DOCSLIB.ORG

Proquest Dissertations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Proquest Dissertations The large-scale structure of the universe in one dimension Item Type text; Dissertation-Reproduction (electronic) Authors Zhan, Hu Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 05/10/2021 21:38:43 Link to Item http://hdl.handle.net/10150/290096 THE LARGE-SCALE STRUCTURE OF THE UNIVERSE IN ONE DIMENSION by Hu Zhan Copyright © Hu Zhan 2004 A Dissertation Submitted to the Faculty of the DEPARTMENT OF PHYSICS In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA 2 0 0 4 UMI Number: 3132278 Copyright 2004 by Zhan, Hu All rights reserved. INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. UMI UMI Microform 3132278 Copyright 2004 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 2 The University of Arizona ® Graduate College As members of the Final Examination Committee, we certify that we have read the dissertation prepared by Hu Zhan entitled The Large-Scale Structure of the Universe in One Dimension and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy David B Date ¥// ^ V Ronaeel Date aniel Eisenstei Date Li-Zhi Fang Date 0 Fulvio Melia Date Michael Shupe Date Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared er my direction and recommend that it be accepted as fulfilling the disseetSti^n requjfe Dissertation Directors; Li-Zhi Fang / David Burstein / Daniel Eisenstein Date 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, pro­ vided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the copyright holder. SIGNED: 4 ACKNOWLEDGMENTS I thank my advisors David Burstein, Daniel Eisenstein, and Li-Zhi Fang for their supervision and guidance over the course of my study. I also thank my mentors and collaborators, Bruce Barrett, Romeel Dave, Xiaohui Fan, Priya Jamkhedkar, Fulvio Meha, Petr Navratil, Andreas Nogga, Michael Shupe, James Vary, Rogier Windhorst, and Ann Zabludoff. Without them, I would probably have spent a lot more time practicing trial and error. I appreciate the financial support from Bruce in the Spring and Summer of 2003, during which ideas in this dissertation germinated. I have benefited from grants from the Theoretical Astrophysics Program and registration fee waivers from the Physics Depart­ ment. I am grateful to my wife, Yue, for her enduring support. DEDICATION This dissertation is dedicated to my parents, Hua'an and Lanzhen, and my wife, Yue. 6 TABLE OF CONTENTS LIST OF FIGURES 8 LIST OF TABLES 11 LIST OF ABBREVIATIONS 12 ABSTRACT 13 CHAPTER 1: INTRODUCTION 14 CHAPTER 2: ONE-DIMENSIONAL POWER SPECTRUM 17 2.1 Three-Dimensional Power Spectrum 17 2.2 One-Dimensional Power Spectrum 19 2.3 Aliasing 21 CHAPTER 3: COVARIANCE OF THE ONE-DIMENSIONAL POWER SPECTRUM: GAUSSIAN RANDOM FIELDS 23 3.1 Covariance of the One-Dimensional Power Spectrum 23 3.2 Line-of-Sight Length 25 CHAPTER 4: COVARIANCE OF THE ONE-DIMENSIONAL POWER SPECTRUM: THE COSMIC DENSITY FIELD 27 4.1 Simulations 27 4.2 Pair-Wise Covariance 29 4.3 Covariance 30 CHAPTER 5: COVARIANCE OF THE ONE-DIMENSIONAL POWER SPECTRUM: THE LYa FOREST 39 5.1 Simulated Lya Forests 39 5.1.1 Hydrodynamical Simulations with Photoionization 39 5.1.2 Pseudo-Hydro Technique 47 5.1.3 Comparison 56 7 TABLE OF CONTENTS - Continued 5.2 Tuning the Lya Forest 61 5.2.1 Equation of State 61 5.2.2 Mean Flux 65 5.3 Mass Statistics vs. Flux Statistics 68 5.3.1 Power Spectrum 68 5.3.2 Covariance 69 5.4 Cosmology 77 CHAPTER 6: INVERTING THE LYa FOREST 80 6.1 Correlation between the Mean Density and Width of Saturated Lya Absorptions 81 6.1.1 Simulations 81 6.1.2 The Correlation 82 6.1.3 A Simple Model 85 6.2 Inversion with A Gaussian Density Profile 87 6.3 Mapping the Power Spectra 91 CHAPTER 7: CONCLUSIONS 96 REFERENCES 98 8 LIST OF FIGURES 4.1 Normalized pair-wise covariance matrices C(A:3,s) in grey scale .... 29 4.2 Diagonal elements of the normalized pair-wise covariance matrices .... 