APPLICATIONS OF TENSOR ANALYSIS
|
|
- Alaina Bennett
- 7 years ago
- Views:
Transcription
1 APPLICATIONS OF TENSOR ANALYSIS (formerly titled: Applications of the Absolute Differential Calculus) by A J McCONNELL Dover Publications, Inc, Neiv York
2 CONTENTS PART I ALGEBRAIC PRELIMINARIES/ CHAPTER I NOTATION AND DEFINITIONS 1 The indioial notation * 1 2 The summation convention Addition, multiplication, and contraction of systems 5 4 Symmetric and skew-symmetric systems 6 5 The skew-symmetric three-systems and the Kroneoker deltas - 7 DETERMINANTS 6 The determinant formed by a double system oj The cofactors of the elements in a determinant Linear equations Corresponding formula for the system a mn Positive definite quadratic forma The determinantal equation 16 CHAPTER H TENSOR ANALYSIS 1 Linear transformations IQ 2 Invariants, contravariant and covariant vectors 20 3 Tensors of any order ' Addition, multiplication and contraction of tensors The quotient law of tensors - -» Relative or weighted tensors General functional transformations Tensors with respect to the general functional transformation r - 32 PART II ALGEBRAIC GEOMETRY CHAPTER m RECTILINEAR COORDINATES 1 Coordinates and tensors Contravariant veotors and displacements The unit points and the geometrical interpretation of rectilinear coordinates vii
3 viii CONTENTS 7-4 The distance between two points and the fundamental double' sor The e-systems 5 The angle between two directions; orthogonality - - "-; 6 Associated tensors - - i, 7 Scalar and vector products of vectors "* 8 Areas and volumes, f CHAPTER IV * THE PLANE \ 1/ The equation of a plane " j 2 The perpendicular distance from a point to a plane \i 3 The intersection of two planes '- \ 4 The intersection of three planes 5 Plane coordinates - - ^ 6 Systems of planes ', : 7 The equation of a point - *'' CHAPTER V THE STRAIGHT LINE 1 The point equations of the straight line * ~- 2 The relations of two straight lines "- 3 The six coordinates of ft straight line, ' 4 The plane equations of a straight line - - T j CHAPTER VI THE QUADRIC CONE AND THE CONIC! 1 The equation of a quadrio cone 'I 2 The equation of a conic - -» v - 3 The tangent plane'to a cone \ 4 Poles and polar planes with respect to a cono 6 The canonical equation of a cone The principal axes of a cone 7 The classification of cones ' CHAPTER VII SYSTEMS OF CONES AND CONICS? of 1 The equation of a system of cones with a common vertex - 2 The common polar directions of a family of cones - 3 The canonical forms of the equation 4 The theory of elementary divisors a family of cones - 5 Analytical discrimination of the cases - - -
4 CONTENTS a CHAPTER VHI CENTRAL QUADRICS 1 The point equation of a central quadric The tangential equation of a central quadric - ' Canonical form of the equation of a quadric Principal axes Classification of the central quadrics Confocal quadrics '110 CHAPTER IX THE GENERAL QUADRIC 1 The general equation of a quadric, The centre , The reduction of the equation of a quadrio ' CHAPTER X AFFINE TRANSFORMATIONS 1 Affine transformations The quadric of a transformation Pure strain Rigid body displacements Infinitesimal deformations 126 PART III DIFFERENTIAL GEOMETRY CHAPTER XI CURVILINEAR COORDINATES 1 General coordinate systems Tensor-fields - - ^ The line-element and the metric tensor The e-systems The angle between two directions 136 CHAPTER XH COVARIANT DIFFERENTIATION 1 A parallel field of vectors The Christoffel symbols The intrinsic and covariant derivation of vectors The intrinsic and covariant derivatives of tensors Conservation of the rules of the ordinary differential calculus Ricci's lemma The divergence and curl of a vector The Laplacian The Riemann-Christoffel tensor The Lame relations - 152
