# MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

Size: px
Start display at page:

Download "MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1"

Transcription

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 detils see ssocited texts on function spces. You my hve lso seen some of this mteril in MATH20122 Metric Spces. Look bck t those notes if you need to do so. However I stress tht this course is methods course nd therefore this mteril is minly bckground mteril necessry to put lter (more pplied) mteril on sound bsis. Note however tht this mteril is exminble - so mke sure you understnd it, especilly prts relting to opertors from pge 4. onwrds. Definition 1.1: A set S of functions forms liner function (vector) spce if 1. f, g S nd α, β R (or C) then αf + βg S, 2. f + g = g + f nd f + (g + h) = (f + g) + h, f, g, h S, 3. zero element 0 such tht f + 0 = f, f S, 4. f S, n element ( f) such tht f + ( f) = 0, 5. (αβ)f = α(βf), f S nd α, β R (or C), 6. (α + β)f = αf + βf nd α(f + g) = αf + αg, f, g S nd α, β R (or C), 7. n element (identity) I S such tht I f = f, f S, with 1. being key property in this course. Exmples: C (, b) - set of ll continuous functions defined on [, b]. C n (, b) - set of ll functions with continuous nth derivtives defined on [, b]. L 2 (, b) - set of ll (Lebesgue) squre integrble functions ( f (x) 2 dx is bounded). Definition 1.2: An inner product spce is liner function spce on which there is defined n inner (sclr) product f, g R (or C) such tht (i) g, f = f, g - rel, (or g, f = f, g - complex - br denotes complex conjugte,) (ii) g, αf 1 + βf 2 = α g, f 1 + β g, f 2, (iii) f, f 0 with equlity f = 0. Notes: () Definition (1.2) extends notion of orthogonlity (but not ngle) to functions, (b) f, f is rel, (c) αf 1 + βf 2, g = ᾱ f 1, g + β f 2, g, (d) If g, f = 0 for ll g S then f = 0, (proof: choose g = f nd use (iii) bove).

2 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 2 Definition 1.3: A normed spce is liner function spce on which there is defined norm f R such tht (i) f 0, (ii) f = 0 f = 0, (iii) αf = α f, (iv) f + g f + g, (tringle inequlity). Note: the norm extends the notion of length, distnce or closeness to functions. It cn be defined in wide vriety of wys. For exmple, we could hve the supremum or uniform norm: f = mx f(x). x [,b] Proposition 1.4: A norm f for function f (x) my be defined to be the non-negtive rel number f = + f, f i.e. we get norm from n inner product. (Note: Cn check tht this is norm.) In this wy it is cler tht we cn lwys define norm from n inner product. However we cnnot lwys define n inner product given norm. In fct it trnspires tht necessry nd sufficient condition tht norm gives rise to n inner product is tht the so-clled Prllelogrm identity must hold, i.e. u + v 2 + u v 2 = 2( u 2 + v 2 ). For finite dimensionl vector spce tke two elements v = (v 1, v 2,...,v n ), w = (w 1, w 2,...,w n ), then the inner product is just the nturl extension of the simple sclr or dot product you hve seen in simple 2-D nd 3-D vector theory to n dimensions: v,w = v w =v 1 w 1 + v 2 w v n w n. Functions re like vectors with infinitely mny components ech component being the function s vlue t distinct point in the dependent vrible. So, imgine tking n intervl of the rel line [, b], nd divide it up into n equl pieces so tht x 0 (= ), x 1, x 2,...,x n (= b) represent the eqully spced points t the ends of ll the line segments. The function f(x) is thus pproximted on [, b] by the vector (f(x 1 ), f(x 2 ), f(x 3 ),...,f(x n )) nd s n the pproximtion gets better. In the simplest cses for function spces we define the inner product s g, f = g, f = f (x) g (x) dx, - relf, g, (1.1) f (x) g (x)dx, - complexf, g, Cre must be tken with regrd to representing functions s infinite vector spces the former hs dependent vrible with n uncountbly infinite domin whilst the ltter hs countbly infinite components. This difference cn be significnt if the function is rther exotic!

3 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 3 or, for more generl situtions, f, g = where the weight function w (x) > 0. w (x) f (x) g (x) dx In mny cses we will consider rel functions nd the inner product, defined bove, i.e. g, f = f (x) g (x) dx nd we note tht the norm induced by this inner product is ( 1/2 f = f (x)dx) 2. Theorem 1.5: In n inner product spce we hve the following sttements which prove very useful (see Sheet 1, Q. 3) () Cuchy-Schwrz inequlity f, g f g (b) Tringle inequlity f + g f + g Convergence of sequences of functions Definition 1.6: A sequence of functions f 1, f 2,...,f n,... is (strongly) convergent, or convergent in norm or uniformly convergent to limit f if f n f 0 s n. The ε definition of uniform convergence is: x (, b) nd ε > 0, N such tht N n, f n f < ε. In prticulr N is dependent only on ε nd NOT x. Exmple: in L 2 (, b) f n (x) f (x) 2 dx 0 s n (convergent in men, men squre convergence). We must distinguish this from pointwise convergence Definition 1.7: A sequence of functions f n (x) converges pointwise to f(x) if f n (x) f (x) 0 s n for ll x [, b]. Pointwise convergence sys tht the N in the definition of uniform convergence bove is dependent not only on ε but lso x. Definition 1.8: A Cuchy sequence is such tht ε > 0, N N such tht for ll m, n > N. f n f m < ε If sequence of functions converges then it is Cuchy sequence. Thus if sequence is NOT Cuchy sequence it will not converge.

4 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 4 Hilbert nd Bnch Spces Note the following distinction () A complete inner product spce is one in which every Cuchy sequence hs strong limit (in tht spce). It is clled Hilbert spce. (b) A complete normed liner spce is clled Bnch spce. It must be stressed tht not ll normed liner function spces re complete. For exmple, the set of continuous functions is complete with respect to the uniform norm, but not with respect to the L 2 norm. Every inner product gives rise to norm, i.e. u = u, u. Thus every Hilbert spce (complete inner product spce) is Bnch spce (complete normed spce) by definition. However, not every Bnch spce cn be Hilbert spce since s we described bove, only those norms which stisfy the prllelogrm equlity cn yield n inner product. Orthogonlity Definition 1.9: (i) f nd g re sid to be orthogonl if f, g = 0. (ii) A sequence of functions φ 1, φ 2,...,φ n,... is sid to be orthogonl if φ i, φ j = 0. for ll i, j N, i j. (iii) An orthogonl sequence φ 1, φ 2,...,φ n,... is sid to be orthonorml if Note: Write φ i, φ i = φ i 2 = 1. φ i, φ j = δ ij = where δ ij is clled the Kronecker delt. Exmple: For n N, the sequence 1 2π, 1 π cosnx, { 1 if i = j 0 if i j 1 π sin nx is orthonorml on [ π, π] nd is fmilir from Fourier series. Definition 1.10: A sequence of functions φ 1, φ 2,...,φ n,... is complete if, for ny f S, f (x) = i φ i (x) i=1 for some sequence of sclrs i, i N.

5 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 5 Theorem 1.11: If the sequence φ 1, φ 2,...,φ n,... is orthonorml then the coefficients i in Definition 1.9 re given by i = φ i, f. Proof: Suppose tht the expnsion in Definition 1.10 holds, then tking the inner product with φ i : φ i, f = φ i, j φ j = = j=1 j δ ij = i. j=1 j φ i, φ j j=1 Opertors Definition 1.12: Let S, T be two liner spces nd let the mpping L : S T be such tht L (αf + βg) = αl (f) + βl (g), then L is clled liner opertor. Exmples: () In finite dimensionl vector spce R n, liner opertors my be represented by mtrices. (b) In function spce in which the functions re sufficiently differentible, the differentil opertor L = n i=0 i (x) di dx i is defined nd is liner opertor. (c) An integrl opertor K defined on L 2 (, b) is given by Kf (x) = K (x, y)f (y)dy where K (x, y) is function of two vribles clled the kernel. Adjoint nd self-djoint opertors Definition 1.13: Let S, T be inner product spces, S, T the inner products on S, T respectively, nd let L : S T be liner opertor. The djoint L : T S of L is defined by g, Lf T = L g, f S, f S nd g T. Theorem 1.14: L exists, is unique nd is liner.

