# 1.3 The Lemma of DuBois-Reymond

Size: px
Start display at page:

Transcription

1 28 CHAPTER 1. INDIRECT METHODS 1.3 The Lemm of DuBois-Reymond We needed extr regulrity to integrte by prts nd obtin the Euler- Lgrnge eqution. The following result shows tht, t lest sometimes, the extr regulrity in such sitution need not be ssumed. Lemm 3 (cf. Lemm 1.8 in BGH). (The lemm of DuBois-Reymond) If f C 0 (, b) nd b f(x)η (x) dx = 0 for every η C c (, b). (1.16) then f c (constnt). Proof: Let ζ Cc (, b) be rbitrry nd tke µ Cc (, b) with µ = 1. Consider ( ) φ = ζ ζ µ = ζ cµ where c = ζ. Note tht φ C c (, b). Also, η(x) = hs η (x) = φ(x) nd (for ǫ > 0 smll) η(+ǫ) = +ǫ x φ(ξ) dξ φ(ξ) dξ = 0 nd η(b ǫ) = b (ζ cµ) dx = c c b µ dx = 0. Thus, η Cc (, b) with φ = η. According to (1.16) we hve ( 0 = fφ = f(ζ cµ) = fζ c fµ = fζ ) ζ c 1 where c 1 = fµ. Therefore, 0 = (f c 1 )ζ for every ζ C c (, b). The fundmentl lemm implies f c 1.

2 1.3. THE LEMMA OF DUBOIS-REYMOND Figure 1.5: For the lemm of DuBois-Reymond, we mollify piecewise smooth function. Another proof of the lemm of DuBois-Rymond Agin the uthors of BGH give different rgument nd more generl result. Lemm 4 (Lemm 1.8 in BGH). If f L 1 loc (, b) nd fη = 0 for every η Cc (, b). (1.17) (,b) then f c (constnt), i.e., there is some constnt c such tht f(x) = c for lmost every x. Proof: Let x nd x be Lebesgue points for f [f 0 ]. We might s well ssume < x < x < b. As suggested in BGH, let us lso tke δ > 0 smll nd fixed so tht η(ξ) = 2χ I (ξ) + [1 + 1δ ] (ξ x) χ T (ξ) [1 1δ ] (ξ x) χ T(ξ) with I = (x + δ, x δ), T = (x δ, x + δ) nd T = ( x δ, x + δ) gives the function with grph indicted in Figure 1.5. The resoning of Exercise 8 shows µ ǫ η Cc (, b) with { 0, ξ (, b)\(x δ ǫ, x + δ + ǫ) µ ǫ η(ξ) = 2, ξ [x + δ + ǫ, x δ ǫ] where µ ǫ is stndrd mollifier with ǫ < δ. Also, d dx (µ ǫ η) = µ ǫ η = 0

3 30 CHAPTER 1. INDIRECT METHODS on the interiors (, b)\[x δ ǫ, x + δ + ǫ] nd (x + δ + ǫ, x δ ǫ) with 1 δ d dx (µ ǫ η) 1 for ll x (, b). δ Let us compute wht hppens in the portions (x δ + ǫ, x + δ ǫ) nd ( x δ + ǫ, x + δ ǫ) of the trnsition intervls T nd T. For ξ in the first intervl µ ǫ η(ξ) = µ ǫ (t) η(ξ t) t = µ ǫ (t) [1 + 1δ ] (ξ t x) t = δ (ξ x) 1 tµ ǫ (t) δ where m(ǫ) = 1 δ t tµ ǫ (t) = 1 δ = η(ξ) m(ǫ) ǫ ǫ t tµ ǫ (t) dt stisfies m(ǫ) < ǫ δ. Similrly for ξ in the intervl round x we hve µ ǫ η(ξ) = η(ξ) + m(ǫ). The hypothesis (1.17) clerly pplies to give f(µ ǫ η) = 0. We wish to tke limit nd conclude f η = 0. (1.18) To this end, let us estimte f(µ ǫ η) f η = f [(µ ǫ η) η ] f (µ ǫ η) η = f (µ ǫ η) η. (1.19) (x δ ǫ, x+δ+ǫ)

4 1.3. THE LEMMA OF DUBOIS-REYMOND 31 In this cse, we do not know f L 2, so the Cuchy-Schwrz inequlity does not help us. We do know, howevever, tht for ǫ smll the intervl J = (x δ ǫ, x+δ+ǫ) (, b) so f L 1 (J). Furthermore, we hve shown the function (µ ǫ η) (ξ) η (ξ) is bounded on ll of (, b) nd stisfies lim (µ ǫ η) (ξ) η (ξ) = 0 ǫ 0 for every ξ (, b). It follows tht the integrnd in (1.19) is bounded independent of ǫ in L 1 (J) nd limits pointwise to zero lmost everywhere. By the Lebesgue dominted convergence theorem, the expression in (1.19) tends to zero, nd we hve estblished (1.18). Since η (ξ) = (1/δ)χ T (ξ) + (1/δ)χ T(ξ) lmost everywhere (i.e., except t the four corner points), we cn rewrite (1.18) s 1 f 1 f = 0. δ (x δ,x+δ) δ (x δ,x+δ) Tking the limit s δ ց 0 nd reclling tht x nd x were Lebesgue points, we get 2f(x) 2f( x) = 0. Tht is, f(x) = f( x), nd it follows tht f is constnt, tking single vlue on its Lebesgue points. Exercises Exercise 11. Show tht if the mollifier µ ǫ is chosen to be even, then the quntity m(ǫ) = tµ ǫ (t) vnishes. Show why this quntity need not be zero when µ ǫ is not even. Exercise 12. Show tht when g hs compct support in (, b), then for ǫ smll enough µ ǫ g lso hs compct support in (, b) nd my therefore be defined on ll of R. Exercise 13. Show directly tht t d dx (µ ǫ g)(x) = µ ǫ g (x) for ll x R

5 32 CHAPTER 1. INDIRECT METHODS when g 1 c (, b) is ny piecewise C1 function with compct support. Use this clcultion to give (new nd different) direct proof tht f η = 0 where f stisfies the hypothesis (1.17) or (1.16) nd η is the function defined in the proof of the DuBois-Reymond lemm from BGH. 1.4 The Euler-Lgrnge Eqution (revisited) Theorem 4 (Corollry 1.10 in BGH). (The Euler-Lgrnge Eqution for wek extremls) If u C 1 (, b) is wek extreml for the functionl b F(x, u(x), u (x)) dx with Lgrngin F C 1 ((, b) R R), then d dx F p(x, u, u ) F z (x, u, u ) = 0 on (, b). (1.20) Proof: This result follows, essentilly, from integrting by prts in the condition for wek extremls (1.1) in the reverse direction: By the fundmentl theorem of clculus ψ(x) = x F z (t, u(t), u (t)) dt is C 1 function with derivtive F z (x, u(x), u (x)). Thus, b F z (x, u, u )φ dx = ψφ b b ψφ dx = b ψφ dx. Combining this expression with the other integrl from (1.1), we get b [F p (x, u, u ) ψ]φ dx = 0 for ll φ C c (, b).