5 * CONTENTS - CHAPTER XIII CURVES IN SPACE 1 The tangent vector to a curve Normal vectors The principal normal and binormal The Frenet fsrmulae - - " Parallel vectors along* a curve The straight line "CHAPTER XIV INTRINSIC GEOMETRY OF A SURFACE 1 Curvilinear coordinates on a surface The conventions regarding Greek indices Surface tensors The element of length" and the metrio tensor Directions on a surface Angle between two directions The equations of a geodesic The transformation of the Christoffel symbols Geodesic coordinates /-"' Parallelism with respect to a surface Intrinsic and covariant differentiation of surface tensors The Riemann-Christoffel tensor The Gaussian ourvature of a surface _ The geodesic curvature of a curve on a surface Beltrami's differential parameters ~ Green's theorem on a surface ' [' CHAPTER XV THE FUNDAMENTAL FORMULAE OF A SURFACE 1 Notation - - _ - - : - ; The tangent vectors^ to a surface The first groundform of a surface The normal vector to the; surface ~- - ' jgg 5 The tensor derivation'of-tensors Gauss's formulae 'The second groundform of a surface Weingarten's formulae The'third groundform of a surface The equations of Gauss and Codazzi - - ' 203 ; CHAPTER XVI CURVES ON A SURFACE 1 The equations of a curve on a surface Meusnier's theorem " " ", ' " " The principal ourvatures Gauss s theorem The lines of curvature, ll 5 The asymptotic lines Enneper's formula - _ ' * 6 The geodesic torsion of a'curve on a surface 2 U
6 CONTENTS xi PART IV APPLIED MATHEMATICS ' CHAPTER XVH DYNAMICS OF A PARTICLE 1 The equations of motion W o r k and energy Lagrange's equations of motion Particle on a curve 223,4 Particle on a surface The principle of least action Trajectories as geodesies 228 CHAPTER DYNAMICS OF RIGID BODIES SECTION A RECTILINEAR COORDINATES 1 Moments of Inertia The equations of motion Moving axes Euler's equations SECTION B THE GEOMETRY OF DYNAMICS 4 Generalised coordinates of a dynamical system The equations of motion in generalised coordinates The manifold of configurations The kinematics! line-element The dynamical trajectories of the manifold of configurations The principle of stationary action The action line-element CHAPTER XTX ELECTRICITY AND MAGNETISM 1 Green's theorem Stokes's theorem The electrostatic field Dielectrics The magnetostatic field The electromagnetic equations CHAPTER XX MECHANICS OF CONTINUOUS MEDIA 1 Infinitesimal strain Analysis of stress Equations of motion for a perfect fluid The equations of elasticity The motion of a viscous fluid - 280
7 xii CONTENTS CHAPTER XXI THE SPECIAL THEORY OF RELATIVITY 1 The four-dimensional manifold Generalised coordinates in space-time r, The principle of special relativity The interval and the fundamental quadratic form - " Local coordinate systems and their transformations Relativistic dynamics of a particle Dynamics of a continuous medium The electromagnetic equations APPENDIX ORTHOGONAL OTEVffilNEAR COORDINATES IN MATHEMATICAL PHYSICS 1 The classical notation The physical components of vectors and tensors Dynamics Electricity Elasticity Hydrodynamics 309 BIBLIOGRAPHY INDEX - 31fi
ANALYSIS OF STRUCTURAL MEMBER SYSTEMS JEROME J. CONNOR NEW YORK : ':,:':,;:::::,,:
ANALYSIS OF JEROME J. CONNOR, Sc.D., Massachusetts Institute of Technology, is Professor of Civil Engineering at Massachusetts Institute of Technology. He has been active in STRUCTURAL MEMBER teaching
More informationDifferential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation
Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of
More informationPre-requisites 2012-2013
Pre-requisites 2012-2013 Engineering Computation The student should be familiar with basic tools in Mathematics and Physics as learned at the High School level and in the first year of Engineering Schools.
More informationLecture L6 - Intrinsic Coordinates
S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6 - Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationWAVES AND FIELDS IN INHOMOGENEOUS MEDIA
WAVES AND FIELDS IN INHOMOGENEOUS MEDIA WENG CHO CHEW UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN IEEE PRESS Series on Electromagnetic Waves Donald G. Dudley, Series Editor IEEE Antennas and Propagation Society,
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE
i93 c J SYSTEMS OF CURVES 695 ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE BY C H. ROWE. Introduction. A system of co 2 curves having been given on a surface, let us consider a variable curvilinear
More informationAsymptotic Analysis of Fields in Multi-Structures
Asymptotic Analysis of Fields in Multi-Structures VLADIMIR KOZLOV Department of Mathematics, Linkoeping University, Sweden VLADIMIR MAZ'YA Department of Mathematics, Linkoeping University, Sweden ALEXANDER
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationTHEORETICAL MECHANICS
PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More information14.11. Geodesic Lines, Local Gauss-Bonnet Theorem
14.11. Geodesic Lines, Local Gauss-Bonnet Theorem Geodesics play a very important role in surface theory and in dynamics. One of the main reasons why geodesics are so important is that they generalize
More informationDimension Theory for Ordinary Differential Equations
Vladimir A. Boichenko, Gennadij A. Leonov, Volker Reitmann Dimension Theory for Ordinary Differential Equations Teubner Contents Singular values, exterior calculus and Lozinskii-norms 15 1 Singular values
More informationLessons on Teaching Undergraduate General Relativity and Differential Geometry Courses
Lessons on Teaching Undergraduate General Relativity and Differential Geometry Courses Russell L. Herman and Gabriel Lugo University of North Carolina Wilmington, Wilmington, NC Abstract We describe the
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com
Table of Contents Variable Forces and Differential Equations... 2 Differential Equations... 3 Second Order Linear Differential Equations with Constant Coefficients... 6 Reduction of Differential Equations
More informationMASTER OF SCIENCE IN PHYSICS MASTER OF SCIENCES IN PHYSICS (MS PHYS) (LIST OF COURSES BY SEMESTER, THESIS OPTION)
MASTER OF SCIENCE IN PHYSICS Admission Requirements 1. Possession of a BS degree from a reputable institution or, for non-physics majors, a GPA of 2.5 or better in at least 15 units in the following advanced
More informationDRAFT. Further mathematics. GCE AS and A level subject content
Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed
More informationSCHWEITZER ENGINEERING LABORATORIES, COMERCIAL LTDA.
Pocket book of Electrical Engineering Formulas Content 1. Elementary Algebra and Geometry 1. Fundamental Properties (real numbers) 1 2. Exponents 2 3. Fractional Exponents 2 4. Irrational Exponents 2 5.
More informationHIGH SCHOOL: GEOMETRY (Page 1 of 4)
HIGH SCHOOL: GEOMETRY (Page 1 of 4) Geometry is a complete college preparatory course of plane and solid geometry. It is recommended that there be a strand of algebra review woven throughout the course
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationMean value theorem, Taylors Theorem, Maxima and Minima.
MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and express-ions. Permutations and Combinations.
More informationChapter 7: Polarization
Chapter 7: Polarization Joaquín Bernal Méndez Group 4 1 Index Introduction Polarization Vector The Electric Displacement Vector Constitutive Laws: Linear Dielectrics Energy in Dielectric Systems Forces
More informationMath Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices.
