DOCSLIB.ORG

General Riemann Integrals and Their Computation Via Domain

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
General Riemann Integrals and Their Computation Via Domain Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1999 General Riemann Integrals and Their omputC ation via Domain. Bin Lu Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Lu, Bin, "General Riemann Integrals and Their omputC ation via Domain." (1999). LSU Historical Dissertations and Theses. 7001. https://digitalcommons.lsu.edu/gradschool_disstheses/7001 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, phnt bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI* Bell & Howell Information and Learning 300 North Zeeb Road, Ann Artx)r, Ml 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. GENERAL RIEMANN INTEGRALS AND THEIR COMPUTATION VIA DOMAIN A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfilment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Bin Lu B.S., Shaanxi Mechanical Engineering Institute, 1982 M.S., Tennessee Teclmological University, 1994 M.S., Louisiana State University, 1996 August 1999 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9945728 UMI Microform 9945728 Copyright 1999, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeh Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS It would be impossible for me to finish this dissertation without the guidance, advice, and encouragement from my doctoral advisor, Professor Jimmie Lawson. I am very grateful for his dedications and efforts that introduced me into this research project and some other research areas, I thank him. Thanks also to my committee members. Professor Delzell, Professor Dorroh, Professor Hildebrant, Professor Rau, and Professor Sundar for their comments, especially thanks to Professor Delzell for his T^pCnical help. I would like to thank Mathematics Depaurtment at LSU for the financial support during the years of graduate studies. Finally I thank all prayers and supports from my family and friends here in this country and in China through many years of graduate studies. u Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS A C K N O W L E D G M E N T S ..................................................................................................ii ABSTRACT ..........................................................................................................................iv CHAPTER 1. INTRODUCTION ..................................................................................... I CHAPTER 2. RIEMANN-LIKE INTEGRATION ........................................................8 2.1. Algebras of Sets and Measures .........................................................................8 2.2. Definitions and Basic R esults .............................................................................9 2.3. Integrability of ^-measurable Functions ......................................................17 2.4. Integration on S u b s e ts ....................................................................................... 22 CHAPTER 3. EXTENSIONS OF THE INTEGRATION .....................................26 3.1. Integration of ^-measurable Functions ..........................................................26 3.2. Integration of ^-measurable Functions ..........................................................31 3.3. Integration of the Crescent Continuous Functions .....................................32 CHAPTER 4. ORDER-PRESERVING FUNCTIONS ............................................. 36 4.1. Integrals with Respect to Simple Valuations ..................................................36 4.1.1. Integrability with Respect to Simple V aluations ............................ 36 4.1.2. Integrals with Respect to Simple Valuations .................................38 4.2. Integrals of Order-Preserving Functions ......................................................39 4.2.1. Basics of the Domain T heory .............................................................. 39 4.2.2. Integrals of the Order-Preserving Functions .....................................42 CHAPTER 5. ON MAXIMAL POINT SPA CES ......................................................52 5.1. Maximal Point Space ....................................................................................... 52 5.2. Probabilistic Power Domain on the Upper Space .........................................55 5.3. The Generalized Riemann Integral .................................................................. 57 5.4. Integration on Maximal Point S pace .............................................................. 60 CHAPTER 6. HECKMANN’S APPROACH .............................................................. 69 6.1. Some Basics ................................................................................... 69 6.2. Heckmann’s A pproacli ....................................................................................... 72 6.3. Equivalence of Integrations ...............................................................................73 REFERENCES .................................................................................................................77 VITA ...................................................................................................................................... 79 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT In this work we extend the domain-theoretic approach of the generalized Riemann integral introduced by A. Edalat in 1995. We begin by laying down a related theory of general Riemann integration for bounded real-valued functions on an arbitrary set X with a finitely additive measure on an algebra of subsets of X. Based on the theory developed we obtain a formula to calculate integral of a bounded function in terms of the regular Riemann integral. By the classical extension theorems on set functions we can further extend this generalized Riemann integral to more general set functions such as valuations on lattices of subsets. For the setting of bounded functions defined on a continuous domain D with a Borel measure for the Scott topology, we can compute the Riemann integral of a function effectively and so the value of the integral can be obtained up to a given accuracy. By invoking the results of J. Lawson we can extend this type of Riemann integral to maximal point spaces, as a special case of the Polish spaces. Furthermore we show that whenX is a compact metric space, our approach of Riemann integration is equivalent to the generalized Riemami integration introduced by A. Edalat in the sense that the two integrals yield the same value. Finally we prove that the approach of integration taken by R. Heckinann is equivalent to our approach, and the values of the integrals are the same. IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. INTRODUCTION The theory of Riemann integration of real-valued functions was developed by Fourier, Cauchy, Dirichlet, Riemann and Darboux, along with other mathematicians of the last century. With its simple and constructive features it became as it is today a foundation of calculus with applications in all fields of sciences. Even shortly after this theory was introduced, mathematicians of the era realized that the integral had serious shortcomings.
Load more

Recommended publications
  • Definition of Riemann Integrals
    The Definition and Existence of the Integral Definition of Riemann Integrals Definition (6.1) Let [a; b] be a given interval. By a partition P of [a; b] we mean a finite set of points x0; x1;:::; xn where a = x0 ≤ x1 ≤ · · · ≤ xn−1 ≤ xn = b We write ∆xi = xi − xi−1 for i = 1;::: n. The Definition and Existence of the Integral Definition of Riemann Integrals Definition (6.1) Now suppose f is a bounded real function defined on [a; b]. Corresponding to each partition P of [a; b] we put Mi = sup f (x)(xi−1 ≤ x ≤ xi ) mi = inf f (x)(xi−1 ≤ x ≤ xi ) k X U(P; f ) = Mi ∆xi i=1 k X L(P; f ) = mi ