes POLAR COORDINATES
PHYS 419: Classical Mechanics Lecture Notes POLAR COORDINATES A vector in two dimensions can be written in Cartesian coordinates as r = xx^ + yy^ (1) where x^ and y^ are unit vectors in the direction of Cartesian axes and x and y are the components of the vector, see also the ¯gure. It is often convenient to use coordinate systems other than the Cartesian system, in particular we will often use polar coordinates. These coordinates are speci¯ed by r = jrj and the angle Á between r and x^, see the ¯gure. The relations between the polar and Cartesian coordinates are very simple: x = r cos Á y = r sin Á and p y r = x2 + y2 Á = arctan : x The unit vectors of polar coordinate system are denoted by r^ and Á^. The former one is de¯ned accordingly as r r^ = (2) r Since r = r cos Á x^ + r sin Á y^; r^ = cos Á x^ + sin Á y^: The simplest way to de¯ne Á^ is to require it to be orthogonal to r^, i.e., to have r^ ¢ Á^ = 0. This gives the condition cos ÁÁx + sin ÁÁy = 0: 1 The simplest solution is Áx = ¡ sin Á and Áy = cos Á or a solution with signs reversed. This gives Á^ = ¡ sin Á x^ + cos Á y^: This vector has unit length Á^ ¢ Á^ = sin2 Á + cos2 Á = 1: The unit vectors are marked on the ¯gure. With our choice of sign, Á^ points in the direc- tion of increasing angle Á. Notice that r^ and Á^ are drawn from the position of the point considered. -
Section 6.7 Polar Coordinates 113
Section 6.7 Polar Coordinates 113 Course Number Section 6.7 Polar Coordinates Instructor Objective: In this lesson you learned how to plot points in the polar coordinate system and write equations in polar form. Date I. Introduction (Pages 476-477) What you should learn How to plot points in the To form the polar coordinate system in the plane, fix a point O, polar coordinate system called the pole or origin , and construct from O an initial ray called the polar axis . Then each point P in the plane can be assigned polar coordinates as follows: 1) r = directed distance from O to P 2) q = directed angle, counterclockwise from polar axis to the segment from O to P In the polar coordinate system, points do not have a unique representation. For instance, the point (r, q) can be represented as (r, q ± 2np) or (- r, q ± (2n + 1)p) , where n is any integer. Moreover, the pole is represented by (0, q), where q is any angle . Example 1: Plot the point (r, q) = (- 2, 11p/4) on the polar py/2 coordinate system. p x0 3p/2 Example 2: Find another polar representation of the point (4, p/6). Answers will vary. One such point is (- 4, 7p/6). Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved. 114 Chapter 6 Topics in Analytic Geometry II. Coordinate Conversion (Pages 477-478) What you should learn How to convert points The polar coordinates (r, q) are related to the rectangular from rectangular to polar coordinates (x, y) as follows . -
1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the position of each point on the plane is determined by an angle and a distance. The distance is usually denoted r and the angle is usually denoted . Thus, in this coordinate system, the position of a point will be given by the ordered pair (r, ). These are called the polar coordinates These two quantities r and , are determined as follows. First, we need some reference points. You may recall that in the Cartesian coordinate system, everything was measured with respect to the coordinate axes. In the polar coordinate system, everything is measured with respect a fixed point called the pole and an axis called the polar axis. The is the equivalent of the origin in the Cartesian coordinate system. The polar axis corresponds to the positive x-axis. Given a point P in the plane, we draw a line from the pole to P . The distance from the pole to P is r, the angle, measured counterclockwise, by which the polar axis has to be rotated in order to go through P is . The polar coordinates of P are then (r, ). Figure 1.7.1 shows two points and their representation in the polar coordinate system. Remark 79 Let us make several remarks. 1. Recall that a positive value of means that we are moving counterclock- wise. But can also be negative. A negative value of means that the polar axis is rotated clockwise to intersect with P . -
PDF Version (433
Math on the Web: A Status Report January, 2002 Focus: Authoring Tools by Robert Miner and Paul Topping, Design Science, Inc. View this paper on-line, where the links and references are live, go to http://www.dessci.com/webmath/status. We plan on updating this report as the world of Math on the Web changes. Join our Math on the Web mailing list and we'll notify you when the report is updated: http://www.dessci.com/webmath. Design Science www.dessci.com How Science Communicates™ Design Science, Inc. 4028 Broadway, Long Beach, CA 90803 USA, Phone: 562.433.0685 Fax: 562.433.6969 1 Math on the Web: A Status Report Focus: Authoring Tools by Robert Miner and Paul Topping, Design Science, Inc. The last six months have seen very significant tions support the idea of developing standards for developments in Math on the Web. Effective, scientific communication, most have little interest ubiquitous support for math notation in in actually implementing math-specific features mainstream web browsers is finally becoming a themselves. As a consequence, the emphasis at reality. This edition of the Status Report is devoted W3C naturally turned toward the development of to taking a closer look at the new generation of general-purpose extension mechanisms that could Math on the Web technology. We begin by accommodate math rendering. While on the examining recent breakthroughs in browser surface, native math support in browsers might support, followed by a rundown of notable news seem preferable, a case can be made that the drive and events for the last six months. -
Open Source Resources for Teaching and Research in Mathematics
