DEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1.
|
|
- Holly Lambert
- 5 years ago
- Views:
Transcription
1 398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 11.1 ORTHOGONAL FUNCTIONS REVIEW MATERIAL The notions of generlized vectors nd vector spces cn e found in ny liner lger text. INTRODUCTION The concepts of geometric vectors in two nd three dimensions, orthogonl or perpendiculr vectors, nd the inner product of two vectors hve een generlized. It is perfectly routine in mthemtics to think of function s vector. In this section we will exmine n inner product tht is different from the one you studied in clculus. Using this new inner product, we define orthogonl functions nd sets of orthogonl functions. Another topic in stndrd clculus course is the expnsion of function f in power series. In this section we will lso see how to expnd suitle function f in terms of n infinite set of orthogonl functions. INNER PRODUCT Recll tht if u nd v re two vectors in 3-spce, then the inner product (u, v) (in clculus this is written s u v) possesses the following properties: (i) (u, v) (v, u), (ii) (ku, v) k(u, v), k sclr, (iii) (u, u) 0 if u 0 nd (u, u) 0 if u 0, (iv) (u v, w) (u, w) (v, w). We expect tht ny generliztion of the inner product concept should hve these sme properties. Suppose tht f 1 nd f 2 re functions defined on n intervl [, ]. * Since definite integrl on [, ] of the product f 1 (x) f 2 (x) possesses the foregoing properties (i) (iv) whenever the integrl exists, we re prompted to mke the following definition. DEFINITION Inner Product of Functions The inner product of two functions f 1 nd f 2 on n intervl [, ] is the numer ( f 1, f 2 ) f 1 (x) f 2 (x) dx. ORTHOGONAL FUNCTIONS Motivted y the fct tht two geometric vectors u nd v re orthogonl whenever their inner product is zero, we define orthogonl functions in similr mnner. DEFINITION Orthogonl Functions Two functions f 1 nd f 2 re orthogonl on n intervl [, ] if ( f 1, f 2 ) f 1 (x) f 2 (x) dx 0. (1) * The intervl could lso e (, ), [0, ), nd so on.
2 11.1 ORTHOGONAL FUNCTIONS 399 For exmple, the functions f 1 (x) x 2 nd f 2 (x) x 3 re orthogonl on the intervl [1, 1], since ( f 1, f 2 ) 1 x 2 x 3 dx x Unlike in vector nlysis, in which the word orthogonl is synonym for perpendiculr, in this present context the term orthogonl nd condition (1) hve no geometric significnce. ORTHOGONAL SETS functions. We re primrily interested in infinite sets of orthogonl DEFINITION Orthogonl Set A set of rel-vlued functions {f 0 (x), f 1 (x), f 2 (x),...} is sid to e orthogonl on n intervl [, ] if (m, n) m(x)n(x) dx 0, m Y n. (2) ORTHONORMAL SETS The norm, or length u, of vector u cn e expressed in terms of the inner product. The expression (u, u) u 2 is clled the squre norm, nd so the norm is u 1(u, u). Similrly, the squre norm of function f n is f n (x) 2 (f n, f n ), nd so the norm, or its generlized length, is f n (x) 1(n, n). In other words, the squre norm nd norm of function f n in n orthogonl set {f n (x)} re, respectively, B f n (x) 2 n 2 (x) dx nd f n (x) f 2 n(x) dx. (3) If {f n (x)} is n orthogonl set of functions on the intervl [, ] with the property tht f n (x) 1 for n 0, 1, 2,..., then {f n (x)} is sid to e n orthonorml set on the intervl. EXAMPLE 1 Orthogonl Set of Functions Show tht the set {1, cos x, cos 2x,...} is orthogonl on the intervl [p, p]. SOLUTION If we mke the identifiction f 0 (x) 1 nd f n (x) cos nx, we must then show tht We hve, in the first cse, ( 0, n) 0(x) n(x) dx 0, n 0, nd 1 n sin nx 0(x) n(x) dx cos nx dx m(x) n(x) dx 0, m n. 1 n [sin n sin(n)] 0, n 0,
3 400 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES nd, in the second, (m, n) m(x)n(x) dx cos mx cos nx dx 1 [cos(m n)x cos(m n)x] dx ; trig identity 2 1 sin (m n)x sin (m n)x 2 m n m n 0, m n. EXAMPLE 2 Norms Find the norm of ech function in the orthogonl set given in Exmple 1. SOLUTION For f 0 (x) 1 we hve, from (3), f 0 (x) 2 so f 0 (x) 12. For f n (x) cos nx, n 0, it follows tht f n (x) 2 Thus for n 0, f n (x) 1. cos 2 nx dx 1 2 Any orthogonl set of nonzero functions {f n (x)}, n 0, 1, 2,... cn e normlized tht is, mde into n orthonorml set y dividing ech function y its norm. It follows from Exmples 1 nd 2 tht the set 1, cos x cos 2x,, is orthonorml on the intervl [p, p]. We shll mke one more nlogy etween vectors nd functions. Suppose v 1, v 2, nd v 3 re three mutully orthogonl nonzero vectors in 3-spce. Such n orthogonl set cn e used s sis for 3-spce; tht is, ny three-dimensionl vector cn e written s liner comintion dx 2, 1 [1 cos 2nx] dx. u c 1 v 1 c 2 v 2 c 3 v 3, (4) where the c i, i 1, 2, 3, re sclrs clled the components of the vector. Ech component c i cn e expressed in terms of u nd the corresponding vector v i. To see this, we tke the inner product of (4) with v 1 : (u, v 1 ) c 1 (v 1, v 1 ) c 2 (v 2, v 1 ) c 3 (v 3, v 1 ) c 1 v 1 2 c 2 0 c 3 0. Hence c 1 (u, v 1) 'v 1 ' 2. In like mnner we find tht the components c 2 nd c 3 re given y c 2 (u, v 2) 'v 2 ' 2 nd c 3 (u, v 3) 'v 3 ' 2.