30 4.3 Estimated one-dimensional mass power spectra and their variance .... 31 4.4 Normalized covariances of one-dimensional mass power spectra averaged over 2000 groups, each of which consists of 64 lines of sight 33 4.5 Normalized covariances of one-dimensional mass power spectra averaged over 256 groups, each of which consists of 1024 lines of sight 34 4.6 Effect of non-Gaussianity and resolution on the covariance of the one- dimensional mass power spectrum 36 4.7 Effect of non-Gaussianity and resolution on the variance of the one-dimensional mass power spectrum 37 4.8 Effect of the length of Hnes of sight 38 5.1 Fourier transform of the assignment function 41 5.2 Thermal broadening of Lya absorptions at z = 0 45 5.3 Thermal broadening of Lya absorptions at z = 3 46 5.4 Correlation between baryon and dark matter densities in the simulation . 48 5.5 Line-of-sight baryon densities and dark matter densities at z = 0 50 5.6 Line-of-sight baryon densities and dark matter densities at z = 3 51 5.7 Three-dimensional and one-dimensional mass power spectra of baryons and dark matter in real space and redshift space 52 5.8 Temperature-density relation of baryons in the simulation 55 5.9 Transmitted Lya flux based on baryon and dark matter distributions at z = 0 57 5.10 Transmitted Lya flux based on baryon and dark matter distributions at 2= 3.... 58 5.11 Flux power spectra of Lya forests generated from baryon and dark matter distributions at z = 0 and z = 3 59 5.12 Effect of thermal broadening on flux power spectra 60 9 LIST OF FIGURES - Continued 5.13 Lya Forests generated from baryon and dark matter distributions with different equations of state at z = 0 and 1 62 5.14 Lya Forests generated from baryon and dark matter distributions with different equations of state at z= 2 and 3 ....... 63 5.15 Flux power spectra of Lya forests generated from dark matter densities using different equations of state 64 5.16 Simulated Lya forests with different mean fluxes 66 5.17 Flux power spectra of simulated Lya forests with different mean fluxes . 67 5.18 Flux power spectra of simulated Lya forests and one-dimensional mass power spectra of the underlying density fields at z = 0 and 1 69 5.19 Flux power spectra of simulated Lya forests and one-dimensional mass power spectra of the underlying density fields at z = 2 and 3 70 5.20 Normalized covariances of one-dimensional baryon and dark matter power spectra at z — 0 71 5.21 Normalized covariances of flux power spectra based on baryon and dark matter distributions at z = Q 72 5.22 Normalized covariances of one-dimensional baryon and dark matter power spectra at 2; =3 ........... 74 5.23 Normalized covariances of flux power spectra based on baryon and dark matter distributions at z = 3 75 5.24 Normalized covariances of observed flux power spectra at z — 3 76 5.25 Mass power spectra of baryons and dark matter and flux power spectra of simulated Lya forests for different cosmological models at 2: = 0 78 5.26 Mass power spectra of baryons and dark matter and flux power spectra of simulated Lya forests for different cosmological models at z = 3 79 6.1 Simulated Lya forests and line-of-sight densities from the LCDM model at z = 3 83 6.2 Correlation between the mean density pa and width w of saturated regions in the Lya forest 84 10 LIST OF FIGURES - Continued 6.3 Best-fittings of the pa.-w relation for four cosmological models from z — 'i to 4.5 86 6.4 One-dimensional mass power spectra of original densities and inverted den­ sities in redshift space for the LCDM model 88 6.5 One-dimensional mass power spectra of original densities and inverted den­ sities in redshift space for OCDM, SCDM, and TCDM models 89 6.6 Recovery of the one-dimensional mass power spectrum with uncertainties in the Pa,--w relation 90 6.7 One-point distribution function of inverted densities and that of the origi­ nal densities 91 6.8 Three-dimensional mass power spectra of the LCDM model at ^ = 3 .
