Applied Partial Differential Equations with Fourier Series and Boundary Value Problems 5th Edition Richard Haberman
|
|
- Bernadette Bates
- 5 years ago
- Views:
Transcription
1 Applied Prtil Differentil Equtions with Fourier Series nd Boundry Vlue Problems 5th Edition Richrd Hbermn
2 Person Eduction Limited Edinburgh Gte Hrlow Essex CM20 2JE Englnd nd Associted Compnies throughout the world Visit us on the World Wide Web t: Person Eduction Limited 2014 All rights reserved. No prt of this publiction my be reproduced, stored in retrievl system, or trnsmitted in ny form or by ny mens, electronic, mechnicl, photocopying, recording or otherwise, without either the prior written permission of the publisher or licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Sffron House, 6 10 Kirby Street, London EC1N 8TS. All trdemrks used herein re the property of their respective owners. The use of ny trdemrk in this text does not vest in the uthor or publisher ny trdemrk ownership rights in such trdemrks, nor does the use of such trdemrks imply ny ffilition with or endorsement of this book by such owners. ISBN 10: X ISBN 10: ISBN 13: ISBN 13: British Librry Ctloguing-in-Publiction Dt A ctlogue record for this book is vilble from the British Librry Printed in the United Sttes of Americ
3 If we directly pply (5.30), then Sturm Liouville Eigenvlue Problems (λ 2 λ 1 )u v =0. Thus, if λ 1 λ 2 (different eigenvlues), the corresponding eigenvectors re orthogonl in the sense tht u v =0. (5.31) We leve s n exercise the proof tht the eigenvlues of symmetric rel mtrix re rel. EXAMPLE 6 2 The eigenvlues of the rel symmetric mtrix re determined from (6 λ) 2 3 (3 λ) 4=λ 2 9λ +14=(λ 7)(λ 2) = 0. For λ = 2, the eigenvector stisfies nd hence x1 x 2 6x 1 +2x 2 =2x 1 nd 2x 1 +3x 2 =2x 2, 1 = x 1. For λ = 7, it follows tht 2 6x 1 +2x 2 =7x 1 nd 2x 1 +3x 2 =7x 2, x1 2 nd the eigenvector is = x x 2. As we hve just proved for ny rel symmetric mtrix, the eigenvectors re orthogonl, =2 2= Eigenvector expnsions. For rel symmetric mtrices it cn be shown tht if n eigenvlue repets R times, there will be R independent eigenvectors corresponding to tht eigenvlue. These eigenvectors re utomticlly orthogonl to ny eigenvectors corresponding to different eigenvlue. The Grm Schmidt procedure cn be pplied so tht ll R eigenvectors corresponding to the sme eigenvlue cn be constructed to be mutully orthogonl. In this mnner, for rel symmetric n n mtrices, n orthogonl eigenvectors cn lwys be obtined. Since these vectors re orthogonl, they spn the n-dimensionl vector spce nd my be chosen s bsis vectors. Any vector v my be represented in series of the eigenvectors: v = c i φ i, (5.32) where φ i is the ith eigenvector. For regulr Sturm Liouville eigenvlue problems, the eigenfunctions re complete, mening tht ny (piecewise smooth) function cn be represented in terms of n eigenfunction expnsion f(x) c i φ i (x). (5.33) 190
4 Sturm Liouville Eigenvlue Problems This is nlogous to (5.32). In (5.33) the Fourier coefficients c i re determined by the orthogonlity of the eigenfunctions. Similrly, the coordintes c i in (5.32) re determined by the orthogonlity of the eigenvectors. We dot Eqution (5.32) into φ m : v φ m = c i φ i φ m = c m φ m φ m, since φ i φ m =0,i m, determining c m. Liner systems. Sturm Liouville eigenvlue problems rise in seprting vribles for prtil differentil equtions. One wy in which the mtrix eigenvlue problem occurs is in seprting liner homogeneous system of ordinry differentil equtions with constnt coefficients. We will be very brief. A liner homogeneous first-order system of differentil equtions my be represented by dv dt = Av, (5.34) where A is n n n mtrix nd v is the desired n-dimensionl vector solution. v usully stisfies given initil conditions, v(0) = v 0. We seek specil solutions of the form of simple exponentils: v(t) =e λt φ, (5.35) where φ is constnt vector. This is nlogous to seeking product solutions by the method of seprtion of vribles. Since dv/dt = λe λt φ, it follows tht Aφ = λφ. (5.36) Thus, there exist solutions to (5.34) of the form (5.35) if λ is n eigenvlue of A nd φ is corresponding eigenvector. We now restrict our ttention to rel symmetric mtrices A. There will lwys be n mutully orthogonl eigenvectors φ i. We hve obtined n specil solutions to the liner homogeneous system (5.34). A principle of superposition exists, nd hence liner combintion of these solutions lso stisfies (5.34): v = c i e λit φ i. (5.37) We ttempt to determine c i so tht (5.37) stisfies the initil conditions, v(0) = v 0 : v 0 = c i φ i. 191