6 1.4. THE EULER-LAGRANGE EQUATION (REVISITED) 33 By the lemm of DuBois-Rymond, there is some constnt c such tht Tht is, F p (x, u, u ) = F p (x, u, u ) ψ = c. x F z (t, u(t), u (t)) dt + c. (1.21) While it my not be true tht F p hs ny higher prtil derivtives nd it my not be true tht u hs ny higher prtil derivtives, we hve shown tht the composition F p (x, u, u ) does hve derivtive: d dx F p(x, u, u ) = F z (x, u, u ). It is importnt to relize tht the Euler-Lgrnge eqution, under these hypotheses, my not llow expnsion of the left side by the chin rule. The eqution (1.21) is clled the DuBois-Rymond eqution or the Euler-Lgrnge eqution in integrted form. The following exmple (Exmple 4 on pge 14 of BGH) shows the weker regulrity llowed by Theorem 4 is sometimes needed. Consider the functionl on 1 1 u 2 (2x u ) 2 dx A = {u C 1 [ 1, 1] : u( 1) = 0, u(1) = 1}. Notice tht F is non-negtive nd F[u 0 ] = 0 where u 0 C 1 [ 1, 1]\C 2 ( 1, 1) is given by { 0, 1 x 0 u 0 (x) = x 2, 0 x 1. We wish to show u 0 is the unique minimizer in A. If u A is ny minimizer, then we must hve 1 1 u 2 (2x u ) 2 dx = 0. This mens tht on ny intervl where u 0, we must hve u = 2x nd u(x) = x 2 + c for some constnt c. In prticulr, integrting from x = 1, we must hve u(x) = 1 + x 1 (2ξ) dξ = x 2 for 0 x 1.

7 34 CHAPTER 1. INDIRECT METHODS If we ssume there is some x 0 with 1 < x 0 < 0 for which u(x 0 ) 0, then there is mximl intervl (, b) with 1 < x 0 < b 0 such tht u(x) = u(x 0 ) + x 2 x for < x < b, but u() = u(b) = 0. Evluting u(x) t x = nd x = b we conclude 2 = b 2 = x 2 0 u(x 0). This contrdicts the fct tht < b 0. Consequently, there is no such point x 0, we hve u(x) 0 for 1 x 0, nd u u 0. Exercise 14. () Find the Euler-Lgrnge eqution for 1 1 u 2 (2x u ) 2 dx, nd show tht u 0 given bove is solution of the eqution. (b) Assume u C 2 [ 1, 1] is clssicl extreml for F, nd use the chin rule (product rule etc.) to write the Euler-Lgrnge eqution s second order qusiliner ODE. Is u 0 lso solution of this eqution? (c) Wht cn you sy bout C 2 [ 1, 1] clssicl extremls for this functionl? 1.5 Exmples We now return to some exmples from the introduction nd write down the ssocited Euler-Lgrnge equtions. We lso mke some elementry observtions bout those exmples nd introduce some dditionl exmples Dirichlet energy Recll tht D[u] = 0 if u c (constnt), nd these re bsolute minimizers in C 1 [, b], but they my not be dmissible. For the Dirichlet energy, the Lgrngin is F(p) = p 2 nd the Euler- Lgrnge eqution is u = 0. (1.22) Notice the rgument of DuBois-Rymond now implies dded regulrity: Any C 1 (wek) extreml must be C 2 (clssicl) extreml. Given the dmissible

8 1.5. EXAMPLES 35 clss A 0 = {u C 1 [, b] : u() = u, u(b) = u b }, it is esy to integrte (1.22) to obtin the unique dmissible extreml: u 0 (x) = u b b (x ) + u. Let s try to show u 0 is the minimizer. Sy u is some other dmissible competitor. Then b { } D[u] D[u 0 ] = [u 0 + (u u 0)] 2 u 02 dx = 2 b b = 2u 0 { (u u 0 )2 + 2u 0 (u u 0 )} dx b u 0 (u u 0 ) dx (u u 0) dx = 2u 0 [u b u b (u u )] = 0. This shows D[u] D[u 0 ] with equlity if nd only if u u 0, which mens u u 0. Tht is, u 0 is the unique minimizer. This pproch suggests the following simple rephrsing of the condition for minimizers: Lemm 5 (Lemm 2.1 in Troutmn). If the universl set U is liner spce (e.g., C 1 [, b] or 1 [, b], etc.) nd u 0 A stisfies F[u 0 + v] F[u 0 ] 0 whenver v U nd u 0 + v A, (1.23) then u 0 is minimizer. Conversely, if u 0 A is minimizer, then (1.23) holds. Furthermore, if equlity holds in (1.23) only for v = 0, then u 0 is the unique minimizer (nd conversely, if u 0 is the unique minimzer, then equlity in (1.23) cn only hold when v = 0.) Poisson s functionl Here we consider 1 0 (u + u 2 ) dx