Load more

Recommended publications
  • Feasibility Study for Teaching Geometry and Other Topics Using Three-Dimensional Printers
    Feasibility Study For Teaching Geometry and Other Topics Using Three-Dimensional Printers Elizabeth Ann Slavkovsky A Thesis in the Field of Mathematics for Teaching for the Degree of Master of Liberal Arts in Extension Studies Harvard University October 2012 Abstract Since 2003, 3D printer technology has shown explosive growth, and has become significantly less expensive and more available. 3D printers at a hobbyist level are available for as little as $550, putting them in reach of individuals and schools. In addition, there are many “pay by the part” 3D printing services available to anyone who can design in three dimensions. 3D graphics programs are also widely available; where 10 years ago few could afford the technology to design in three dimensions, now anyone with a computer can download Google SketchUp or Blender for free. Many jobs now require more 3D skills, including medical, mining, video game design, and countless other fields. Because of this, the 3D printer has found its way into the classroom, particularly for STEM (science, technology, engineering, and math) programs in all grade levels. However, most of these programs focus mainly on the design and engineering possibilities for students. This thesis project was to explore the difficulty and benefits of the technology in the mathematics classroom. For this thesis project we researched the technology available and purchased a hobby-level 3D printer to see how well it might work for someone without extensive technology background. We sent designed parts away. In addition, we tried out Google SketchUp, Blender, Mathematica, and other programs for designing parts. We came up with several lessons and demos around the printer design.
    [Show full text]
  • Intersecting Cylinders: from Archimedes and Zu Chongzhi to Steinmetz and Beyond Project Report∗
    Intersecting Cylinders: From Archimedes and Zu Chongzhi to Steinmetz and Beyond Project Report∗ Lingyi Kong, Luvsandondov Lkhamsuren, Abigail Turner, Aananya Uppal, A.J. Hildebrand (Faculty Mentor) University of Illinois at Urbana-Champaign April 25, 2013 1 Background Our work was motivated by the following problem, which can be found in many calculus texts as a challenging example of computing volumes by multiple integrals. Intersecting Cylinders Problem. What is the volume of the region of intersection of two or three pairwise perpendicular cylinders of unit radius (\Steinmetz solids")? Figure 1: The classical 2-cylinder and 3-cylinder Steinmetz solids. ∗This project is part of an ongoing research project \Adventures with n-dimensional integrals", carried out at the Illinois Geometry Lab, www.math.illinois.edu/igl. An expanded version of this report is expected to be prepared for possible publication. 1 The above solids are named after Charles Proteus Steinmetz (1865{1923), a brilliant electrical engineer and mathematician who is said to solved the two-cylinder version of this problem in two minutes at a dinner party. The two-cylinder version of the problem was first studied more than two thousand years by Archimedes and the Chinese mathematician Zu Chongzhi, who proved that the volume of a two- cylinder Steinmetz solid is 16=3. The three-cylinder version was first considered in the 1970s by Moore [3], a physicist who was motivated by applications in crystallography. Usingp calculus methods, Moore showed that the volume of a three-cylinder Steinmetz solid is 16 − 8 2. The problem has been popularized by Martin Gardner in his Scientific American column [1], and the two-dimensional Steinmetz solid is featured on the cover of one of Gardner's books [2].
    [Show full text]
  • Research and Practice of Integrating Ideological and Political Education Into Calculus Teaching
    US-China Foreign Language, August 2021, Vol. 19, No. 8, 208-216 doi:10.17265/1539-8080/2021.08.002 D DAVID PUBLISHING Research and Practice of Integrating Ideological and Political Education Into Calculus Teaching HAO Shunli Beijing International Studies University, Beijing, China This paper first analyses the reasons for the low effectiveness of ideological and political education in the current calculus teaching, and then puts forward the contents, methods, and approaches of integrating ideological and political education into calculus teaching on this basis, and finally finds out the points for needing attention in the integrating ideological and political education into calculus teaching. Keywords: ideological and political education, calculus teaching, contents of ideological and political theories teaching in all courses, methods of ideological and political theories teaching in all courses, approaches of ideological and political theories teaching in all courses Introduction Ideological and political theories teaching in all courses is a new idea. It is a comprehensive education idea that constructs the pattern of “San Quan” Education, integrates non-ideological and political theory courses and ideological and political theory courses together to form a synergistic effect, and takes strengthening morality and cultivating young persons as the fundamental task of education. It integrates the elements of ideological and political education (including theoretical knowledge, value idea, spiritual pursuit, and so on of ideological and political education) into the non-ideological and political theory courses, and it has a subtle influence on students’ ideology and behaviour. Ideological and political theories teaching in all courses is a new method. It is a method to reflect Marx’s guiding position and practice the socialist core values in the process of implementing the fundamental task of strengthening morality and cultivating young persons.
    [Show full text]
  • MATH 2300 - Homework 4 Instructions: This Homework Is Due on Friday, June 29Th
    MATH 2300 - Homework 4 Instructions: This homework is due on Friday, June 29th. You may work with other students, however each person is responsible for writing their own solutions. Please write the names of any students who helped you. x2 y2 1. Consider the circle x2 + y2 = b2 sitting inside the ellipse + = 1, a2 b2 b b a (a) Find the area between the circle and the ellipse (in green) by using the methods of section 7.4 (see Example 8, p. 356-357): in the first quadrant, the area is given by the q p 2 b2x2 2 2 integral of the difference of the functions y = b − a2 and y = b − x , so just multiply this result by 4 to get the total area. (b) Find the area between the circle and the ellipse (in green) by using the method of (horizontal) slicing (section 8.1). You may work in either top or bottom half and multiply the result by 2 if you wish, or in the first quadrant and multiply the result by 4. (c) Find the area between the circle and the ellipse (in green) by using polar coordinates (section 8.3). [Hint: Work in the first quadrant and multiply the result by 4. Plug 2 2 x = r cos θ and y = r sinθ into the equation x + y = 1 and solve for r to get r as a a2 b2 p arctan( α+1 tan θ) function of θ, r = f(θ). You may use the fact that R 1 dθ = p + C. 1+αsinθ α+1 Note: When evaluating this between 0 and π=2, you have to evaluate it between 0 and c, taking the limit as c ! π=2, because tan(π=2) is not defined.] x2 y2 2.
    [Show full text]
  • View This Volume's Front and Back Matter
    Titles in This Series Volume 8 Kare n Hunger Parshall and David £. Rowe The emergenc e o f th e America n mathematica l researc h community , 1876-1900: J . J. Sylvester, Felix Klein, and E. H. Moore 1994 7 Hen k J. M. Bos Lectures in the history of mathematic s 1993 6 Smilk a Zdravkovska and Peter L. Duren, Editors Golden years of Moscow mathematic s 1993 5 Georg e W. Mackey The scop e an d histor y o f commutativ e an d noncommutativ e harmoni c analysis 1992 4 Charle s W. McArthur Operations analysis in the U.S. Army Eighth Air Force in World War II 1990 3 Pete r L. Duren, editor, et al. A century of mathematics in America, part III 1989 2 Pete r L. Duren, editor, et al. A century of mathematics in America, part II 1989 1 Pete r L. Duren, editor, et al. A century of mathematics in America, part I 1988 This page intentionally left blank https://doi.org/10.1090/hmath/008 History of Mathematics Volume 8 The Emergence o f the American Mathematical Research Community, 1876-1900: J . J. Sylvester, Felix Klein, and E. H. Moor e Karen Hunger Parshall David E. Rowe American Mathematical Societ y London Mathematical Societ y 1991 Mathematics Subject Classification. Primary 01A55 , 01A72, 01A73; Secondary 01A60 , 01A74, 01A80. Photographs o n th e cove r ar e (clockwis e fro m right ) th e Gottinge n Mathematisch e Ges - selschafft, Feli x Klein, J. J. Sylvester, and E. H. Moore.