5 Sturm Liouville Eigenvlue Problems Here, the orthogonlity of the eigenvectors is helpful, nd thus, s before, c i = v 0 φ i φ i φ i. EXERCISES 5 APPENDIX 5A.1. Prove tht the eigenvlues of rel symmetric mtrices re rel. 5A.2. () Show tht the mtrix 1 0 A = 2 1 hs only one independent eigenvector. (b) Show tht the mtrix 1 0 A = 0 1 hs two independent eigenvectors. 5A.3. Consider the eigenvectors of the mtrix A = () Show tht the eigenvectors re not orthogonl. (b) If the dot product of two vectors is defined s follows,. b = 1 4 1b b 2, show tht the eigenvectors re orthogonl with this dot product. 5A.4. Solve dv/dt = Av using mtrix methods if *() A =, v(0) = (b) A =, v(0) = A.5. Show tht the eigenvlues re rel nd the eigenvectors orthogonl: 2 1 () A = i *(b) A = (see Exercise 5A.6) 1+i 1 5A.6. FormtrixA whose entries re complex numbers, the complex conjugte of the trnspose is denoted by A H. For mtrices in which A H = A (clled Hermitin): () Prove tht the eigenvlues re rel. (b) Prove tht eigenvectors corresponding to different eigenvlues re orthogonl (in the sense tht φ i φ m =0,where denotes the complex conjugte). 192
6 Sturm Liouville Eigenvlue Problems 6 RAYLEIGH QUOTIENT The Ryleigh quotient cn be derived from the Sturm Liouville differentil eqution, d p(x) dφ + q(x)φ + λσ(x)φ =0, (6.1) by multiplying (6.1) by φ nd integrting: φ d ( p dφ ) + qφ 2 Since φ2 σ>0, we cn solve for λ: λ = φ d ( p dφ + λ φ2 σ ) + qφ 2 φ 2 σ=0.. (6.2) Integrtion by prts udv= uv vdu, where u = φ, dv = d/(pdφ/) nd hence du = dφ/, v = pdφ/ yields n expression involving the function φ evluted t the boundry: λ = pφ dφ b + b p ( ) 2 dφ qφ 2 φ2 σ, (6.3) known s the Ryleigh quotient. In Sections 3 nd 4 we hve indicted some pplictions of this result. Further discussion will be given in Section 7. Nonnegtive eigenvlues. Often in physicl problems, the sign of λ is quite importnt. As shown in Section 2.1, dh/dt + λh = 0 in certin het flow problems. Thus, positive λ corresponds to exponentil decy in time, while negtive λ corresponds to exponentil growth. On the other hnd, in certin vibrtion problems (see Section 7), d 2 h/dt 2 = λh. There, only positive λ corresponds to the usully expected oscilltions. Thus, in both types of problems we often expect λ 0: The Ryleigh quotient (6.3) directly proves tht λ 0if () pφ dφ 0, nd (b) q 0. b (6.4) We clim tht both () nd (b) re physiclly resonble conditions for nonnegtive λ. Consider the boundry constrint, pφ dφ/ b 0. The simplest types of homogeneous boundry conditions, φ = 0nddφ/ = 0, do not contribute to this boundry 193
7 Sturm Liouville Eigenvlue Problems term, stisfying (). The condition dφ/ = hφ (for the physicl cses of Newton s lw of cooling or the elstic boundry condition) hs h>0 t the left end, x =. Thus,it will hve positive contribution t x =. The sign switch t the right end, which occurs for this type of boundry condition, will lso cuse positive contribution. The periodic boundry condition e.g., φ() = φ(b) nd p() dφ/() = p(b) dφ/(b) s well s the singulrity condition φ() bounded, if p() = 0 lso do not contribute. Thus, in ll these cses pφ dφ/ b 0. The source constrint q 0 lso hs mening in physicl problems. For het flow problems, q 0 corresponds (q = α, Q = αu) to n energy-bsorbing (endothermic) rection, while for vibrtion problems, q 0 corresponds (q = α, Q = αu) to restoring force. Minimiztion principle. The Ryleigh quotient cnnot be used to determine explicitly the eigenvlue (since φ is unknown). Nonetheless, it cn be quite useful in estimting the eigenvlues. This is becuse of the following theorem: The minimum vlue of the Ryleigh quotient for ll continuous functions stisfying the boundry conditions (but not necessrily the differentil eqution) is the lowest eigenvlue: λ 1 = min pu du/ b p + (du/) 2 