6 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 6 Proof: None for existence nd uniqueness. Linerity: for ny f L (αg 1 + βg 2 ),f S = αg 1 + βg 2, Lf T by Definition 1.13 = ᾱ g 1, Lf T + β g 2, Lf T by conjugte linerity = ᾱ L g 1, f S + β L g 2, f S by Definition 1.13 = αl g 1 + βl g 2, f S by conjugte linerity. So L (αg 1 + βg 2 ) = αl g 1 + βl g 2. Definition 1.15: If S = T then (i) L is clled self-djoint if L = L, (ii) L is clled skew-djoint if L = L. Theorem 1.16: (i) (L ) = L, (ii) 1 2 (L + L ) is self-djoint, (iii) 1 2 (L L ) is skew-djoint, (iv) Any liner opertor L my be expressed s the sum of self- nd skew-djoint opertor. Proof: (i) By definition of (L ) : S T, f S, g T, (L ) f, g T = f, L g S pplying 1.13 to L = L g, f S by 1.2(i) = g, Lf T by 1.13 = Lf, g T by 1.2(i) So (L ) f = Lf, f S nd so (L ) = L. (ii) ( ) 1 2 (L + L ) = 1 2 (L + (L ) ) = 1 2 (L + L). (iii) (iv) ( ) 1 2 (L L ) = 1 2 (L (L ) ) = 1 2 (L L) = 1 2 (L L ). L = 1 2 (L + L ) (L L ). Why study the djoint opertor? Well, it trnspires tht it is importnt in vrious theorems regrding the existence nd uniqueness of solutions to boundry vlue problems (BVPs). In prticulr Self-Adjoint opertors possess very nice properties s we shll see lter. Exmple 1: Suppose tht f () = f (b) = g () = g (b) = 0 nd let L = d dx.

7 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 7 Show tht i.e. L is skew-djoint. L = d dx, See Exmples Sheet 1. NOTE: If ssumptions bout boundry conditions re required to prove self-djointness, the opertor is sid to be formlly self-djoint. Exmple 2: Given homogeneous boundry conditions (e.g. f () = 0 or f () = 0 nd similrly for g) show tht the Sturm-Liouville opertor L is formlly self-djoint, where L = 1 { [ d p (x) d ] } + q (x), r (x) dx dx r (x) > 0 nd See Exmples Sheet 1. g, f = r (x) f (x) g (x) dx. Definition 1.17: A non-trivil function φ S is clled n eigenfunction of L if there exists λ R (or C) such tht Lφ = λφ. The prmeter λ is clled n eigenvlue of L corresponding to φ. Notes: (i) φ 0 is lwys solution of this homogeneous eqution, for ny λ. Some vlues of λ my llow non-trivil φ. (ii) In finite dimensionl vector spce L my be represented by mtrix (φ is clled n eigenvector). When the mtrix is symmetric the following theorem looks very fmilir. Theorem 1.18: Let L be self-djoint opertor then (i) The eigenvlues of L re rel, (ii) The eigenfunctions φ corresponding to distinct eigenvlues re orthogonl. Proof: (i) Since L is self-djoint, by 1.13, φ, Lφ = Lφ, φ = φ, Lφ by definition 1.2(i). Thus the complex number φ, Lφ is equl to its complex conjugte nd hence is rel. Also, by Definition 1.17, Lφ = λφ so tking the inner product of both sides gives φ, Lφ = φ, λφ = λ φ, φ = λ φ 2. (1.2)