9 36 CHAPTER 1. INDIRECT METHODS on A 0 = {u C 1 [, b] : u(0) = 0 = u(1)}. Here the function u 0 is dmissible nd gives 0, but this is not the minimizer. The Lgrngin is F(z, p) = z + p 2 nd the Euler-Lgrnge eqution is 2u 1 = 0. Agin, we hve extr regulrity for extremls, nd the unique extreml is u 0 = 1 x(x 1). 4 Computing F[u 0 ] we get [ x2 1 4 x + 1 ( x 1 ) ] 2 dx = 1 ( ) = Thus, F[u 0 ] < 0. See Exercise Arclength functionl Here is n instnce where C 2 regulrity is not immedite from the Euler- Lgrnge eqution. The Lgrngin is 1 + p 2, nd the Euler-Lgrnge eqution is ( ) d u = 0. dx 1 + u 2 With dditionl regulrity, the differentition my be crried out so tht the left side is immeditely recognizble s the curvture of the grph of u: k = u (1 + u 2 ) 3/2. Of course, if the curvture vnishes, the grph is stright line. See Exercise Potentil eqution An ordinry differentil eqution (ODE) centrl to the study of mny physicl phenomen is the second order liner potentil eqution u + c(x)u = 0

10 1.6. EXERCISES 37 ssocited with the vritionl integrl b [ u 2 + c(x)u 2] dx. The function c = c(x) is clled the potentil. See Exercise 19. This is Exmple 2 on pge 14 of BGH. 1.6 Exercises Exercise 15. Compute the Dirichlet energy of the sequence of functions u j A 0 = {u C 1 [0, 1] : u(0) = 0, u(1) = 1} which converge in L 1 (0, 1) to the constnt zero function. Exercise 16. Show u 0 = x(x 1)/4 is the unique minimizer of Poisson s functionl on 1 0 (u + u 2 ) dx A 0 = {u C 1 [0, 0] : u(0) = 0 = u(1)}. Exercise 17. Explicitly integrte the Euler-Lgrnge eqution for the rclength functionl to verify ll extreml solutions must be ffine, hving the form u(x) = αx + β for some constnts α nd β. Exercise 18. Find the extremls for the totl vrition functionl. Exercise 19. Show the potentil eqution is the Euler-Lgrnge eqution of the functionl b [ u 2 + c(x)u 2] dx. Wht cn you sy bout the regulrity of solutions in C 1 [, b]? Exercise 20 (project exercise). Consider cylindricl continer (modeled by) {(x, y, z) : x 2 + y 2 = 1, z 0} {(x, y, 0) : x 2 + y 2 1}. Assume the continer contins volume V = π of liquid in grvity field so tht the volume initilly occupies the spce {(x, y, z) : x 2 + y 2 < 1, 0 < z < 1}.

