Integrals along Curves.


 Randall Glenn
 8 months ago
 Views:
Transcription
1 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 prmetriztion of the curve nd the imge (or the pth) of the curve. The point () is clled the initil (end) point nd (b) is clled the finl (end) point. There re such continuous functions whose imge do not t ll look like wht we think of curve. For exmple, there re spcefilling curves in the plne whose imges contin solid rectngle. Such curves hve, intuitively speking, n infinite length. So our first restriction will be: Definition. A curve : [, b] R n is sid to be rectifible if there exists number K such tht for every prtition P of the domin, P : = t 0 < t 1 < < t 1 we hve l P := (t j ) (t j 1 ) K 2. If curve is rectifible then we define its length to be l() := sup{l P : P is prtition of [, b] }. Definition We sy tht curve : [, b] R n is smooth curve if is continuously differentible 3. We sy tht it is piecewise smooth curve if there exists finite prtition P : = t 0 < t 1 < < t N such tht the restrictions of to the intervls [t i 1, t i ] re ll smooth curves. Using this ide we my, more generlly define the integrl of bounded function f : R long curve. For given prtition P let τ i [t i 1, t i ] ( mrking of the prtition). Let P := mx i t i t i 1. We my define the Riemnn sums nd the Riemnn integrl s limit of such sums: f(x) ds := lim f((τ j )) (t j ) (t j 1 ) P 0 whenever this limit exists. It cn be shown tht this limit exists if, for exmple, f is continuous. We could lterntively use upper nd lower sums to define integrbility, but show tht both methods yield the sme result. All this is good to know for your generl knowledge of mthemtics, however we will usully sty wy from such generlity nd restrict ourselves to piecewise smooth curves. If the curve is smooth then (see the theorem below): b f(x) ds = f((t)) (t) dt, with the obvious generliztion to integrls long piecewise smooth curves s sums of such integrls long smooth curves. 1 N = since the number of prtition points depends on the prtition. 2 x y denotes ordinry Eucliden distnce between the points x nd y nd so l P is the length of the polygonl line connecting the points (t j) 3 At the endpoints we, of course, interpret differentible in the onesided sense. 1
2 These ides trnslte in n obvious wy to curves in C nd integrls of complex vlued functions long curves. Definition. A curve : [, b] C is sid to be rectifible if there exists number K such tht for every prtition P of the domin, P : = t 0 < t 1 < < t we hve l P := (t j ) (t j 1 ) K. If curve is rectifible then we define its length to be l() := sup{l P : P is prtition of [, b] }. Definition We sy tht curve : [, b] C is smooth curve if is continuously differentible nd is nowhere zero. We sy tht it is piecewise smooth curve if there exists finite prtition P : = t 0 < t 1 < < t N such tht the restrictions of to the intervls [t i 1, t i ] re ll smooth curves. Suppose tht f : C is bounded function defined long curve. For given prtition P let τ i [t i 1, t i ] be mrking of the prtition. Let P := mx i t i t i 1. We my define f(z) ds := lim f((τ j )) (t j ) (t j 1 ) P 0 whenever this limit exists. Of course, if we write f in terms of its rel nd imginry prts, f = g+ih, then f(z) ds = g(x + iy) ds + i h(x + iy) ds. If the curve is smooth then (see the theorem below): f(x) ds = b f((t)) (t) dt, with the obvious generliztion to integrls long piecewise smooth curves. Definitions. We sy tht the curves : [, b] R n nd β : [c, d] R n re equivlent, nd we write δ, if there exists continuous, strictly incresing function µ : [, b] [c, d] such tht = β µ. If we let [] be the equivlence clss of curve, then we let [] be the equivlence clss of β where β : [, b] R n defined by β(t) = ( + b t). Tht is, the sme curve ( = β ) but with the opposite orienttion. If α : [p, q] R n nd : [, b] R n re two curves such tht the terminl point of α is the initil point of then we cn define [β] = [α] + [] in nturl wy, for exmple { α(t) if p t q β(t) = (t + q) if q t q + b, 2
3 so tht the pth of β is the pth of α, followed by the pth of. If the initil point of curve is lso its terminl point then we sy the curve is closed. If the curve does not intersect itself, except possibly tht the initil point is equl to the terminl point, then we sy tht the curve is simple. Definition Suppose tht the curve : [, b] R n is differentible t t 0 nd tht (t 0 ) 0. Any vector of the form c (t 0 ), c 0, is clled tngent vector t the point (t 0 ). The line x = (t 0 ) + s (t 0 ), s R is clled the tngent line to the curve t the point (t 0 ). The plne (or line, or hyperplne, depending on the dimension) through this point nd norml to (t 0 ) is clled the norml plne. It hs the eqution (t 0 ) (x (t 0 )) = 0. Theorem 1 If : [, b] R n is smooth curve nd f is continuous function on then l() = f(x) ds = b b (t) dt, f((t)) (t) dt. The proof is lengthy (pp [1]) but n excellent exmple of proof in nlysis, nd you re encourged to red it. Note tht integrls long equivlent curves hve the sme vlue, i.e. the integrl does not depend on the prmeteriztion. This is simple consequence of the chin rule. 2. RiemnnStieltjes integrls. There is simple generliztion of the Riemnn integrl, clled the RiemnnStieltjes integrl. The RiemnnStieltjes integrl of relvlued function f of rel vrible with respect to rel function g is denoted by b f(t)dg(t) nd defined to be the limit, s the mesh of the prtition P := { = t 0 < t 1 < < t N = b} of the intervl [, b] pproches zero, of the pproximting sum N S(P, f, g) := f(τ i )[g(t i ) g(t i 1 )] i=1 where τ i [t i 1, t i ]. The two functions f nd g re respectively clled the integrnd nd the integrtor. The limit is here understood to be number A (the vlue of the RiemnnStieltjes integrl) such tht for every ɛ > 0 there exists δ > 0 such tht for every prtition P with mesh P < δ, nd for every choice of points τ i [t i 1, t i ], we hve S(P, f, g) A < ɛ. If g is nondecresing function on [, b], then the generlized RiemnnStieltjes of with respect to g exists if nd only if, for every ɛ > 0, there exists prtition P such tht U(P, f, g) L(P, f, g) < ɛ (the Riemnn lemm). If g is continuously differentible then this integrl my be reduced to n ordinry Riemnn integrl: b f(t)dg(t) = b f(t)g (t)dt. 3
4 Exercise. Prove the bove ssertion. However, g my hve jump discontinuities, or my hve derivtive zero lmost everywhere while still being continuous nd incresing, in either of which cses the RiemnnStieltjes integrl is not cptured by ny expression involving derivtives of g. Exmple. Let f(t) = 1/t 2 nd g(t) = t, i.e. g(t) is the lrgest integer tht is less thn or equl to t. Then for ny positive integer N 3. Line Integrls. N 0 f(t)dg(t) := N n=1 1 n 2. Definition. Let : [, b] U R n be rectifible curve, x i (t) = i (t), i = 1, 2,, n nd let f : U R n, f = (f 1, f 2,, f n ), be piecewise continuous function, then for 1 k n we define f k dx k := b f k ((t))d k (t) f k (x)dx k nd f dx = n i=1 f i (x) dx i. Exercise. Show tht if the curve is piecewise smooth then f dx = n i=1 b f i ((t)) dx i dt dt. From now on we will only del with curves tht re smooth or t lest piecewise smooth. Properties. f dx = f dx δ+ f dx = δ f dx + f dx. Definitions. We sy tht f is grdient field in D R n if φ C 1 (D) such tht f = φ in D. In this cse φ is sid to be potentil function for f. If for ech : [, b] D tht is piecewise smooth the vlue of the line integrl f dx depends only on the endpoints then we sy tht line integrls of f re pth independent. In this cse we sy tht the field f is conservtive. If f is grdient field then the differentil form f 1 dx 1 + f 2 dx f n dx n is sid to be exct. 4
5 Theorem 2. Suppose tht D is domin in R n nd tht f C(D, R n ). then the following re equivlent: f is grdient field. f is conservtive. For ny piecewise smooth closed curve we hve f dx = 0. The mjor prt of the proof is to show tht if f is conservtive then given ny point x 0 D, φ(x) := x x 0 f dx long ny piecewise smooth curve from x 0 to x is well defined nd is potentil for f. Definition. A set S R n is sid to be convex if the line segment uv S whenever u, v S. Let us consider the twodimensionl sitution. If f(x.y) := (P (x, y), Q(x, y)) is continuously differentible grdient field defined on some open set D, then there exists (twice continuously differentible) potentil function φ such tht P = φ x nd Q := φ y. Therefore P y = φ xy = φ yx = Q x 4. In generl the differentil form P (x, y)dx + Q(x, y)dy is sid to be closed if P y = Q x. Hence P (x, y)dx+q(x, y)dy is closed if it is exct. The converse is not true nd is, in fct, highly dependent on the geometry of the domin D. A creful study of closed versus exct leds us to the field of Algebric Topology. Green s Theorem. Let Ω R 2 be convex nd bounded, nd suppose tht its boundry, Ω, is piecewise smooth. Suppose tht P nd Q re continuously differentible on n open set U Ω Ω. Then ( Q P dx + Qdy = x P ) dx dy, y D where the line integrl is tken in the counterclockwise sense 5. Ω Corollry 1. Let Ω R 2 be convex nd bounded, nd suppose tht its boundry, Ω, is piecewise smooth. Suppose tht P nd Q re continuously differentible on n open set U Ω Ω. Suppose lso tht P y = Q x in Ω. Then P dx + Qdy = 0. D 4 Recll tht for twice continuously differentible functions the mixed second order derivtives re independent of the order in which the derivtives re tken. 5 The requirement tht Ω be convex cn be gretly relxed. For exmple it is sufficient to ssume tht Ω is simply connected. 5
6 4. Complex Line Integrls. Definition Let D be n open set in C nd f : D C bounded function. Suppose tht : [, b] D is rectifible curve in D. Let P := {(t j ) = t 0 < t 1 < < t = b} denote generic prtition of the curve nd let τ j [t j 1, t j ] denote mrking of the prtition. We define f(z) dz := lim f((τ j ))((t j ) (t j 1 )) P 0 whenever this limit exists (irrespective of the choice of the mrking). It cn be shown tht this limit exists if, for exmple, f is continuous. Letting z j := (t j ) nd ζ j := (t j ) we hve the more compct eqution f(z) dz := lim f(ζ j )(z j z j 1 ). P 0 Note tht this is simply complex RiemnnStieltjes integrl. To simplify mtters we will restrict ourselves to piecewise smooth curves tht re mde up of finitely mny sections of smooth curve. We my write f(z) = u(z) + iv(z). If is smooth curve then (t) = 1 (t) + i 2 (t) nd there exist, by the Men Vlue Theorem, τ j, σ j [t j 1, t j ] so tht z j z j 1 = [ 1 (σ j) + i 2 (τ j)](t j t j 1 ). This mens tht we cn write the Riemnn sums [u((τ j )) 1(τ j ) + iu((σ j )) 2(σ j ) + iv((τ j )) 1(τ j ) v((σ j )) 2(τ j )](t j t j 1 ). Letting P tend to zero we see tht f(z) dz = b [u((t) 1(t) + iu((t)) 2(t) + iv((t)) 1(t) v((t)) 2(t)] dt = So we hve the sme formul s the definition of line integrl in the text, [2]: f(z) dz = b b f((t)) (t) dt. f((t)) (t) dt. (1) 5. Properties of Line Integrls in C. Theorem 3 Suppose f nd g re integrble long the curves nd δ, nd let, b C then () (b) [f(z) + bg(z)] dz = f(z) dz + b g(z) dz. f(z) dz = f(z) dz. 6
7 (c) (d) +δ f(z) dz f(z) dz = f(z) dz + f(z) dz. δ f(z) ds Ml(), where M := sup f(z). The proofs of these, for the cse where the integrnds re continuous nd where the pths re smooth, re given in the text [2], but they esily follow from the definition of line integrls of integrble functions long rectifible curves. Theorem 4. Suppose f is differentible on the curve : [α, β] C then f (z) dz = f((β)) f((α)). Corollry 2. Suppose f hs continuous ntiderivtive F, i.e. F (z) = f(z), on the simple closed curve : [α, β] C then f(z) dz = 0. Theorem 5. Let Ω C be convex, open, nd bounded, nd suppose tht its boundry, Ω, is smooth. Suppose tht f re continuously differentible on n open set U Ω Ω. Then f(z) dz = 0. (2) D The proof follows immeditely from Green s theorem nd the CuchyRiemnn equtions (esy exercise). Note the difference in the sttements of theorem 5 nd corollry 2 6. To illustrte the difference consider the following exmple Exmple. Let B be the unit disk centered t the origin. Then B =, the unit circle. We orient counterclockwise. We cn show tht 1 dz = 2πi. z Note tht 1/z is not differentible on ll of B. Corollry 2 implies tht there pprently does not exist n ntiderivtive for 1/z on ll of. We will understnd this more when we study the (multivlued) function ln(z). own. 6 The text [2] does not use Green s Theorem nd so the proof there is lengthier, but hs the dvntge of stnding on its 7
8 Note. The strtegy for the proof in the text [2] is not to use Green s Theorem but to follow the following steps: () Consider the cse of stndrd rectngles in C, tht is to sy rectngles whose sides re prllel to the xes, nd to initilly only consider entire functions f, i.e. functions tht re nlytic everywhere in C. (b) Show tht f(z) dz = 0 if is stndrd rectngle. (c) Prove tht if f is entire then we cn construct n nlytic function F such tht F = f. (d) To show tht f(z) dz = 0 for ny piecewise smooth simple closed curve. References [1] Friedmn, Avner. Advnced Clculus, Dover Publictions. [2] Bk, Joseph nd Donld Newmn. Complex Anlysis, 3rd edition, Springer. 8
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
More informationMapping 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
More informationWeek 10: Riemann integral and its properties
Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the
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 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 informationAnonymous Math 361: Homework 5. x i = 1 (1 u i )
Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define
More informationUNIFORM 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 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 informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More information1 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
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationMATH 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 informationRiemann Stieltjes Integration  Definition and Existence of Integral
 Definition nd Existence of Integrl Dr. Adity Kushik Directorte of Distnce Eduction Kurukshetr University, Kurukshetr Hryn 136119 Indi. Prtition Riemnn Stieltjes Sums Refinement Definition Given closed
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 HENSTOCKKURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When relvlued
More informationMath 4200: Homework Problems
Mth 4200: Homework Problems Gregor Kovčič 1. Prove the following properties of the binomil coefficients ( n ) ( n ) (i) 1 + + + + 1 2 ( n ) (ii) 1 ( n ) ( n ) + 2 + 3 + + n 2 3 ( ) n ( n + = 2 n 1 n) n,
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 informationThe problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.
ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion
More informationFINALTERM EXAMINATION 2011 Calculus &. Analytical GeometryI
FINALTERM EXAMINATION 011 Clculus &. Anlyticl GeometryI Question No: 1 { Mrks: 1 )  Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More informationMath 324 Course Notes: Brief description
Brief description These re notes for Mth 324, n introductory course in Mesure nd Integrtion. Students re dvised to go through ll sections in detil nd ttempt ll problems. These notes will be modified nd
More informationNecessary and Sufficient Conditions for Differentiating Under the Integral Sign
Necessry nd Sufficient Conditions for Differentiting Under the Integrl Sign Erik Tlvil 1. INTRODUCTION. When we hve n integrl tht depends on prmeter, sy F(x f (x, y dy, it is often importnt to know when
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More information7.2 The Definition of the Riemann Integral. Outline
7.2 The Definition of the Riemnn Integrl Tom Lewis Fll Semester 2014 Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility Outline Upper nd
More informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
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 information5: The Definite Integral
5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce
More informationConvex 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
More information2 Definitions and Basic Properties of Extended Riemann Stieltjes Integrals
2 Definitions nd Bsic Properties of Extended Riemnn Stieltjes Integrls 2.1 Regulted nd Intervl Functions Regulted functions Let X be Bnch spce, nd let J be nonempty intervl in R, which my be bounded or
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationPractice final exam solutions
University of Pennsylvni Deprtment of Mthemtics Mth 26 Honors Clculus II Spring Semester 29 Prof. Grssi, T.A. Asher Auel Prctice finl exm solutions 1. Let F : 2 2 be defined by F (x, y (x + y, x y. If
More informationMATH Summary of Chapter 13
MATH 21259 ummry of hpter 13 1. Vector Fields re vector functions of two or three vribles. Typiclly, vector field is denoted by F(x, y, z) = P (x, y, z)i+q(x, y, z)j+r(x, y, z)k where P, Q, R re clled
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 informationMath 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 informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationWeek 7 Riemann Stieltjes Integration: Lectures 1921
Week 7 Riemnn Stieltjes Integrtion: Lectures 1921 Lecture 19 Throughout this section α will denote monotoniclly incresing function on n intervl [, b]. Let f be bounded function on [, b]. Let P = { = 0
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 n+2 a n+1 M n a 2 a 1. (2)
Rel Anlysis Fll 004 Tke Home Finl Key 1. Suppose tht f is uniformly continuous on set S R nd {x n } is Cuchy sequence in S. Prove tht {f(x n )} is Cuchy sequence. (f is not ssumed to be continuous outside
More informationChapter 4. Lebesgue Integration
4.2. Lebesgue Integrtion 1 Chpter 4. Lebesgue Integrtion Section 4.2. Lebesgue Integrtion Note. Simple functions ply the sme role to Lebesgue integrls s step functions ply to Riemnn integrtion. Definition.
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be relvlues nd smooth The pproximtion of n integrl by numericl
More informationPDE 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.....................................
More information7  Continuous random variables
71 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7  Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationA short introduction to local fractional complex analysis
A short introduction to locl rctionl complex nlysis Yng XioJun Deprtment o Mthemtics Mechnics, hin University o Mining Technology, Xuhou mpus, Xuhou, Jingsu, 228, P R dyngxiojun@63com This pper presents
More informationHenstock Kurzweil delta and nabla integrals
Henstock Kurzweil delt nd nbl integrls Alln Peterson nd Bevn Thompson Deprtment of Mthemtics nd Sttistics, University of NebrskLincoln Lincoln, NE 685880323 peterso@mth.unl.edu Mthemtics, SPS, The University
More information5.5 The Substitution Rule
5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n ntiderivtive is not esily recognizble, then we re in
More informationSection 4.4. Green s Theorem
The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higherdimensionl nlogues) with the definite integrls
More informationThe RiemannStieltjes integral. and some applications in complex analysis and probability theory. Klara Leffler
The RiemnnStieltjes integrl nd some pplictions in complex nlysis nd probbility theory Klr Leffler VT 2014 Exmensrbete, 15hp Kndidtexmen i mtemtik, 180hp Institutionen för mtemtik och mtemtisk sttistik
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More information(0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35
7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More information3.4 Numerical integration
3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,
More informationLine Integrals and Entire Functions
Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
More informationx = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x " 0 :
Clculus II MAT 146 Integrtion Applictions: Volumes of 3D Solids Our gol is to determine volumes of vrious shpes. Some of the shpes re the result of rotting curve out n xis nd other shpes re simply given
More informationProblem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:
(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one
More informationDiscrete Leastsquares Approximations
Discrete Lestsqures 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 informationA. Limits  L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More informationC1M14. Integrals as Area Accumulators
CM Integrls s Are Accumultors Most tetbooks do good job of developing the integrl nd this is not the plce to provide tht development. We will show how Mple presents Riemnn Sums nd the ccompnying digrms
More information1 Error Analysis of Simple Rules for Numerical Integration
cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion
More informationNumerical Integration
Chpter 1 Numericl Integrtion Numericl differentition methods compute pproximtions to the derivtive of function from known vlues of the function. Numericl integrtion uses the sme informtion to compute numericl
More informationMATH 174A: PROBLEM SET 5. Suggested Solution
MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous relvlued function on I), nd let L 1 (I) denote the completion
More informationTHIELE CENTRE. Linear stochastic differential equations with anticipating initial conditions
THIELE CENTRE for pplied mthemtics in nturl science Liner stochstic differentil equtions with nticipting initil conditions Nrjess Khlif, HuiHsiung Kuo, Hbib Ouerdine nd Benedykt Szozd Reserch Report No.
More informationTHE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS
THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS CARLOS SUERO, MAURICIO ALMANZAR CONTENTS 1 Introduction 1 2 Proof of Gussin Qudrture 6 3 Iterted 2Dimensionl Gussin Qudrture 20 4
More informationChapter 4. Additional Variational Concepts
Chpter 4 Additionl Vritionl Concepts 137 In the previous chpter we considered clculus o vrition problems which hd ixed boundry conditions. Tht is, in one dimension the end point conditions were speciied.