8 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 8 Now, φ 0 so λ = φ, Lφ φ 2 which is rel. (ii) Let φ 1, φ 2 be eigenfunctions corresponding to distinct eigenvlues λ, µ, i.e. λ µ, Lφ 1 = λφ 1, Lφ 2 = µφ 2 nd µ φ 1, φ 2 = φ 1, µφ 2 linerity in 2nd rgument of, = φ 1, Lφ 2 eigenvlue definition 1.17 = Lφ 1, φ 2 L is self-djoint = λφ 1, φ 2 eigenvlue definition 1.17 = λ φ 1, φ 2 conjugte linerity in 1st rgument of, = λ φ 1, φ 2 λ is rel. But µ λ so φ 1, φ 2 = 0. Note: More thn one linerly independent eigenfunction my correspond to the sme eigenvlue - these eigenfunctions spn subspce S of S. The dimension of S is clled the multiplicity of the eigenvlue. An orthogonl bsis for S exists (Grm-Schmidt orthogonlistion). So, we my ssume ll linerly independent eigenfunctions re orthogonl. L my be such tht there exists denumerble sequence λ 1, λ 2,...,λ n,... of eigenvlues nd corresponding sequence of orthonorml eigenfunctions φ 1, φ 2,...φ n,.... If this set is complete then ny function f (x) S my be expnded in n infinite series s in Definition Definition 1.19: An opertor L is sid to be positive if f, Lf is rel nd positive for ll f S. Proposition 1.20: A positive opertor (i) is self-djoint, (ii) hs rel nd positive eigenvlues. No proof, see Exmples Sheet 1. Exmple: Show tht the generl solution of the inhomogeneous eqution Lu = f where L is self-djoint liner opertor nd L possesses complete orthonorml set of eigenfunctions φ 1, φ 2, φ 3,... with corresponding eigenvlues λ 1, λ 2, λ 3,..., is given by u = φ n, f λ n φ n.

9 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 9 Solution: Since the φ n form complete set, expnd the solution in the form u = n φ n, where the coefficients n re to be found. Then ( ) f = Lu = L n φ n = = n Lφ n by linerity n λ n φ n, by Definition 1.17 of eigenvlue/eigenfunction. Thus, tking the inner product with φ m, m = 1, 2,... φ m, f = φ m, n λ n φ n = n λ n φ m, φ n = n λ n δ mn = m λ m. Thus, provided λ m 0, Hence u = m = φ m, f λ m. φ n, f λ n φ n.

### Best 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 information

### STURM-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 information

### NOTES 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 information

### 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 information

### Physics 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 information

### Abstract 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 information

### Analytical Methods Exam: Preparatory Exercises

Anlyticl Methods Exm: Preprtory Exercises Question. Wht does it men tht (X, F, µ) is mesure spce? Show tht µ is monotone, tht is: if E F re mesurble sets then µ(e) µ(f). Question. Discuss if ech of the

More information

### The 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 information

### Advanced 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 information

### MATH 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 information

### Best 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 information

### Linearity, 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 information

### Theoretical 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 information

### Orthogonal 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 information

### Math 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 information

### MATH 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 information

### 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.

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

More information

### Hilbert 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 information

### Inner-product spaces

Inner-product spces Definition: Let V be rel or complex liner spce over F (here R or C). An inner product is n opertion between two elements of V which results in sclr. It is denoted by u, v nd stisfies:

More information

### Functional Analysis I Solutions to Exercises. James C. Robinson

Functionl Anlysis I Solutions to Exercises Jmes C. Robinson Contents 1 Exmples I pge 1 2 Exmples II 5 3 Exmples III 9 4 Exmples IV 15 iii 1 Exmples I 1. Suppose tht v α j e j nd v m β k f k. with α j,

More information

### Chapter 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 information

### Green 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 information

### STUDY GUIDE FOR BASIC EXAM

STUDY GUIDE FOR BASIC EXAM BRYON ARAGAM This is prtil list of theorems tht frequently show up on the bsic exm. In mny cses, you my be sked to directly prove one of these theorems or these vrints. There

More information

### Sturm-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 information

### Sturm-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 information

### c 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 information

### Math 61CM - Solutions to homework 9

Mth 61CM - Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ

More information

### Review 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 information

### Math Solutions to homework 1

Mth 75 - Solutions to homework Cédric De Groote October 5, 07 Problem, prt : This problem explores the reltionship between norms nd inner products Let X be rel vector spce ) Suppose tht is norm on X tht

More information

### g 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 information

### Main topics for the First Midterm

Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

More information

### ODE: 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 information

### Advanced 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 information

### 1 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 information

### Lecture 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 information

### Lecture 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 information

### STURM-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 information

### UNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3

UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,

More information

### Lecture 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 information

### Review 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 information

### Variational 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 information

### UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence

More information

### W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying

Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)

More information

### STURM-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 information

### M597K: Solution to Homework Assignment 7

M597K: Solution to Homework Assignment 7 The following problems re on the specified pges of the text book by Keener (2nd Edition, i.e., revised nd updted version) Problems 3 nd 4 of Section 2.1 on p.94;

More information

### How can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?

Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those

More information

### 1 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 information

### CMDA 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 information

### The Banach algebra of functions of bounded variation and the pointwise Helly selection theorem

The Bnch lgebr of functions of bounded vrition nd the pointwise Helly selection theorem Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics, University of Toronto Jnury, 015 1 BV [, b] Let < b. For f

More information

### 1 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 information

### Regulated functions and the regulated integral

Regulted functions nd the regulted integrl Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics University of Toronto April 3 2014 1 Regulted functions nd step functions Let = [ b] nd let X be normed

More information

### LECTURE 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 information

### Fourier series. Preliminary material on inner products. Suppose V is vector space over C and (, )

Fourier series. Preliminry mteril on inner products. Suppose V is vector spce over C nd (, ) is Hermitin inner product on V. This mens, by definition, tht (, ) : V V C nd tht the following four conditions

More information

### Chapter 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 information

### Chapter 3 Polynomials

Dr M DRAIEF As described in the introduction of Chpter 1, pplictions of solving liner equtions rise in number of different settings In prticulr, we will in this chpter focus on the problem of modelling

More information

### 11 An introduction to Riemann Integration

11 An introduction to Riemnn Integrtion The PROOFS of the stndrd lemms nd theorems concerning the Riemnn Integrl re NEB, nd you will not be sked to reproduce proofs of these in full in the exmintion in

More information

### MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals.

MATH 409 Advnced Clculus I Lecture 19: Riemnn sums. Properties of integrls. Drboux sums Let P = {x 0,x 1,...,x n } be prtition of n intervl [,b], where x 0 = < x 1 < < x n = b. Let f : [,b] R be bounded

More information

### 1 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 information

### MAA 4212 Improper Integrals

Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which

More information

### 1 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 information

### Lecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)

Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of

More information

### SOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL (1 + µ(f n )) f(x) =. But we don t need the exact bound.) Set

SOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL 28 Nottion: N {, 2, 3,...}. (Tht is, N.. Let (X, M be mesurble spce with σ-finite positive mesure µ. Prove tht there is finite positive mesure ν on (X, M such

More information

### MATRICES 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 information

### ACM 105: Applied Real and Functional Analysis. Solutions to Homework # 2.

ACM 05: Applied Rel nd Functionl Anlysis. Solutions to Homework # 2. Andy Greenberg, Alexei Novikov Problem. Riemnn-Lebesgue Theorem. Theorem (G.F.B. Riemnn, H.L. Lebesgue). If f is n integrble function

More information

### Separation 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 information

### MA 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 information

### Math Advanced Calculus II

Mth 452 - Advnced Clculus II Line Integrls nd Green s Theorem The min gol of this chpter is to prove Stoke s theorem, which is the multivrible version of the fundmentl theorem of clculus. We will be focused