    [Show full text]
  • It Slices, It Dices
    URL: http://opinionator.blogs.nytimes.com/2010/04/18/it-slices-it-dices/ April 18, 2010, 5:00 pm It Slices, It Dices By STEVEN STROGATZ Mathematical signs and symbols are often cryptic, but the best of them offer visual clues to their own meaning. The symbols for zero, one and infinity aptly resemble an empty hole, a single mark and an endless loop: 0, 1, ∞. And the equals sign, =, is formed by two parallel lines because, in the words of its originator, Welsh mathematician Robert Recorde in 1557, “no two things can be more equal.” In calculus the most recognizable icon is the integral sign: Its graceful lines are evocative of a musical clef or a violin’s f-hole — a fitting coincidence, given that some of the most enchanting harmonies in mathematics are expressed by integrals. But the real reason that Leibniz chose this symbol is much less poetic. It’s simply a long-necked S, for “summation.” As for what’s being summed, that depends on context. In astronomy, the gravitational pull of the sun on the earth is described by an integral. It represents the collective effect of all the miniscule forces generated by each solar atom at their varying distances from the earth. In oncology, the growing mass of a solid tumor can be modeled by an integral. So can the cumulative amount of drug administered during the course of a chemotherapy regimen. Historically, integrals arose first in geometry, in connection with the problem of finding the areas of curved shapes. As we saw two weeks ago, the area of a circle can be viewed as the sum of many thin pie slices.
    [Show full text]
  • Supporting Information to Imaging Catalytic Hotspots on Single Plasmonic Nanostructures Via Correlated Super-Resolution and Electron Microscopy
    Supporting Information to Imaging Catalytic Hotspots on Single Plasmonic Nanostructures via Correlated Super-Resolution and Electron Microscopy Ningmu Zou, Guanqun Chen, Xianwen Mao, Hao Shen, Eric Choudhary, Xiaochun Zhou, and Peng Chen* Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York, 14853 Table of Content S1. Catalysts preparation and characterization ....................................................................................................... 2 S1.1. Synthesis of Au nanorods ................................................................................................................................................... 2 S1.2. Synthesis of linked Au-Au nanorod-nanorod and Au-Ag nanorod-nanoparticle nanostructures ........................................ 2 S1.3. Encapsulation of linked nanostructures with mesoporous silica and subsequent UV-ozone treatment ............................... 3 S1.4. SEM imaging and nanostructure gap size measurement ..................................................................................................... 4 S1.5. EDX elemental analysis ...................................................................................................................................................... 5 S1.6. Biotin-streptavidin linkage increases the yield of linked nanostructures by ~7 times ......................................................... 6 S1.7. Ascorbic acid treatment can reduce surface Ag2O formed during UV-ozone treatment ....................................................
    [Show full text]
  • Duke University Dissertation Template
    GaAsBi Synthesis: From Band Structure Modification to Nanostructure Formation by Kristen N. Collar Department of Physics Duke University Date:______________________ Approved: ___________________________ Harold Baranger, Chair ___________________________ April Brown, Advisor/ Co-Chair ___________________________ Ronen Plesser ___________________________ Maiken Mikkelsen ___________________________ Patrick Charbonneau Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics in the Graduate School of Duke University 2017 i v ABSTRACT GaAsBi Synthesis: From Band Structure Modification to Nanostructure Formation by Kristen N. Collar Department of Physics Duke University Date:_______________________ Approved: ___________________________ Harold Baranger, Chair ___________________________ April Brown, Advisor/Co-Chair ___________________________ Maiken Mikkelsen ___________________________ Ronen Plesser ___________________________ Patrick Charbonneau An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics in the Graduate School of Duke University 2017 i v Copyright by Kristen N. Collar 2017 Abstract Research and development of bismides have proven bismides to be a promising field for material science with important applications in optoelectronics. However, the development of a complete description of the electrical and material properties of bismide ternaries is not comprehensive or straightforward. One of the main benefits of this ternary system is the opportunity for bandgap tuning, which opens doors to new applications. Tuning the bandgap is achieved by varying the composition; this allows access to a wider energy spectrum with applications in long wavelength emitters and detectors. In addition to bandgap tuning, Bi provides an opportunity to decrease lasing threshold currents, to decrease the temperature sensitivity and to decrease a major loss mechanism of today’s telecom lasers.