Math Placement Test Study Guide General Characteristics of the Test 1. All items are to be completed by all students. The items are roughly ordered from elementary to advanced. The expectation is that
More informationGravity Field and Dynamics of the Earth
Milan Bursa Karel Pec Gravity Field and Dynamics of the Earth With 89 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest Preface v Introduction 1 1 Fundamentals
More informationSome Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)
Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving
More informationName Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More informationDetermine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s
Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,
More informationTeaching Electromagnetic Field Theory Using Differential Forms
IEEE TRANSACTIONS ON EDUCATION, VOL. 40, NO. 1, FEBRUARY 1997 53 Teaching Electromagnetic Field Theory Using Differential Forms Karl F. Warnick, Richard H. Selfridge, Member, IEEE, and David V. Arnold
More information(Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7)
Chapter 4. Lagrangian Dynamics (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7 4.1 Important Notes on Notation In this chapter, unless otherwise stated, the following
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationGeometric description of the cross product of the vectors u and v. The cross product of two vectors is a vector! u x v is perpendicular to u and v
12.4 Cross Product Geometric description of the cross product of the vectors u and v The cross product of two vectors is a vector! u x v is perpendicular to u and v The length of u x v is uv u v sin The
More informationRARITAN VALLEY COMMUNITY COLLEGE ACADEMIC COURSE OUTLINE MATH 251 CALCULUS III
RARITAN VALLEY COMMUNITY COLLEGE ACADEMIC COURSE OUTLINE MATH 251 CALCULUS III I. Basic Course Information A. Course Number and Title: MATH 251 Calculus III B. New or Modified Course: Modified Course C.
More informationMATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!
MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Pre-algebra Algebra Pre-calculus Calculus Statistics
More informationRAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A
RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:
More informationElasticity Theory Basics
G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold
More informationUNIVERSITY OF PUNE, PUNE 411007. BOARD OF STUDIES IN MATHEMATICS SYLLABUS
UNIVERSITY OF PUNE, PUNE 411007. BOARD OF STUDIES IN MATHEMATICS SYLLABUS F.Y.B.Sc (MATHEMATICS) PAPER 1 ALGEBRA AND GEOMETRY FIRST TERM 1) Sets (4 Lectures) 1.1 Power set of a set, Product of two sets.
More informationMetrics on SO(3) and Inverse Kinematics
Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction
More informationOn Motion of Robot End-Effector using the Curvature Theory of Timelike Ruled Surfaces with Timelike Directrix
Malaysian Journal of Mathematical Sciences 8(2): 89-204 (204) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal On Motion of Robot End-Effector using the Curvature
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationFRACTIONAL INTEGRALS AND DERIVATIVES. Theory and Applications
FRACTIONAL INTEGRALS AND DERIVATIVES Theory and Applications Stefan G. Samko Rostov State University, Russia Anatoly A. Kilbas Belorussian State University, Minsk, Belarus Oleg I. Marichev Belorussian
More informationMathematics I, II and III (9465, 9470, and 9475)
Mathematics I, II and III (9465, 9470, and 9475) General Introduction There are two syllabuses, one for Mathematics I and Mathematics II, the other for Mathematics III. The syllabus for Mathematics I and
More informationVector Calculus: a quick review
Appendi A Vector Calculus: a quick review Selected Reading H.M. Sche,. Div, Grad, Curl and all that: An informal Tet on Vector Calculus, W.W. Norton and Co., (1973). (Good phsical introduction to the subject)
More informationExtrinsic geometric flows
On joint work with Vladimir Rovenski from Haifa Paweł Walczak Uniwersytet Łódzki CRM, Bellaterra, July 16, 2010 Setting Throughout this talk: (M, F, g 0 ) is a (compact, complete, any) foliated, Riemannian
More information12.510 Introduction to Seismology Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 04/30/2008 Today s
More informationGymnázium, Brno, Slovanské nám. 7, SCHEME OF WORK Mathematics SCHEME OF WORK. http://agb.gymnaslo. cz
SCHEME OF WORK Subject: Mathematics Year: Third grade, 3.X School year:../ List of topics Topics Time period 1. Revision (functions, plane geometry) September 2. Constructive geometry in the plane October
More informationVectors and Tensors in Engineering Physics
Module Description Vectors and Tensors in Engineering Physics General Information Number of ECTS Credits 3 Abbreviation FTP_Tensors Version 19.02.2015 Responsible of module Christoph Meier, BFH Language
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More informationDistinguished Professor George Washington University. Graw Hill
Mechanics of Fluids Fourth Edition Irving H. Shames Distinguished Professor George Washington University Graw Hill Boston Burr Ridge, IL Dubuque, IA Madison, Wl New York San Francisco St. Louis Bangkok