qu 2 u 2 σ, (6.5) where λ 1 represents the smllest eigenvlue. The minimiztion includes ll continuous functions tht stisfy the boundry conditions. The minimum is obtined only for u = φ 1 (x), the lowest eigenfunction. For exmple, the lowest eigenvlue is importnt in het flow problems (see Section 4). Tril functions. Before proving (6.5), we will indicte how (6.5) is pplied to obtin bounds on the lowest eigenvlue. Eqution (6.5) is difficult to pply directly since we do not know how to minimize over ll functions. However, let u T be ny continuous function stisfying the boundry conditions; u T is known s tril function. We compute the Ryleigh quotient of this tril function, RQu T : pu T du T / b + b p (du T /) 2 qu 2 T λ 1 RQu T = u2 T σ. (6.6) We hve noted tht λ 1 must be less thn or equl to the quotient since λ 1 is the minimum of the rtio for ll functions. Eqution (6.6) gives n upper bound for the lowest eigenvlue. 194
Math Theory of Partial Differential Equations Lecture 2-9: Sturm-Liouville eigenvalue problems (continued).
Mth 412-501 Theory of Prtil Differentil Equtions Lecture 2-9: Sturm-Liouville eigenvlue problems (continued). Regulr Sturm-Liouville eigenvlue problem: d ( p dφ ) + qφ + λσφ = 0 ( < x < b), dx dx β 1 φ()
More information1 2-D Second Order Equations: Separation of Variables
Chpter 12 PDEs in Rectngles 1 2-D Second Order Equtions: Seprtion of Vribles 1. A second order liner prtil differentil eqution in two vribles x nd y is A 2 u x + B 2 u 2 x y + C 2 u y + D u 2 x + E u +
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
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 information4 Sturm-Liouville Boundary Value Problems
4 Sturm-Liouville Boundry Vlue Problems We hve seen tht trigonometric functions nd specil functions re the solutions of differentil equtions. These solutions give orthogonl sets of functions which cn be
More informationIntroductory Statistics Neil A. Weiss Ninth Edition
Introductory Statistics Neil A. Weiss Ninth Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at:
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationSTURM-LIOUVILLE BOUNDARY VALUE PROBLEMS
STURM-LIOUVILLE 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 informationMath Fall 2006 Sample problems for the final exam: Solutions
Mth 42-5 Fll 26 Smple problems for the finl exm: Solutions Any problem my be ltered or replced by different one! Some possibly useful informtion Prsevl s equlity for the complex form of the Fourier series
More informationThe Practice Book for Conceptual Physics. Paul G. Hewitt Eleventh Edition
The Practice Book for Conceptual Physics Paul G. Hewitt Eleventh Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on
More informationElementary Linear Algebra with Applications Bernard Kolman David Hill Ninth Edition
Elementary Linear Algebra with Applications Bernard Kolman David Hill Ninth Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM JE England and Associated Companies throughout the world Visit
More information1 E3102: a study guide and review, Version 1.0
1 E3102: study guide nd review, Version 1.0 Here is list of subjects tht I think we ve covered in clss (your milege my vry). If you understnd nd cn do the bsic problems in this guide you should be in very
More informationAMS 212A Applied Mathematical Methods I Lecture 06 Copyright by Hongyun Wang, UCSC. ( ), v (that is, 1 ( ) L i
AMS A Applied Mthemticl Methods I Lecture 6 Copyright y Hongyun Wng, UCSC Recp of Lecture 5 Clssifiction of oundry conditions Dirichlet eumnn Mixed Adjoint opertor, self-djoint opertor Sturm-Liouville
More informationPartial Differential Equations
Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for One-Dimensionl Eqution The reen s function provides complete solution to boundry
More informationVariational Techniques for Sturm-Liouville Eigenvalue Problems
Vritionl Techniques for Sturm-Liouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment
More informationLinearity, linear operators, and self adjoint eigenvalue problems
Linerity, liner opertors, nd self djoint eigenvlue problems 1 Elements of liner lgebr The study of liner prtil differentil equtions utilizes, unsurprisingly, mny concepts from liner lgebr nd liner ordinry
More informationPearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world
Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk Pearson Education Limited 204
More informationDifferential Equations and Linear Algebra C. Henry Edwards David E. Penney Third Edition
Differential Equations and Linear Algebra C. Henry Edwards David E. Penney Third Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world
More information18 Sturm-Liouville Eigenvalue Problems
18 Sturm-Liouville Eigenvlue Problems Up until now ll our eigenvlue problems hve been of the form d 2 φ + λφ = 0, 0 < x < l (1) dx2 plus mix of boundry conditions, generlly being Dirichlet or Neumnn type.
More informationWilliam R. Wade Fourth Edition
Introduction to Analysis William R. Wade Fourth Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web
More information1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation
1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview
More informationPhysics for Scientists & Engineers with Modern Physics Douglas C. Giancoli Fourth Edition
Physics for Scientists & Engineers with Modern Physics Douglas C. Giancoli Fourth Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the
More information21.6 Green Functions for First Order Equations
21.6 Green Functions for First Order Equtions Consider the first order inhomogeneous eqution subject to homogeneous initil condition, B[y] y() = 0. The Green function G( ξ) is defined s the solution to
More informationReview SOLUTIONS: Exam 2
Review SOUTIONS: Exm. True or Flse? (And give short nswer ( If f(x is piecewise smooth on [, ], we cn find series representtion using either sine or cosine series. SOUTION: TRUE. If we use sine series,
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationSTURM-LIOUVILLE THEORY, VARIATIONAL APPROACH
STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH XIAO-BIAO LIN. Qudrtic functionl nd the Euler-Jcobi Eqution The purpose of this note is to study the Sturm-Liouville problem. We use the vritionl problem s
More informationBest Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 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 informationElementary Statistics in Social Research Essentials Jack Levin James Alan Fox Third Edition
Elementary Statistics in Social Research Essentials Jack Levin James Alan Fox Third Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the
More informationSTURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES
STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES 1. Regulr Sturm-Liouville Problem The method of seprtion of vribles to solve boundry vlue problems leds to ordinry differentil equtions on intervls
More informationNumerical Linear Algebra Assignment 008
Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com
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 informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationSturm-Liouville Eigenvalue problem: Let p(x) > 0, q(x) 0, r(x) 0 in I = (a, b). Here we assume b > a. Let X C 2 1
Ch.4. INTEGRAL EQUATIONS AND GREEN S FUNCTIONS Ronld B Guenther nd John W Lee, Prtil Differentil Equtions of Mthemticl Physics nd Integrl Equtions. Hildebrnd, Methods of Applied Mthemtics, second edition
More informationThe Algebra (al-jabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
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 information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationIntroductory Chemistry Essentials Nivaldo J. Tro Fourth Edition
Introductory Chemistry Essentials Nivaldo J. Tro Fourth Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationu t = k 2 u x 2 (1) a n sin nπx sin 2 L e k(nπ/l) t f(x) = sin nπx f(x) sin nπx dx (6) 2 L f(x 0 ) sin nπx 0 2 L sin nπx 0 nπx
Chpter 9: Green s functions for time-independent problems Introductory emples One-dimensionl het eqution Consider the one-dimensionl het eqution with boundry conditions nd initil condition We lredy know
More information1 E3102: A study guide and review, Version 1.2
1 E3102: A study guide nd review, Version 1.2 Here is list of subjects tht I think we ve covered in clss (your milege my vry). If you understnd nd cn do the bsic problems in this guide you should be in