    [Show full text]
  • Jean-Baptiste Hiriart-Urruty Volume 1 (For Undergraduates)
    Springer Undergraduate Mathematics Series Jean-Baptiste Hiriart-Urruty Mathematical Tapas Volume 1 (for Undergraduates) Springer Undergraduate Mathematics Series Advisory Board M.A.J. Chaplain, University of St. Andrews, St. Andrews, Scotland, UK A. MacIntyre, Queen Mary University of London, London, England, UK S. Scott, King’s College London, London, England, UK N. Snashall, University of Leicester, England, UK E. Süli, University of Oxford, Oxford, England, UK M.R. Tehranchi, University of Cambridge, Cambridge, England, UK J.F. Toland, University of Cambridge, Cambridge, England, UK The Springer Undergraduate Mathematics Series (SUMS) is a series designed for undergraduates in mathematics and the sciences worldwide. From core founda- tional material to final year topics, SUMS books take a fresh and modern approach. Textual explanations are supported by a wealth of examples, problems and fully-worked solutions, with particular attention paid to universal areas of difficulty. These practical and concise texts are designed for a one- or two-semester course but the self-study approach makes them ideal for independent use. More information about this series at http://www.springer.com/series/3423 Jean-Baptiste Hiriart-Urruty Mathematical Tapas Volume 1 (for Undergraduates) 123 Jean-Baptiste Hiriart-Urruty Institut de mathématiques Université Paul Sabatier Toulouse Cedex 9 France ISSN 1615-2085 ISSN 2197-4144 (electronic) Springer Undergraduate Mathematics Series ISBN 978-3-319-42185-8 ISBN 978-3-319-42186-5 (eBook) DOI 10.1007/978-3-319-42186-5 Library of Congress Control Number: 2016945771 Mathematics Subject Classification (2010): 00A07, 26Axx, 26Bxx, 26Dxx, 15-XX, 52Axx, 90Cxx © Springer International Publishing Switzerland 2016 This work is subject to copyright.
    [Show full text]
  • Course Notes for Calculus IV
    Course Notes for Calculus IV Remkes Kooistra The King's University February 21, 2019 Contents 1 Introduction 1 2 Integration 2 2.1 The Indefinite Integral . 2 2.2 The Definite Integral . 3 2.3 Iterated Integrals . 6 2.4 Integration over Arbitrary Sets . 9 2.5 Change of Variables . 17 2.6 Polar Coordinates . 24 2.7 Cylindrical and Spherical Coordinates . 27 2.8 Applications of Multiple Integration . 31 3 Vector Calculus 47 3.1 Vector Fields . 47 3.2 Integral Curves . 49 3.3 Vector Operations on Vector Fields . 52 3.4 Conservative Vector Fields . 56 3.5 Line Integrals of Vector Fields . 57 3.6 Parametric Surfaces . 63 3.7 Surface Integrals . 64 3.8 Flux Integrals . 66 3.9 The Major Results . 69 3.10 Vector Calculus and Maxwell's Equations . 76 1 4 Differential Forms and Differential Geometry 81 4.1 Manifolds and Coordinate Functions . 81 4.2 Functions on Manifolds . 83 4.3 Differential Forms . 83 4.4 Pullbacks . 88 4.5 Integration of Differential Forms . 89 4.6 Stokes' Theorem . 91 2 License This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. Chapter 1 Introduction Calculus III was primarily devoted to extending the derivative to multivariable scalar functions. This course starts with the parallel program of extending the integral to multivariable scalar functions. Unlike the derivative, the definition of the multivariable (definite) integral is relativley similar to the single variable case. The challenge of integration is in the geometry of the subsets of Rn involved in the integration.