More informationSPECIFICATION. Mathematics 6360 2014. General Certificate of Education
Version 1.0: 0913 General Certificate of Education Mathematics 6360 014 Material accompanying this Specification Specimen and Past Papers and Mark Schemes Reports on the Examination Teachers Guide SPECIFICATION
More informationScalars, Vectors and Tensors
Scalars, Vectors and Tensors A scalar is a physical quantity that it represented by a dimensional number at a particular point in space and time. Examples are hydrostatic pressure and temperature. A vector
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More informationMathematics Notes for Class 12 chapter 10. Vector Algebra
1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is non-negative
More informationFINAL EXAM SOLUTIONS Math 21a, Spring 03
INAL EXAM SOLUIONS Math 21a, Spring 3 Name: Start by printing your name in the above box and check your section in the box to the left. MW1 Ken Chung MW1 Weiyang Qiu MW11 Oliver Knill h1 Mark Lucianovic
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationA Correlation of Pearson Texas Geometry Digital, 2015
A Correlation of Pearson Texas Geometry Digital, 2015 To the Texas Essential Knowledge and Skills (TEKS) for Geometry, High School, and the Texas English Language Proficiency Standards (ELPS) Correlations
More informationFixed Point Theory. With 14 Illustrations. %1 Springer
Andrzej Granas James Dugundji Fixed Point Theory With 14 Illustrations %1 Springer Contents Preface vii 0. Introduction 1 1. Fixed Point Spaces 1 2. Forming New Fixed Point Spaces from Old 3 3. Topological
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationSpecial Theory of Relativity
June 1, 2010 1 1 J.D.Jackson, Classical Electrodynamics, 3rd Edition, Chapter 11 Introduction Einstein s theory of special relativity is based on the assumption (which might be a deep-rooted superstition
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More information376 CURRICULUM AND SYLLABUS for Classes XI & XII
376 CURRICULUM AND SYLLABUS for Classes XI & XII MATHEMATICS CLASS - XI One Paper Time : 3 Hours 100 Marks Units Unitwise Weightage Marks Periods I. Sets Relations and Functions [9 marks] 1. Sets Relations
More informationWorksheet to Review Vector and Scalar Properties
Worksheet to Review Vector and Scalar Properties 1. Differentiate between vectors and scalar quantities 2. Know what is being requested when the question asks for the magnitude of a quantity 3. Define
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More informationExample SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross
CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal
More informationMATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationVectors Math 122 Calculus III D Joyce, Fall 2012
Vectors Math 122 Calculus III D Joyce, Fall 2012 Vectors in the plane R 2. A vector v can be interpreted as an arro in the plane R 2 ith a certain length and a certain direction. The same vector can be
More informationBending Stress in Beams
936-73-600 Bending Stress in Beams Derive a relationship for bending stress in a beam: Basic Assumptions:. Deflections are very small with respect to the depth of the beam. Plane sections before bending
More informationStructural Axial, Shear and Bending Moments
Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More informationJUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson
JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3
More informationIncenter Circumcenter
TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is
More informationCONTINUUM MECHANICS. (Lecture Notes) Zdeněk Martinec
CONTINUUM MECHANICS (Lecture Notes) Zdeněk Martinec Department of Geophysics Faculty of Mathematics and Physics Charles University in Prague V Holešovičkách 2, 180 00 Prague 8 Czech Republic e-mail: zm@karel.troja.mff.cuni.cz
More informationUnit 3 (Review of) Language of Stress/Strain Analysis
Unit 3 (Review of) Language of Stress/Strain Analysis Readings: B, M, P A.2, A.3, A.6 Rivello 2.1, 2.2 T & G Ch. 1 (especially 1.7) Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering
More informationMathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 Pre-Algebra 4 Hours
MAT 051 Pre-Algebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT
More informationJEE (Main) 2016. A Detailed Analysis by Resonance
JEE (Main) 2016 JEE (Main)-2016 A Detailed by Resonance On 3-April-2016, JEE (Main) 2016 was conducted for students opting to take this exam by offline mode. This is the fourth edition of this Exam after
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationJEE (Main) 2014. A Detailed Analysis by Resonance
JEE (Main) 2014 A Detailed by Resonance OVERALL MARKS DISTRIBUTION OVERALL DIFFICULTY LEVEL ANALYSIS Difficulty Level Overall Physics Mathematics Chemistry 1.70 1.74 1.76 1.76 1.66 1.68 1.70 1.72 1.74
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationStress Analysis, Strain Analysis, and Shearing of Soils
C H A P T E R 4 Stress Analysis, Strain Analysis, and Shearing of Soils Ut tensio sic vis (strains and stresses are related linearly). Robert Hooke So I think we really have to, first, make some new kind
More informationIn order to describe motion you need to describe the following properties.