11 38 CHAPTER 1. INDIRECT METHODS (Ignore surfce tension nd wetting energy.) Assume, more generlly, tht the liquid is bounded by the grph of n even function z = u(r) for 1 < r < 1 (in polr coordintes) in the dmissible clss A = { u C 1 [0, 1] : u (0) = 0, 2π 1 0 } ru(r) dr = π. If the continer (long with the liquid it contins) is rotting t constnt ngulr velocity ω, clculte the physicl energy of the system two wys: () The physicl energy is the kinetic energy of rottion. (b) With respect to rotting frme fixed to the continer, the physicl energy is the potentil energy with respect to the pprent field. Find (nd solve) the Euler-Lgrnge eqution for the interfce. Wht hppens if the continer hs some other shpe (but still rottes with given volume of liquid round some xis)? Solutions Solution 1 (Exercise 8). Let f(x) = χ ( δ,δ) ( x x0 δ nd consider the convolution integrl µ ǫ f(x) = Show the following: () µ ǫ f = f µ ǫ. (b) µ ǫ f C c (R). (c) 0 µ ǫ f(x) 1 for ll x R. ξ R ) µ(ξ)f(ξ x). (d) When ǫ < δ µ ǫ f(x) = { 0, x δ + ǫ 1, x δ ǫ.

### 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

### 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

### 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

### 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

### Calculus of Variations: The Direct Approach

Clculus of Vritions: The Direct Approch Lecture by Andrejs Treibergs, Notes by Bryn Wilson June 7, 2010 The originl lecture slides re vilble online t: http://www.mth.uth.edu/~treiberg/directmethodslides.pdf

### 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

### f(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all

3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the

### Introduction to the Calculus of Variations

Introduction to the Clculus of Vritions Jim Fischer Mrch 20, 1999 Abstrct This is self-contined pper which introduces fundmentl problem in the clculus of vritions, the problem of finding extreme vlues

### 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

### The Basic Functional 2 1

2 The Bsic Functionl 2 1 Chpter 2: THE BASIC FUNCTIONAL TABLE OF CONTENTS Pge 2.1 Introduction..................... 2 3 2.2 The First Vrition.................. 2 3 2.3 The Euler Eqution..................

### 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

### 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

### Math 360: A primitive integral and elementary functions

Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

### 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.

### 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

### Notes on length and conformal metrics

Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued

### 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)

### 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

### 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

### 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

### 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() =

### MatFys. Week 2, Nov , 2005, revised Nov. 23

MtFys Week 2, Nov. 21-27, 2005, revised Nov. 23 Lectures This week s lectures will be bsed on Ch.3 of the text book, VIA. Mondy Nov. 21 The fundmentls of the clculus of vritions in Eucliden spce nd its

### 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

### 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

### 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

### n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1

The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the

### 63. 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 =

### Conservation 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

### II. Integration and Cauchy s Theorem

MTH6111 Complex Anlysis 2009-10 Lecture Notes c Shun Bullett QMUL 2009 II. Integrtion nd Cuchy s Theorem 1. Pths nd integrtion Wrning Different uthors hve different definitions for terms like pth nd curve.

### FUNDAMENTALS OF REAL ANALYSIS by. III.1. Measurable functions. f 1 (

FUNDAMNTALS OF RAL ANALYSIS by Doğn Çömez III. MASURABL FUNCTIONS AND LBSGU INTGRAL III.. Mesurble functions Hving the Lebesgue mesure define, in this chpter, we will identify the collection of functions

### Definite integral. Mathematics FRDIS MENDELU

Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the

### Convex Sets and Functions

B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line

### Math 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

### 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

### Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30

Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function

### Solution to HW 4, Ma 1c Prac 2016

Solution to HW 4 M c Prc 6 Remrk: every function ppering in this homework set is sufficiently nice t lest C following the jrgon from the textbook we cn pply ll kinds of theorems from the textbook without

### NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

### 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

### Math 115 ( ) Yum-Tong Siu 1. Lagrange Multipliers and Variational Problems with Constraints. F (x,y,y )dx

Mth 5 2006-2007) Yum-Tong Siu Lgrnge Multipliers nd Vritionl Problems with Constrints Integrl Constrints. Consider the vritionl problem of finding the extremls for the functionl J[y] = F x,y,y )dx with