More information

### Math 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 information

### Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1

Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution

More information

### Chapter 0. What is the Lebesgue integral about?

Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous

More information

### 2 Fundamentals of Functional Analysis

Fchgruppe Angewndte Anlysis und Numerik Dr. Mrtin Gutting 22. October 2015 2 Fundmentls of Functionl Anlysis This short introduction to the bsics of functionl nlysis shll give n overview of the results

More information

### Inner Product Space. u u, v u, v u, v.

Inner Product Spce Definition Assume tht V is ector spce oer field of sclrs F in our usge this will e. Then we define inry opertor.. :V V F [once gin in our usge this will e ] so tht the following properties

More information

### Calculus of Variations

Clculus of Vritions Com S 477/577 Notes) Yn-Bin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function

More information

### Math 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 information

### 1 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 information

### 2. 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 information

### LECTURE. INTEGRATION AND ANTIDERIVATIVE.

ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE. INTEGRATION AND ANTIDERIVATIVE. ROTHSCHILD CAESARIA COURSE, 2015/6 1. Integrtion Historiclly, it ws the problem of computing res nd volumes, tht triggered development

More information

### Presentation Problems 5

Presenttion Problems 5 21-355 A For these problems, ssume ll sets re subsets of R unless otherwise specified. 1. Let P nd Q be prtitions of [, b] such tht P Q. Then U(f, P ) U(f, Q) nd L(f, P ) L(f, Q).

More information

### Problem Set 4: Solutions Math 201A: Fall 2016

Problem Set 4: s Mth 20A: Fll 206 Problem. Let f : X Y be one-to-one, onto mp between metric spces X, Y. () If f is continuous nd X is compct, prove tht f is homeomorphism. Does this result remin true

More information

### 2 Sturm Liouville Theory

2 Sturm Liouville Theory So fr, we ve exmined the Fourier decomposition of functions defined on some intervl (often scled to be from π to π). We viewed this expnsion s n infinite dimensionl nlogue of expnding

More information

### Notes on the Eigenfunction Method for solving differential equations

Notes on the Eigenfunction Metho for solving ifferentil equtions Reminer: Wereconsieringtheinfinite-imensionlHilbertspceL 2 ([, b] of ll squre-integrble functions over the intervl [, b] (ie, b f(x 2

More information

### 4. 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 information

### Chapter 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 information

### THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem

More information

### 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 information

### Partial 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 information

### Applied Partial Differential Equations with Fourier Series and Boundary Value Problems 5th Edition Richard Haberman

Applied Prtil Differentil Equtions with Fourier Series nd Boundry Vlue Problems 5th Edition Richrd Hbermn Person Eduction Limited Edinburgh Gte Hrlow Essex CM20 2JE Englnd nd Associted Compnies throughout

More information

### Chapter 6. Infinite series

Chpter 6 Infinite series We briefly review this chpter in order to study series of functions in chpter 7. We cover from the beginning to Theorem 6.7 in the text excluding Theorem 6.6 nd Rbbe s test (Theorem

More information

### Bernoulli Numbers Jeff Morton

Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f

More information

### 1.9 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 information

### Numerical Analysis: Trapezoidal and Simpson s Rule

nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =

More information

### A 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 information

### The Henstock-Kurzweil integral

fculteit Wiskunde en Ntuurwetenschppen The Henstock-Kurzweil integrl Bchelorthesis Mthemtics June 2014 Student: E. vn Dijk First supervisor: Dr. A.E. Sterk Second supervisor: Prof. dr. A. vn der Schft

More information

### 1. 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 information

### MORE 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 information

### Elementary Linear Algebra

Elementry Liner Algebr Anton & Rorres, 1 th Edition Lecture Set 5 Chpter 4: Prt II Generl Vector Spces 163 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 163 คณตศาสตรวศวกรรม 3 สาขาวชาวศวกรรมคอมพวเตอร

More information

### 4 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 information

### 221B 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

### Math 8 Winter 2015 Applications of Integration

Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

### The 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 information