4 11.1 ORTHOGONAL FUNCTIONS 401 Hence (4) cn e expressed s u (u, v 1) 'v 1 ' 2 v 1 (u, v 2) 'v 2 ' 2 v 2 (u, v 3) 'v 3 ' 2 v 3 3 n1 (u, v n ) 'v n ' 2 v n. (5) ORTHOGONAL SERIES EXPANSION Suppose {f n (x)} is n infinite orthogonl set of functions on n intervl [, ]. We sk: If y f (x) is function defined on the intervl [, ], is it possile to determine set of coefficients c n, n 0, 1, 2,..., for which f (x) c 00(x) c 11(x) c nn(x)? (6) As in the foregoing discussion on finding components of vector we cn find the coefficients c n y utilizing the inner product. Multiplying (6) y f m (x) nd integrting over the intervl [, ] gives f (x)m(x) dx c 0 0(x)m(x) dx c 1 1(x)m(x) dx c n n(x)m(x) dx c 0 (0, m) c 1 (1, m) c n (n, m). By orthogonlity ech term on the right-hnd side of the lst eqution is zero except when m n. In this cse we hve f (x)n(x) dx c n It follows tht the required coefficients re c n f (x)n(x) dx, n 0, 1, 2, n(x)dx In other words, f (x) c nn(x), (7) n0 where c n f (x)n(x) dx. 'n(x)' 2 (8) With inner product nottion, (7) ecomes f (x) ( f, n) n0 'n(x)' 2 n(x). (9) Thus (9) is seen to e the function nlogue of the vector result given in (5). 2 n(x) dx. DEFINITION Orthogonl Set/Weight Function A set of rel-vlued functions {f 0 (x), f 1 (x), f 2 (x),...} is sid to e orthogonl with respect to weight function w(x) on n intervl [, ] if w(x)m(x)n(x) dx 0, m n. The usul ssumption is tht w(x) 0 on the intervl of orthogonlity [, ]. The set {1, cos x, cos 2x,...} in Exmple 1 is orthogonl with respect to the weight function w(x) 1 on the intervl [p, p]. If {f n (x)} is orthogonl with respect to weight function w(x) on the intervl [, ], then multiplying (6) y w(x)f n (x) nd integrting yields c n f (x) w(x)n(x) dx 'n(x)' 2, (10)
5 402 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES where f n (x) 2 2 w(x)n(x) dx. (11) The series (7) with coefficients given y either (8) or (10) is sid to e n orthogonl series expnsion of f or generlized Fourier series. COMPLETE SETS The procedure outlined for determining the coefficients c n ws forml; tht is, sic questions out whether or not n orthogonl series expnsion such s (7) is ctully possile were ignored. Also, to expnd f in series of orthogonl functions, it is certinly necessry tht f not e orthogonl to ech f n of the orthogonl set {f n (x)}. (If f were orthogonl to every f n, then c n 0, n 0, 1, 2,....) To void the ltter prolem, we shll ssume, for the reminder of the discussion, tht n orthogonl set is complete. This mens tht the only function tht is orthogonl to ech memer of the set is the zero function. EXERCISES 11.1 In Prolems 1 6 show tht the given functions re orthogonl on the indicted intervl. 1. f 1 (x) x, f 2 (x) x 2 ; [2, 2] 2. f 1 (x) x 3, f 2 (x) x 2 1; [1, 1] 3. f 1 (x) e x, f 2 (x) xe x e x ; [0, 2] 4. f 1 (x) cos x, f 2 (x) sin 2 x; [0, p] 5. f 1 (x) x, f 2 (x) cos 2x; [p2, p2] 6. f 1 (x) e x, f 2 (x) sin x; [p4, 5p4] In Prolems 7 12 show tht the given set of functions is orthogonl on the indicted intervl. Find the norm of ech function in the set. 7. {sin x, sin 3x, sin 5x,...}; [0, p2] 8. {cos x, cos 3x, cos 5x,...}; [0, p 2] 9. {sin nx}, n 1, 2, 3,...; [0,p] n 10. sin ; [0, p] p x, n 1, 2, 3, n 1, cos p x, n 1, 2, 3,... ; [0, p] n m 1, cos x, sin p p x, n 1, 2, 3,..., m 1, 2, 3,... ; [p, p] In Prolems 13 nd 14 verify y direct integrtion tht the functions re orthogonl with respect to the indicted weight function on the given intervl. 13. H 0 (x) 1, H 1 (x) 2x, H 2 (x) 4x 2 2; w(x) e x2, (, ) 14. L 0 (x) 1, L 1 (x) x 1, L 2 (x) 1 w(x) e x 2 x2 2x 1;, [0, ) Answers to selected odd-numered prolems egin on pge ANS Let {f n (x)} e n orthogonl set of functions on [, ] such tht f 0 (x) 1. Show tht n(x) dx 0 for n 1, 2, Let {f n (x)} e n orthogonl set of functions on [, ] such tht f 0 (x) 1 nd f 1 (x) x. Show tht (x )n(x) dx 0 for n 2, 3,... nd ny constnts nd. 17. Let {f n (x)} e n orthogonl set of functions on [, ]. Show tht f m (x) f n (x) 2 f m (x) 2 f n (x) 2, m n. 18. From Prolem 1 we know tht f 1 (x) x nd f 2 (x) x 2 re orthogonl on the intervl [2, 2]. Find constnts c 1 nd c 2 such tht f 3 (x) x c 1 x 2 c 2 x 3 is orthogonl to oth f 1 nd f 2 on the sme intervl. 19. The set of functions {sin nx}, n 1, 2, 3,..., is orthogonl on the intervl [p, p]. Show tht the set is not complete. 20. Suppose f 1, f 2, nd f 3 re functions continuous on the intervl [, ]. Show tht ( f 1 f 2, f 3 ) ( f 1, f 3 ) ( f 2, f 3 ). Discussion Prolems 21. A rel-vlued function f is sid to e periodic with period T if f (x T ) f (x). For exmple, 4p is period of sin x, since sin(x 4p) sin x. The smllest vlue of T for which f (x T ) f (x) holds is clled the fundmentl period of f. For exmple, the fundmentl period of f (x) sin x is T 2p. Wht is the fundmentl period of ech of the following functions? () f (x) cos 2px () f (x) sin 4 L x (c) f (x) sin x sin 2x (d) f (x) sin 2x cos 4x (e) f (x) sin 3x cos 2x (f) f (x) A 0 A n cos n n1 p x B n sin n p x, A n nd B n depend only on n
