Accelerator Physics. G. A. Krafft Jefferson Lab Old Dominion University Lecture 5

 Bertina Bailey
 11 months ago
 Views:
Transcription
1 Accelertor Phyic G. A. Krfft Jefferon L Old Dominion Univerity Lecture 5 ODU Accelertor Phyic Spring 15
2 Inhomogeneou Hill Eqution Fundmentl trnvere eqution of motion in prticle ccelertor for mll devition from deign trjectory d d y 1 B 1 p d B p 1 B 1 p y d y B y p ρ rdiu of curvture for end, B' trnvere field grdient for mgnet tht focu (poitive correpond to horizontl focuing), Δp/p momentum devition from deign momentum. Homogeneou eqution i nd order liner ordinry differentil eqution. ODU Accelertor Phyic Spring 15
3 Diperion From theory of liner ordinry differentil eqution, the generl olution to the inhomogeneou eqution i the um of ny olution to the inhomogeneou eqution, clled the prticulr integrl, plu two linerly independent olution to the homogeneou eqution, whoe mplitude my e djuted to ccount for oundry condition on the prolem. = = A B y y A y B y p 1 p y 1 y Becue the inhomogeneou term re proportionl to Δp/p, the prticulr olution cn generlly e written p D p yp Dy p where the diperion function tify = = d B d B dd B ddy B D Dy y y p p ODU Accelertor Phyic Spring 15
4 M 56 In ddition to the trnvere effect of the diperion, there re importnt effect of the diperion long the direction of motion. The primry effect i to chnge the timeofrrivl of the offmomentum prticle compred to the onmomentum prticle which trvere the deign trjectory. = d z D p p d d p z D d p d D p p Deign Trjectory Dipered Trjectory M 56 1 D D y y d ODU Accelertor Phyic Spring 15
5 Dipole Solution Homogeneou Eqn. co / in / i i i d d in / / co / i i i d d Drift i 1 i d d 1 i d d ODU Accelertor Phyic Spring 15
6 Qudrupole in the focuing direction k B/ B i i co k in k / k i d d k in k co i i k i d d Thin Focuing Len (limiting ce when rgument goe to zero!) 1 d d 1/ f 1 d d Thin Defocuing Len: chnge ign of f ODU Accelertor Phyic Spring 15
7 Trnfer Mtrice Dipole with end Θ (put coordinte of finl poition in olution) fter efore co in d in / co d fter efore d d Drift fter efore 1 Ldrift d 1 d fter efore d d ODU Accelertor Phyic Spring 15
8 Qudrupole in the focuing direction length L Qudrupole in the defocuing direction length L fter co k L in k L / k efore d in co d fter k k L k L efore d d fter coh k L inh k L / k efore d inh co d fter k k L k L efore d d Wille: pg. 71 ODU Accelertor Phyic Spring 15
9 Thin Lene f f Thin Focuing Len (limiting ce when rgument goe to zero!) len len 1 d d 1/ f 1 len len d d Thin Defocuing Len: chnge ign of f ODU Accelertor Phyic Spring 15
10 Compoition Rule: Mtri Multipliction! Element 1 Element 1 Rememer: Firt element frthet RIGHT 1 M1 1 More generlly 1 M 1 MM 1 M M M... M M tot N N 1 1 ODU Accelertor Phyic Spring 15
11 Some Geometry of Ellipe Eqution for n upright ellipe y y 1 In em optic, the eqution for ellipe re normlized (y multipliction of the ellipe eqution y ) o tht the re of the ellipe divided y π pper on the RHS of the defining eqution. For generl ellipe A By Cy D ODU Accelertor Phyic Spring 15
12 The re i eily computed to e Are AC D B Eqn. (1) So the eqution i equivlently y y AC A B, AC B B, nd AC C B ODU Accelertor Phyic Spring 15
13 When normlized in thi mnner, the eqution coefficient clerly tify 1 Emple: the defining eqution for the upright ellipe my e rewritten in following uggetive wy y β = / nd γ = /, note, m y m ODU Accelertor Phyic Spring 15
14 Generl Tilted Ellipe Need 3 prmeter for complete decription. One wy y y= y where i lope prmeter, i the mimum etent in the direction, nd the yintercept occur t ±, nd gin ε i the re of the ellipe divided y π 1 y y ODU Accelertor Phyic Spring 15
15 ODU Accelertor Phyic Spring 15 Identify,, 1 Note tht βγ α = 1 utomticlly, nd tht the eqution for ellipe ecome y y eliminting the (redundnt!) prmeter γ
16 Ellipe Dimenion in the βfunction Decription y, y== α / β /, A for the upright ellipe m, y m Wille: pge 81 ODU Accelertor Phyic Spring 15
17 Are Theorem for Liner Optic Under generl liner trnformtion ' M y' M 11 1 y n ellipe i trnformed into nother ellipe. Furthermore, if det (M) = 1, the re of the ellipe fter the trnformtion i the me tht efore the trnformtion. M M 1 Pf: Let the initil ellipe, normlized ove, e y y ODU Accelertor Phyic Spring 15
18 Let the finl ellipe e y y Effect of Trnformtion, y y y The trnformed coordinte mut olve thi eqution. M 1 M y, The trnformed coordinte mut lo olve the initil eqution trnformed. y y ' M11 M1 y' M1 M y 1 1 M11 M y M1 M y ODU Accelertor Phyic Spring 15
19 Becue 1 1 M M 11 1 ' y 1 1 M M y' 1 The trnformed initil ellipe i y y M M M M M M M M M M M M M M M M 1 1 ODU Accelertor Phyic Spring 15
20 Becue (verify!) M M M M 1 1 M M 1 1 M M det M the re of the trnformed ellipe (divided y π) i, y Eqn. (1) Are det M 1 det M ODU Accelertor Phyic Spring 15
21 ODU Accelertor Phyic Spring 15 Tilted ellipe from the upright ellipe In the tilted ellipe the ycoordinte i ried y the lope with repect to the untilted ellipe y y 1 1 ' ',, 1 1,,, M Becue det (M)=1, the tilted ellipe h the me re the upright ellipe, i.e., ε = ε.
22 Phe Advnce of Unimodulr Mtri Any twoytwo unimodulr (Det (M) = 1) mtri with Tr M < cn e written in the form M 1 co 1 in The phe dvnce of the mtri, μ, give the eigenvlue of the mtri λ = e ±iμ, nd co μ = (Tr M)/. Furthermore βγ α =1 Pf: The eqution for the eigenvlue of M i M M 1 11 ODU Accelertor Phyic Spring 15
23 Becue M i rel, oth λ nd λ* re olution of the qudrtic. Becue Tr M i 1 TrM / For Tr M <, λ λ* =1 nd o λ 1, = e ±iμ. Conequently co μ = (Tr M)/. Now the following mtri i trcefree. M 1 co 1 M 11 M M 1 M M 1 M 11 ODU Accelertor Phyic Spring 15
24 Simply chooe M M in, M1, in 11 M 1 in nd the ign of μ to properly mtch the individul mtri element with β >. It i eily verified tht βγ α = 1. Now M M n 1 nd more generlly 1 co 1 co 1 in n inn ODU Accelertor Phyic Spring 15
25 Therefore, ecue in nd co re oth ounded function, the mtri element of ny power of M remin ounded long Tr (M) <. NB, in ome em dynmic literture it i (incorrectly!) tted tht the le tringent Tr (M) enure oundedne nd/or tility. Tht equlity cnnot e llowed cn e immeditely demontrted y counteremple. The upper tringulr or lower tringulr ugroup of the twoytwo unimodulr mtrice, i.e., mtrice of the form clerly hve unounded power if i not equl to. or ODU Accelertor Phyic Spring 15
Matrix Eigenvalues and Eigenvectors September 13, 2017
Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues
More informationSTABILITY and RouthHurwitz Stability Criterion
Krdeniz Technicl Univerity Deprtment of Electricl nd Electronic Engineering 6080 Trbzon, Turkey Chpter 8 nd RouthHurwitz Stbility Criterion Bu der notlrı dece bu deri ln öğrencilerin kullnımın çık olup,
More informationINTRODUCTION TO LINEAR ALGEBRA
ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More informationu(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.
Lecture 4 Complex Integrtion MATHGA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More information8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.
8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd GussJordn elimintion to solve systems of liner
More informationChapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY
Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in
More information44 Efield Calculations using Coulomb s Law
1/11/5 ection_4_4_efield_clcultion_uing_coulomb_lw_empty.doc 1/1 44 Efield Clcultion uing Coulomb Lw Reding Aignment: pp. 998 Specificlly: 1. HO: The Uniform, Infinite Line Chrge. HO: The Uniform Dik
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationMATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC
FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot
More informationMath 270A: Numerical Linear Algebra
Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner
More informationECON 331 Lecture Notes: Ch 4 and Ch 5
Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve
More informationPartial Differential Equations
Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for OneDimensionl Eqution The reen s function provides complete solution to boundry
More informationMath 100 Review Sheet
Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s
More informationSolutions Problem Set 2. Problem (a) Let M denote the DFA constructed by swapping the accept and nonaccepting state in M.
Solution Prolem Set 2 Prolem.4 () Let M denote the DFA contructed y wpping the ccept nd nonccepting tte in M. For ny tring w B, w will e ccepted y M, tht i, fter conuming the tring w, M will e in n ccepting
More informationConducting Ellipsoid and Circular Disk
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More informationProblemSolving Companion
ProblemSolving Compnion To ccompny Bic Engineering Circuit Anlyi Eight Edition J. Dvid Irwin Auburn Univerity JOHN WILEY & SONS, INC. Executive Editor Bill Zobrit Aitnt Editor Kelly Boyle Mrketing Mnger
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More informationAnalytically, vectors will be represented by lowercase boldface Latin letters, e.g. a, r, q.
1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples
More informationHomework Assignment 3 Solution Set
Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More informationCSCI 5525 Machine Learning
CSCI 555 Mchine Lerning Some Deini*ons Qudrtic Form : nn squre mtri R n n : n vector R n the qudrtic orm: It is sclr vlue. We oten implicitly ssume tht is symmetric since / / I we write it s the elements
More informationP 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
More information1 From NFA to regular expression
Note 1: How to convert DFA/NFA to regulr expression Version: 1.0 S/EE 374, Fll 2017 Septemer 11, 2017 In this note, we show tht ny DFA cn e converted into regulr expression. Our construction would work
More information8 factors of x. For our second example, let s raise a power to a power:
CH 5 THE FIVE LAWS OF EXPONENTS EXPONENTS WITH VARIABLES It s no time for chnge in tctics, in order to give us deeper understnding of eponents. For ech of the folloing five emples, e ill stretch nd squish,
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationApproximation of continuoustime systems with discretetime systems
Approximtion of continuoutime ytem with icretetime ytem he continuoutime ytem re replce by icretetime ytem even for the proceing of continuoutime ignl.. Impule invrince metho 2. Step invrince metho
More informationLaplace s equation in Cylindrical Coordinates
Prof. Dr. I. Ner Phy 571, T131 Oct13 Lplce eqution in Cylindricl Coordinte 1 Circulr cylindricl coordinte The circulr cylindricl coordinte (, φ, z ) re relted to the rectngulr Crtein coordinte ( x,
More information9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes
The Vector Product 9.4 Introduction In this section we descrie how to find the vector product of two vectors. Like the sclr product its definition my seem strnge when first met ut the definition is chosen
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More informationdx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.
Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More informationThe Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5
The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle
More informationApplied Physics Introduction to Vibrations and Waves (with a focus on elastic waves) Course Outline
Applied Physics Introduction to Vibrtions nd Wves (with focus on elstic wves) Course Outline Simple Hrmonic Motion && + ω 0 ω k /m k elstic property of the oscilltor Elstic properties of terils Stretching,
More informationHomework Solution  Set 5 Due: Friday 10/03/08
CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution  et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte nonfinl.
More informationEnergy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon
Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,
More informationThe Basic Functional 2 1
2 The Bsic Functionl 2 1 Chpter 2: THE BASIC FUNCTIONAL TABLE OF CONTENTS Pge 2.1 Introduction..................... 2 3 2.2 The First Vrition.................. 2 3 2.3 The Euler Eqution..................
More informationTransfer Functions. Chapter 5. Transfer Functions. Derivation of a Transfer Function. Transfer Functions
5/4/6 PM : Trnfer Function Chpter 5 Trnfer Function Defined G() = Y()/U() preent normlized model of proce, i.e., cn be ued with n input. Y() nd U() re both written in devition vrible form. The form of
More informationLINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nthorder
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationModule 9: The Method of Green s Functions
Module 9: The Method of Green s Functions The method of Green s functions is n importnt technique for solving oundry vlue nd, initil nd oundry vlue prolems for prtil differentil equtions. In this module,
More informationMATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35
MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.
More informationLinear Systems with Constant Coefficients
Liner Systems with Constnt Coefficients 4305 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system
More informationCHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx
CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twicedifferentile function of x, then t
More informationPreSession Review. Part 1: Basic Algebra; Linear Functions and Graphs
PreSession Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationMT Integral equations
MT58  Integrl equtions Introduction Integrl equtions occur in vriety of pplictions, often eing otined from differentil eqution. The reson for doing this is tht it my mke solution of the prolem esier or,
More informationAN020. a a a. cos. cos. cos. Orientations and Rotations. Introduction. Orientations
AN020 Orienttions nd Rottions Introduction The fct tht ccelerometers re sensitive to the grvittionl force on the device llows them to be used to determine the ttitude of the sensor with respect to the
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationOrdinary differential equations
Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview IntroductionModelling Bsic concepts to understnd n ODE. Description nd properties
More information. The set of these fractions is then obviously Q, and we can define addition and multiplication on it in the expected way by
50 Andre Gthmnn 6. LOCALIZATION Locliztion i very powerful technique in commuttive lgebr tht often llow to reduce quetion on ring nd module to union of mller locl problem. It cn eily be motivted both from
More informationMATH 423 Linear Algebra II Lecture 28: Inner product spaces.
MATH 423 Liner Algebr II Lecture 28: Inner product spces. Norm The notion of norm generlizes the notion of length of vector in R 3. Definition. Let V be vector spce over F, where F = R or C. A function
More informationChapter 2. Determinants
Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if dbc0. The expression dbc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is
More informationDeterminants Chapter 3
Determinnts hpter Specil se : x Mtrix Definition : the determinnt is sclr quntity defined for ny squre n x n mtrix nd denoted y or det(). x se ecll : this expression ppers in the formul for x mtrix inverse!
More informationFORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81
FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first
More informationChapter 6 Continuous Random Variables and Distributions
Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete
More informationECE 451 Automated Microwave Measurements. TRL Calibration
utomted Microwve Mesurements Clibrtion Jose E. Schuttine Electricl & Computer Engineering University of Illinois jose@emlb.uiuc.edu Copyright by Jose E. Schutt ine, ll ights eserved Coxil Microstrip rnsition
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More informationOverview. Before beginning this module, you should be able to: After completing this module, you should be able to:
Module.: Differentil Equtions for First Order Electricl Circuits evision: My 26, 2007 Produced in coopertion with www.digilentinc.com Overview This module provides brief review of time domin nlysis of
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10
University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin
More informationMagnetic forces on a moving charge. EE Lecture 26. Lorentz Force Law and forces on currents. Laws of magnetostatics
Mgnetic forces on moving chrge o fr we ve studied electric forces between chrges t rest, nd the currents tht cn result in conducting medium 1. Mgnetic forces on chrge 2. Lws of mgnetosttics 3. Mgnetic
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationENGI 3424 Engineering Mathematics Five Tutorial Examples of Partial Fractions
ENGI 44 Engineering Mthemtics Five Tutoril Exmples o Prtil Frctions 1. Express x in prtil rctions: x 4 x 4 x 4 b x x x x Both denomintors re liner nonrepeted ctors. The coverup rule my be used: 4 4 4
More informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
More informationMATH 174A: PROBLEM SET 5. Suggested Solution
MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous relvlued function on I), nd let L 1 (I) denote the completion
More informationWaveguide Guide: A and V. Ross L. Spencer
Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it
More informationu( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 218, pp 4448): Determine the equation of the following graph.
nlyzing Dmped Oscilltions Prolem (Medor, exmple 218, pp 4448): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $
More informationUSA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year
1/1/21. Fill in the circles in the picture t right with the digits 18, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationDETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ
All Mthemticl truths re reltive nd conditionl. C.P. STEINMETZ 4. Introduction DETERMINANTS In the previous chpter, we hve studied bout mtrices nd lgebr of mtrices. We hve lso lernt tht system of lgebric
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationSturmLiouville Theory
LECTURE 1 SturmLiouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory
More informationMPE Review Section I: Algebra
MPE Review Section I: lger t Colordo Stte Universit, the College lger sequence etensivel uses the grphing fetures of the Tes Instruments TI8 or TI8 grphing clcultor. Whenever possile, the questions on
More informationSTURMLIOUVILLE BOUNDARY VALUE PROBLEMS
STURMLIOUVILLE BOUNDARY VALUE PROBLEMS Throughout, we let [, b] be bounded intervl in R. C 2 ([, b]) denotes the spce of functions with derivtives of second order continuous up to the endpoints. Cc 2
More informationPhys 7221, Fall 2006: Homework # 6
Phys 7221, Fll 2006: Homework # 6 Gbriel González October 29, 2006 Problem 37 In the lbortory system, the scttering ngle of the incident prticle is ϑ, nd tht of the initilly sttionry trget prticle, which
More informationAMATH 731: Applied Functional Analysis Fall Some basics of integral equations
AMATH 731: Applied Functionl Anlysis Fll 2009 1 Introduction Some bsics of integrl equtions An integrl eqution is n eqution in which the unknown function u(t) ppers under n integrl sign, e.g., K(t, s)u(s)
More informationThermal Diffusivity. Paul Hughes. Department of Physics and Astronomy The University of Manchester Manchester M13 9PL. Second Year Laboratory Report
Therml iffusivity Pul Hughes eprtment of Physics nd Astronomy The University of nchester nchester 3 9PL Second Yer Lbortory Report Nov 4 Abstrct We investigted the therml diffusivity of cylindricl block
More informationOn the Adders with Minimum Tests
Proceeding of the 5th Ain Tet Sympoium (ATS '97) On the Adder with Minimum Tet Seiji Kjihr nd Tutomu So Dept. of Computer Science nd Electronic, Kyuhu Intitute of Technology Atrct Thi pper conider two
More informationPART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.
PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationChapter 2 Organizing and Summarizing Data. Chapter 3 Numerically Summarizing Data. Chapter 4 Describing the Relation between Two Variables
Copyright 013 Peron Eduction, Inc. Tble nd Formul for Sullivn, Sttitic: Informed Deciion Uing Dt 013 Peron Eduction, Inc Chpter Orgnizing nd Summrizing Dt Reltive frequency = frequency um of ll frequencie
More informationMinimum Energy State of Plasmas with an Internal Transport Barrier
Minimum Energy Stte of Plsms with n Internl Trnsport Brrier T. Tmno ), I. Ktnum ), Y. Skmoto ) ) Formerly, Plsm Reserch Center, University of Tsukub, Tsukub, Ibrki, Jpn ) Plsm Reserch Center, University
More informationCS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)
CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts
More informationChapter 6. Section 6.1. Chapter 6 Opener. Big Ideas Math Blue WorkedOut Solutions. 6.1 Activity (pp ) Try It Yourself (p.
Chpter 6 Opener Try It Yourelf (p. 9). Becue 0 i equl to,.0 i equl to.. 0 So,.0.. i le thn. Becue 8 So,.
More informationFormal languages, automata, and theory of computation
Mälrdlen University TEN1 DVA337 2015 School of Innovtion, Design nd Engineering Forml lnguges, utomt, nd theory of computtion Thursdy, Novemer 5, 14:1018:30 Techer: Dniel Hedin, phone 021107052 The exm
More informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationMTH 505: Number Theory Spring 2017
MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c
More informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More informationThinPlate Splines. Contents
ThinPlte Splines Dvid Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Cretive Commons Attribution 4.0 Interntionl License. To view copy of this
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationalong the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate
L8 VECTOR EQUATIONS OF LINES HL Mth  Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne
More informationMASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS
MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK WRITTEN EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/MCUPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More information