    [Show full text]
  • Sri Chaitanya IIT Academy, India
    GRAND TEST-6 Sri Chaitanya IIT Academy, India A.P, TELANGANA, KARNATAKA, TAMILNADU, MAHARASHTRA, DELHI, RANCHI A right Choice for the Real Aspirant ICON CENTRAL OFFICE, MADHAPUR-HYD Sec: Sr.IPLCO/IC/ISB/LIIT Dt: 24-04-16 Time: 09:00 AM to 12:00 Noon Max.Marks: 222 Name of the Student: ___________________ I.D. NO: PAPER-I 24-04-16_ Sr.IPLCO/IC/ISB/LIIT_GTA-6_Weekend Syllabus Mathematics : Total Syllabus Physics : Total Syllabus Chemistry : Total Syllabus Sri Chaitanya IIT Academy 24-04-16_ Sr.IPLCO/IC/ISB/LIIT _JEE-ADV_(New Model-3_P1)_GTA-6_Q’Paper JEE-ADVANCED-New Model -3–P1 Time: 09.00 AM to 12.00 Noon IMPORTANT INSTRUCTIONS Max Marks: 222 PHYSICS: +Ve - Ve No.of Total Section Question Type Marks Marks Qs marks Sec – I(Q.N : 1 – 10) Questions with Single Correct Choice 4 -2 10 40 Questions with Comprehension Type Integer Sec – II(Q.N : 11 – 16) (3 Comprehensions – 2 +2+2 = 6Q) 3 -1 6 18 Sec – III(Q.N : 17 – 20) Matrix Matching Type 4 -2 4 16 Total 20 74 MATHEMATICS: +Ve - Ve No.of Total Section Question Type Marks Marks Qs marks Sec – I(Q.N : 21 – 30) Questions with Single Correct Choice 4 -2 10 40 Questions with Comprehension Type Integer Sec – II(Q.N : 31 – 36) (3 Comprehensions – 2 +2+2 = 6Q) 3 -1 6 18 Sec – III(Q.N : 37 – 40) Matrix Matching Type 4 -2 4 16 Total 20 74 CHEMISTRY: +Ve - Ve No.of Total Section Question Type Marks Marks Qs marks Sec – I(Q.N : 41 – 50) Questions with Single Correct Choice 4 -2 10 40 Questions with Comprehension Type Integer Sec – II(Q.N : 51 – 56) (3 Comprehensions – 2 +2+2 = 6Q) 3 -1 6 18 Sec – III(Q.N : 57 – 60) Matrix Matching Type 4 -2 4 16 Total 20 74 Sec: Sr.IPLCO/IC/ISB/LIIT space for rough work Page 2 Sri Chaitanya IIT Academy 24-04-16_ Sr.IPLCO/IC/ISB/LIIT _JEE-ADV_(New Model-3_P1)_GTA-6_Q’Paper PHYSICS Max Marks : 74 Section-1 (Only one Option correct Type) This section contains 10 Multiple Choice questions.
    [Show full text]
  • MATH2070: LAB 10: Quadrature 1 Introduction 2 Matlab Hint
    MATH2070: LAB 10: Quadrature Introduction Exercise 1 Matlab hint Exercise 2 The Midpoint Method Exercise 3 Reporting Errors Exercise 4 Exactness Exercise 5 The Trapezoid Method Exercise 6 Singular Integrals Exercise 7 Newton-Cotes Rules Exercise 8 Gauss Quadrature Exercise 9 Adaptive quadrature Exercise 10 Integration by Monte Carlo methods (Extra) Exercise 11 Exercise 12 Exercise 13 Exercise 14 Exercise 15 Extra Credit 1 Introduction The term \numerical quadrature" refers to the estimation of an area, or, more generally, any integral. (You might hear the term \cubature" referring to estimating a volume in three dimensions, but most people use the term \quadrature.") We might want to integrate some function f(x) or a set of tabulated data. The domain might be a finite or infinite interval, it might be a rectangle or irregular shape, it might be a multi-dimensional volume. We first discuss the \degree of exactness" (sometimes called the \degree of precision") of a quadrature rule, and its relation to the \order of accuracy." We consider some simple tests to measure the degree of exactness and the order of accuracy of a rule, and then describe (simple versions of) the midpoint and trapezoidal rules. Then we consider the Newton-Cotes and Gauss-Legendre families of rules, and discuss how to get more accurate approximations by either increasing the order of the rule or subdividing the interval. Finally, we will consider a way of adapting the details of the integration method to meet a desired error estimate. In the majority of the exercises below, we will be more interested in the error (difference between calculated and known value) than in the calculated value itself.
    [Show full text]