Open Source Resources for Teaching and Research in Mathematics Dr. Russell Herman Dr. Gabriel Lugo University of North Carolina Wilmington Open Source Resources, ICTCM 2008, San Antonio 1 Outline History Definition General Applications Open Source Mathematics Applications Environments Comments Open Source Resources, ICTCM 2008, San Antonio 2 In the Beginning ... then there were Unix, GNU, and Linux 1969 UNIX was born, Portable OS (PDP-7 to PDP-11) – in new “C” Ken Thompson, Dennis Ritchie, and J.F. Ossanna Mailed OS => Unix hackers Berkeley Unix - BSD (Berkeley Systems Distribution) 1970-80's MIT Hackers Public Domain projects => commercial RMS – Richard M. Stallman EMACS, GNU - GNU's Not Unix, GPL Open Source Resources, ICTCM 2008, San Antonio 3 History Free Software Movement – 1983 RMS - GNU Project – 1983 GNU GPL – GNU General Public License Free Software Foundation (FSF) – 1985 Free = “free speech not free beer” Open Source Software (OSS) – 1998 Netscape released Mozilla source code Open Source Initiative (OSI) – 1998 Eric S. Raymond and Bruce Perens The Cathedral and the Bazaar 1997 - Raymond Open Source Resources, ICTCM 2008, San Antonio 4 The Cathedral and the Bazaar The Cathedral model, source code is available with each software release, code developed between releases is restricted to an exclusive group of software developers. GNU Emacs and GCC are examples. The Bazaar model, code is developed over the Internet in public view Raymond credits Linus Torvalds, Linux leader, as the inventor of this process. http://en.wikipedia.org/wiki/The_Cathedral_and_the_Bazaar Open Source Resources, ICTCM 2008, San Antonio 5 Given Enough Eyeballs ... central thesis is that "given enough eyeballs, all bugs are shallow" the more widely available the source code is for public testing, scrutiny, and experimentation, the more rapidly all forms of bugs will be discovered. -
1 Basic Introduction
1 Basic Introduction If you are new to LiveMath, you don’t necessarily want to drudge through a big software manual to just get started doing some math. So this first chapter is just to get started. 1.1 Why is this program called “LiveMath” ? LiveMath is a computer algebra and graphing program. These types of programs were originally called “CAS” = computer algebra systems. The biggest such program is Mathematica, with others being Maple, MatLab, and other minor computational engines, both large and small, free and commercial. Nearly all of these math software programs, with the exception of the MathCad program, are closely tied to a programming language - using command-line syntax to perform computations, make graphs, etc.: type in a command in some sort of text syntax, then “hit Enter” to send those commands to the computation engine, and wait for the result. LiveMath was originally called Theorist, and was developed as a computational engine on top of the math equation editor Expressionist. This original Theorist was not created as an “input/output” engine, but rather as a steady-state machine. This means that when you open up a LiveMath notebook, it will do computations, graphs, etc. to achieve a steady-state. If you change something inside of the notebook - a number, or a variable, or something - the LiveMath engine will bring the notebook back to a steady-state, and do the computations, graphs, etc necessary that have been affected by your change. In this sense, the notebook is ready to do live mathematics. Imagine you had some equations written on a blackboard - a series of The LiveMath Book Page 2 computations. -
Section Summary: Polar Coordinates A. Definitions B. Theorems C
Section Summary: Polar Coordinates a. Definitions The polar coordinate system (introduced by Newton) is an alter- native to the Cartesian coordinate system (named after Descartes) in which every point in the plane is expressed by its distance and direction (angle) from the origin, called the pole. The polar axis plays the role formerly played by the positive x-axis. The polar coordinates are often given as (r, θ), where r is the distance from the pole, and • θ is the angle between the polar axis and the ray passing through • the point of interest. The graph of a polar equation contains all points P that have at least one polar representation (r, θ) whose coordinates solve the equa- tion. b. Theorems None to speak of. c. Properties/Tricks/Hints/Etc. Given r and θ it’s easy to find the corresponding x and y: x = r cos θ y = r sin θ It’s not quite so easy to go in the opposite direction, unless we restrict r and θ: r [0, ) ∈ ∞ θ [0, 2π) ∈ gives a unique representation in polar coordinates (except for the pole, which has equation r = 0 regardless of the value of θ). 1 As a convenience we let r take negative values: so ( r, θ)=(r, θ + π) − In any event, we see that, by contrast with the Cartesian coordinate system, the polar coordinate system allows points to have multiple representations. To find the Cartesian coordinates from the polar coordinates, we can use the equations r = √x2 + y2 −1 y θ = tan x Be careful, however, as there are two values of θ that solve these equa- tions in each interval of 2π, and one must choose the proper value based on the quadrant in which (x, y) lies... -
The Algebra Curriculum and Personal Technology
The Algebra Curriculum and Personal Technology: Exploring the Links Barry Kissane Murdoch University, Australia <[email protected]> Abstract: Prior to the availability of personal technology in the form of graphics calculators, the algebra curriculum was rarely influenced by either computers or calculators. In this paper, some of the ways in which both the teaching and learning of algebra might be related to technology are identified, exemplified and briefly explored. Some of the links concern conceptual development of the ideas of variables and functions, the solution of equations and inequalities and the place of symbolic manipulation generally. There are many powerful, interesting and enticing technologies presently available for school mathematics in general, and algebra in particular. These include educational computer software such as Derive and Cabri Geometry, powerful new developments in telecommunications such as multimedia pages accessible via Internet web browsers, and mixtures of the two such as LiveMath. Exciting and important as such developments are, they share the common weakness of requiring relatively expensive computer and telecommunications facilities, so that it will be some time before they can be genuinely regarded as accessible to pupils at large in a particular state or country. In this paper, the focus is on personal technology, here regarded as technology that is likely to be available at the personal level to all pupils involved in studying mathematics (Kissane, 1995). The significance of such technology