Chapter 2 One Dimensional Kinematics How would you describe the following motion? Ex: random 1-D path speeding up and slowing down In order to describe motion you need to describe the following properties.
More informationKinematics of Robots. Alba Perez Gracia
Kinematics of Robots Alba Perez Gracia c Draft date August 31, 2007 Contents Contents i 1 Motion: An Introduction 3 1.1 Overview.......................................... 3 1.2 Introduction.........................................
More informationPCHS ALGEBRA PLACEMENT TEST
MATHEMATICS Students must pass all math courses with a C or better to advance to the next math level. Only classes passed with a C or better will count towards meeting college entrance requirements. If
More informationFLUID MECHANICS IM0235 DIFFERENTIAL EQUATIONS - CB0235 2014_1
COURSE CODE INTENSITY PRE-REQUISITE CO-REQUISITE CREDITS ACTUALIZATION DATE FLUID MECHANICS IM0235 3 LECTURE HOURS PER WEEK 48 HOURS CLASSROOM ON 16 WEEKS, 32 HOURS LABORATORY, 112 HOURS OF INDEPENDENT
More informationm i: is the mass of each particle
Center of Mass (CM): The center of mass is a point which locates the resultant mass of a system of particles or body. It can be within the object (like a human standing straight) or outside the object
More informationGradient, Divergence and Curl in Curvilinear Coordinates
Gradient, Divergence and Curl in Curvilinear Coordinates Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found more convenient to work with other coordinate systems.
More informationNumber Sense and Operations
Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents
More informationNotes on the representational possibilities of projective quadrics in four dimensions
bacso 2006/6/22 18:13 page 167 #1 4/1 (2006), 167 177 tmcs@inf.unideb.hu http://tmcs.math.klte.hu Notes on the representational possibilities of projective quadrics in four dimensions Sándor Bácsó and
More informationPlates and Shells: Theory and Computation - 4D9 - Dr Fehmi Cirak (fc286@) Office: Inglis building mezzanine level (INO 31)
Plates and Shells: Theory and Computation - 4D9 - Dr Fehmi Cirak (fc286@) Office: Inglis building mezzanine level (INO 31) Outline -1-! This part of the module consists of seven lectures and will focus
More informationMATHEMATICAL METHODS OF STATISTICS
MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS
More informationKyu-Jung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A.
MECHANICS: STATICS AND DYNAMICS Kyu-Jung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A. Keywords: mechanics, statics, dynamics, equilibrium, kinematics,
More informationApplied Linear Algebra
Applied Linear Algebra OTTO BRETSCHER http://www.prenhall.com/bretscher Chapter 7 Eigenvalues and Eigenvectors Chia-Hui Chang Email: chia@csie.ncu.edu.tw National Central University, Taiwan 7.1 DYNAMICAL
More informationAN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc.,
More informationGAME ENGINE DESIGN. A Practical Approach to Real-Time Computer Graphics. ahhb. DAVID H. EBERLY Geometrie Tools, Inc.
3D GAME ENGINE DESIGN A Practical Approach to Real-Time Computer Graphics SECOND EDITION DAVID H. EBERLY Geometrie Tools, Inc. ahhb _ jfw H NEW YORK-OXFORD-PARIS-SAN DIEGO fl^^h ' 4M arfcrgsbjlilhg, SAN
More information