k and v = v 1 j + u 3 i + v 2
ORTHOGONAL FUNCTIONS AND FOURIER SERIES Orthogonl functions A function cn e considered to e generliztion of vector. Thus the vector concets like the inner roduct nd orthogonlity of vectors cn e extended
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
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 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 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 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 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 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 informationAMATH 731: Applied Functional Analysis Fall Additional notes on Fréchet derivatives
AMATH 731: Applied Functionl Anlysis Fll 214 Additionl notes on Fréchet derivtives (To ccompny Section 3.1 of the AMATH 731 Course Notes) Let X,Y be normed liner spces. The Fréchet derivtive of n opertor
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 informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationENGI 9420 Lecture Notes 7 - Fourier Series Page 7.01
ENGI 940 ecture Notes 7 - Fourier Series Pge 7.0 7. Fourier Series nd Fourier Trnsforms Fourier series hve multiple purposes, including the provision of series solutions to some liner prtil differentil
More informationThe Evaluation Theorem
These notes closely follow the presenttion of the mteril given in Jmes Stewrt s textook Clculus, Concepts nd Contexts (2nd edition) These notes re intended primrily for in-clss presenttion nd should not
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
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 information1. Gauss-Jacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. Guss-Jcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
More information4. Calculus of Variations
4. Clculus of Vritions Introduction - Typicl Problems The clculus of vritions generlises the theory of mxim nd minim. Exmple (): Shortest distnce between two points. On given surfce (e.g. plne), nd the
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 informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
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 informationDiscrete Least-squares Approximations
Discrete Lest-squres Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve
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 informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationOrthogonal Polynomials
Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils
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 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 informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
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 informationFundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
More informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationLecture 2e Orthogonal Complement (pages )
Lecture 2e Orthogonl Complement (pges -) We hve now seen tht n orthonorml sis is nice wy to descrie suspce, ut knowing tht we wnt n orthonorml sis doesn t mke one fll into our lp. In theory, the process
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
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. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More information1 The Lagrange interpolation formula
Notes on Qudrture 1 The Lgrnge interpoltion formul We briefly recll the Lgrnge interpoltion formul. The strting point is collection of N + 1 rel points (x 0, y 0 ), (x 1, y 1 ),..., (x N, y N ), with x
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
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 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 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 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 informationLECTURE 3. Orthogonal Functions. n X. It should be noted, however, that the vectors f i need not be orthogonal nor need they have unit length for
ECTURE 3 Orthogonl Functions 1. Orthogonl Bses The pproprite setting for our iscussion of orthogonl functions is tht of liner lgebr. So let me recll some relevnt fcts bout nite imensionl vector spces.