More information(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.
Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time
More informationOrthogonal Polynomials and LeastSquares Approximations to Functions
Chpter Orthogonl Polynomils nd LestSqures Approximtions to Functions **4/5/3 ET. Discrete LestSqures Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
More informationBIFURCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS
BIFRCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible
More informationUniversitaireWiskundeCompetitie. Problem 2005/4A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that
Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de
More informationMath 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two
More informationNumerical Integration
Numericl Integrtion Wouter J. Den Hn London School of Economics c 2011 by Wouter J. Den Hn June 3, 2011 Qudrture techniques I = f (x)dx n n w i f (x i ) = w i f i i=1 i=1 Nodes: x i Weights: w i Qudrture
More informationImproper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
More information7.6 The Use of Definite Integrals in Physics and Engineering
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems
More informationOn the Generalized Weighted QuasiArithmetic Integral Mean 1
Int. Journl of Mth. Anlysis, Vol. 7, 2013, no. 41, 20392048 HIKARI Ltd, www.mhikri.com http://dx.doi.org/10.12988/ijm.2013.3499 On the Generlized Weighted QusiArithmetic Integrl Men 1 Hui Sun School
More informationConducting Ellipsoid and Circular Disk
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
More informationLecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics  A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
More informationKeywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.
Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. LmiAthens Lmi 3500 Greece Abstrct Using
More informationThe Definite Integral
CHAPTER 3 The Definite Integrl Key Words nd Concepts: Definite Integrl Questions to Consider: How do we use slicing to turn problem sttement into definite integrl? How re definite nd indefinite integrls
More informationThe Basic Functional 2 1
2 The Bsic Functionl 2 1 Chpter 2: THE BASIC FUNCTIONAL TABLE OF CONTENTS Pge 2.1 Introduction..................... 2 3 2.2 The First Vrition.................. 2 3 2.3 The Euler Eqution..................
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationMATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC
FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot
More informationInnerproduct spaces
Innerproduct 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 informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationPartial Differential Equations
Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for OneDimensionl Eqution The reen s function provides complete solution to boundry
More informationEnergy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon
Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,
More informationProblem Set 3
14.102 Problem Set 3 Due Tuesdy, October 18, in clss 1. Lecture Notes Exercise 208: Find R b log(t)dt,where0
More informationIf u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du
Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
More informationMATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35
MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.
More 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 informationMEAN VALUE PROBLEMS OF FLETT TYPE FOR A VOLTERRA OPERATOR
Electronic Journl of Differentil Equtions, Vol. 213 (213, No. 53, pp. 1 7. ISSN: 1726691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu MEAN VALUE PROBLEMS OF FLETT
More informationSummary of Elementary Calculus
Summry of Elementry Clculus Notes by Wlter Noll (1971) 1 The rel numbers The set of rel numbers is denoted by R. The set R is often visulized geometriclly s numberline nd its elements re often referred
More informationFunctions of bounded variation
Division for Mthemtics Mrtin Lind Functions of bounded vrition Mthemtics Clevel thesis Dte: 20060130 Supervisor: Viktor Kold Exminer: Thoms Mrtinsson Krlstds universitet 651 88 Krlstd Tfn 054700 10
More information1 Sequences. 2 Series. 2 SERIES Analysis Study Guide
2 SERIES Anlysis Study Guide 1 Sequences Def: An ordered field is field F nd totl order < (for ll x, y, z F ): (i) x < y, y < x or x = y, (ii) x < y, y < z x < z (iii) x < y x + z < y + z (iv) 0 < y, x
More informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More informationCommunications inmathematicalanalysis Volume 6, Number 2, pp (2009) ISSN
Communictions inmthemticlanlysis Volume 6, Number, pp. 33 41 009) ISSN 19389787 www.communmthnl.org A SHARP GRÜSS TYPE INEQUALITY ON TIME SCALES AND APPLICATION TO THE SHARP OSTROWSKIGRÜSS INEQUALITY
More informationThe Dirac distribution
A DIRAC DISTRIBUTION A The Dirc distribution A Definition of the Dirc distribution The Dirc distribution δx cn be introduced by three equivlent wys Dirc [] defined it by reltions δx dx, δx if x The distribution
More information