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationc n φ n (x), 0 < x < L, (1) n=1
SECTION : Fourier Series. MATH4. In section 4, we will study method clled Seprtion of Vribles for finding exct solutions to certin clss of prtil differentil equtions (PDEs. To do this, it will be necessry
More informationOutline. Math Partial Differential Equations. Rayleigh Quotient. Rayleigh Quotient. Sturm-Liouville Problems Part C
Mth 53 - Prtil Differentil Equtions Sturm-Liouville Problems Prt C Outline Tril Functions Joseph M. Mhffy, jmhffy@mil.sdsu.edu Deprtment of Mthemtics nd Sttistics Dynmicl Systems Group Computtionl Sciences
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 rel-vlued function on I), nd let L 1 (I) denote the completion
More informationBoundary-value problems
226 Chpter 10 Boundry-vlue problems The initil-vlue problem is chrcterized by the imposition of uxiliry dt t single point: if the eqution is of the nth order, the n otherwise rbitrry constnts in its solution
More informationMATH STUDENT BOOK. 10th Grade Unit 5
MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY
More informationMAC-solutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationSturm-Liouville Theory
LECTURE 1 Sturm-Liouville 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 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 informationStudent Workbook for Physics for Scientists and Engineers: A Strategic Approach with Modern Physics Randall D. Knight Third Edition
Student Workbook for Physics for Scientists and Engineers: A Strategic Approach with Modern Physics Randall D. Knight Third Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England
More informationMATRICES AND VECTORS SPACE
MATRICES AND VECTORS SPACE MATRICES AND MATRIX OPERATIONS SYSTEM OF LINEAR EQUATIONS DETERMINANTS VECTORS IN -SPACE AND -SPACE GENERAL VECTOR SPACES INNER PRODUCT SPACES EIGENVALUES, EIGENVECTORS LINEAR
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationSeparation of Variables in Linear PDE
Seprtion of Vribles in Liner PDE Now we pply the theory of Hilbert spces to liner differentil equtions with prtil derivtives (PDE). We strt with prticulr exmple, the one-dimensionl (1D) wve eqution 2 u
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More information221B Lecture Notes WKB Method
Clssicl Limit B Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using
More information(9) P (x)u + Q(x)u + R(x)u =0
STURM-LIOUVILLE THEORY 7 2. Second order liner ordinry differentil equtions 2.1. Recll some sic results. A second order liner ordinry differentil eqution (ODE) hs the form (9) P (x)u + Q(x)u + R(x)u =0
More informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationPearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world
Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk Pearson Education Limited 2014
More informationSTURM-LIOUVILLE PROBLEMS
STURM-LIOUVILLE PROBLEMS Mrch 8, 24 We hve seen tht in the process of solving certin liner evolution equtions such s the het or wve equtions we re led in very nturl wy to n eigenvlue problem for second
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
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 informationREGULAR TWO-POINT BOUNDARY VALUE PROBLEMS
REGUAR TWO-POINT BOUNDARY VAUE PROBEMS PHIIP WWAKER Supposethtndrerelnumerswith
More information1 1D heat and wave equations on a finite interval
1 1D het nd wve equtions on finite intervl In this section we consider generl method of seprtion of vribles nd its pplictions to solving het eqution nd wve eqution on finite intervl ( 1, 2. Since by trnsltion
More informationMatrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24
Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationWave Equation on a Two Dimensional Rectangle
Wve Eqution on Two Dimensionl Rectngle In these notes we re concerned with ppliction of the method of seprtion of vriles pplied to the wve eqution in two dimensionl rectngle. Thus we consider u tt = c
More informationINTRODUCTION. The three general approaches to the solution of kinetics problems are:
INTRODUCTION According to Newton s lw, prticle will ccelerte when it is subjected to unblnced forces. Kinetics is the study of the reltions between unblnced forces nd the resulting chnges in motion. The
More informationMatrices, Moments and Quadrature, cont d
Jim Lmbers MAT 285 Summer Session 2015-16 Lecture 2 Notes Mtrices, Moments nd Qudrture, cont d We hve described how Jcobi mtrices cn be used to compute nodes nd weights for Gussin qudrture rules for generl
More informationChapter 5. , r = r 1 r 2 (1) µ = m 1 m 2. r, r 2 = R µ m 2. R(m 1 + m 2 ) + m 2 r = r 1. m 2. r = r 1. R + µ m 1
Tor Kjellsson Stockholm University Chpter 5 5. Strting with the following informtion: R = m r + m r m + m, r = r r we wnt to derive: µ = m m m + m r = R + µ m r, r = R µ m r 3 = µ m R + r, = µ m R r. 4
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 informationg i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f
1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationPressure Wave Analysis of a Cylindrical Drum
Pressure Wve Anlysis of Cylindricl Drum Chris Clrk, Brin Anderson, Brin Thoms, nd Josh Symonds Deprtment of Mthemtics The University of Rochester, Rochester, NY 4627 (Dted: December, 24 In this pper, hypotheticl
More informationThe Riemann-Lebesgue Lemma
Physics 215 Winter 218 The Riemnn-Lebesgue Lemm The Riemnn Lebesgue Lemm is one of the most importnt results of Fourier nlysis nd symptotic nlysis. It hs mny physics pplictions, especilly in studies of
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationPearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world
Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk Pearson Education Limited 2014
More informationQuantum Physics II (8.05) Fall 2013 Assignment 2
Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.
More information1. On the line, i.e., on R, i.e., 0 x L, in general, a x b. Here, Laplace s equation assumes the simple form. dx2 u(x) = C 1 x + C 2.
Lecture 16 Lplce s eqution - finl comments To summrize, we hve investigted Lplce s eqution, 2 = 0, for few simple cses, nmely, 1. On the line, i.e., on R, i.e., 0 x L, in generl, x b. Here, Lplce s eqution
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationStudent Workbook for College Physics: A Strategic Approach Volume 2 Knight Jones Field Andrews Second Edition
Student Workbook for College Physics: A Strategic Approach Volume 2 Knight Jones Field Andrews Second Edition Pearson Education Limited Edinburgh Gate Harlow Esse CM2 2JE England and Associated Companies
More informationMatrix 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 informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationLecture Solution of a System of Linear Equation
ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville - D. Keffer, 5/9/98 (updted /) Lecture 8- - Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner
More informationContinuous Quantum Systems
Chpter 8 Continuous Quntum Systems 8.1 The wvefunction So fr, we hve been tlking bout finite dimensionl Hilbert spces: if our system hs k qubits, then our Hilbert spce hs n dimensions, nd is equivlent
More informationChapter 28. Fourier Series An Eigenvalue Problem.
Chpter 28 Fourier Series Every time I close my eyes The noise inside me mplifies I cn t escpe I relive every moment of the dy Every misstep I hve mde Finds wy it cn invde My every thought And this is why
More informationGreen function and Eigenfunctions
Green function nd Eigenfunctions Let L e regulr Sturm-Liouville opertor on n intervl (, ) together with regulr oundry conditions. We denote y, φ ( n, x ) the eigenvlues nd corresponding normlized eigenfunctions
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationConsequently, the temperature must be the same at each point in the cross section at x. Let:
HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the
More informationJack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah
1. Born-Oppenheimer pprox.- energy surfces 2. Men-field (Hrtree-Fock) theory- orbitls 3. Pros nd cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usully does HF-how? 6. Bsis sets nd nottions 7. MPn,
More information