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd
More informationReview on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.
Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5. - 5.3) Remrks on the course. Slide Review: Sec. 5.-5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description
More information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
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 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 information1 Linear Least Squares
Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving
More informationCAAM 453 NUMERICAL ANALYSIS I Examination There are four questions, plus a bonus. Do not look at them until you begin the exam.
Exmintion 1 Posted 23 October 2002. Due no lter thn 5pm on Mondy, 28 October 2002. Instructions: 1. Time limit: 3 uninterrupted hours. 2. There re four questions, plus bonus. Do not look t them until you
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
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 informationLecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationdf dt f () b f () a dt
Vector lculus 16.7 tokes Theorem Nme: toke's Theorem is higher dimensionl nlogue to Green's Theorem nd the Fundmentl Theorem of clculus. Why, you sk? Well, let us revisit these theorems. Fundmentl Theorem
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
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 informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationHilbert Spaces. Chapter Inner product spaces
Chpter 4 Hilbert Spces 4.1 Inner product spces In the following we will discuss both complex nd rel vector spces. With L denoting either R or C we recll tht vector spce over L is set E equipped with ddition,
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
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 informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationMath 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 information(4.1) D r v(t) ω(t, v(t))
1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution
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 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 informationMATH1050 Cauchy-Schwarz Inequality and Triangle Inequality
MATH050 Cuchy-Schwrz Inequlity nd Tringle Inequlity 0 Refer to the Hndout Qudrtic polynomils Definition (Asolute extrem for rel-vlued functions of one rel vrile) Let I e n intervl, nd h : D R e rel-vlued
More informationMA Handout 2: Notation and Background Concepts from Analysis
MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationLecture 3: Curves in Calculus. Table of contents
Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up
More informationNOTES ON HILBERT SPACE
NOTES ON HILBERT SPACE 1 DEFINITION: by Prof C-I Tn Deprtment of Physics Brown University A Hilbert spce is n inner product spce which, s metric spce, is complete We will not present n exhustive mthemticl
More informationOrthogonal Polynomials and Least-Squares Approximations to Functions
Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
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 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 informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationChapter Five - Eigenvalues, Eigenfunctions, and All That
Chpter Five - Eigenvlues, Eigenfunctions, n All Tht The prtil ifferentil eqution methos escrie in the previous chpter is specil cse of more generl setting in which we hve n eqution of the form L 1 xux,tl
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 nth-order
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
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 informationA product convergence theorem for Henstock Kurzweil integrals
A product convergence theorem for Henstock Kurzweil integrls Prsr Mohnty Erik Tlvil 1 Deprtment of Mthemticl nd Sttisticl Sciences University of Albert Edmonton AB Cnd T6G 2G1 pmohnty@mth.ulbert.c etlvil@mth.ulbert.c
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
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 informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
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 informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
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 information1. On some properties of definite integrals. We prove
This short collection of notes is intended to complement the textbook Anlisi Mtemtic 2 by Crl Mdern, published by Città Studi Editore, [M]. We refer to [M] for nottion nd the logicl stremline of the rguments.
More informationSection 7.1 Area of a Region Between Two Curves
Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region
More informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
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 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 informationCalculus of variations with fractional derivatives and fractional integrals
Anis do CNMAC v.2 ISSN 1984-820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810-193 Aveiro, Portugl
More informationThe Fundamental Theorem of Calculus
The Fundmentl Theorem of Clculus MATH 151 Clculus for Mngement J. Robert Buchnn Deprtment of Mthemtics Fll 2018 Objectives Define nd evlute definite integrls using the concept of re. Evlute definite integrls
More information