### 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

### Continuous 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

### Math 6455 Oct 10, Differential Geometry I Fall 2006, Georgia Tech

Mth 6455 Oct 10, 2006 1 Differentil Geometry I Fll 2006, Georgi Tech Lecture Notes 12 Riemnnin Metrics 0.1 Definition If M is smooth mnifold then by Riemnnin metric g on M we men smooth ssignment of n

### 7.2 Riemann Integrable Functions

7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous

### Properties 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

### 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

### Integrals along Curves.

Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the

### 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 +

### Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

### Euler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )

Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s

### Math& 152 Section Integration by Parts

Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible

### 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,

### 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

### Overview of Calculus I

Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,

### 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

### 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

### 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

### Stuff You Need to Know From Calculus

Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you

### INTRODUCTION TO INTEGRATION

INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide

### a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1

Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the

### Improper 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

### 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

### Problem set 1: Solutions Math 207B, Winter 2016

Problem set 1: Solutions Mth 27B, Winter 216 1. Define f : R 2 R by f(,) = nd f(x,y) = xy3 x 2 +y 6 if (x,y) (,). ()Show tht thedirectionl derivtives of f t (,)exist inevery direction. Wht is its Gâteux

### 7.2 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

### Line Integrals. Partitioning the Curve. Estimating the Mass

Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to

### Phil Wertheimer UMD Math Qualifying Exam Solutions Analysis - January, 2015

Problem 1 Let m denote the Lebesgue mesure restricted to the compct intervl [, b]. () Prove tht function f defined on the compct intervl [, b] is Lipschitz if nd only if there is constct c nd function

### 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,

### 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

### 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

### The 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

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

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

### Math 116 Calculus II

Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................

### 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

### International Jour. of Diff. Eq. and Appl., 3, N1, (2001),

Interntionl Jour. of Diff. Eq. nd Appl., 3, N1, (2001), 31-37. 1 New proof of Weyl s theorem A.G. Rmm Mthemtics Deprtment, Knss Stte University, Mnhttn, KS 66506-2602, USA rmm@mth.ksu.edu http://www.mth.ksu.edu/

### 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

### Homework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.

Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points

### Part III. Calculus of variations

Prt III. Clculus of vritions Lecture notes for MA342H P. Krgeorgis pete@mths.tcd.ie 1/43 Introduction There re severl pplictions tht involve expressions of the form Jy) = Lx,yx),y x))dx. For instnce, Jy)

### The Riemann Integral

Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function

### 1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q.

Chpter 6 Integrtion In this chpter we define the integrl. Intuitively, it should be the re under curve. Not surprisingly, fter mny exmples, counter exmples, exceptions, generliztions, the concept of the

### 8 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

### Section 6.1 Definite Integral

Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

### 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

### 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

### 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

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

### MA 124 January 18, Derivatives are. Integrals are.

MA 124 Jnury 18, 2018 Prof PB s one-minute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,

### f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral

Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one

### Note 16. Stokes theorem Differential Geometry, 2005

Note 16. Stokes theorem ifferentil Geometry, 2005 Stokes theorem is the centrl result in the theory of integrtion on mnifolds. It gives the reltion between exterior differentition (see Note 14) nd integrtion

### Notes on the Calculus of Variations and Optimization. Preliminary Lecture Notes

Notes on the Clculus of Vritions nd Optimiztion Preliminry Lecture Notes Adolfo J. Rumbos c Drft dte November 14, 17 Contents 1 Prefce 5 Vritionl Problems 7.1 Miniml Surfces...........................

### Test 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).

Test 3 Review Jiwen He Test 3 Test 3: Dec. 4-6 in CASA Mteril - Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 14-17 in CASA You Might Be Interested

### Lecture 1: Introduction to integration theory and bounded variation

Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. You

### (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

### 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

### Math 100 Review Sheet

Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s

### Chapters 4 & 5 Integrals & Applications

Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions

### PDE Notes. Paul Carnig. January ODE s vs PDE s 1

PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................

### 10 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

### Mapping the delta function and other Radon measures

Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support

### 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

### Czechoslovak 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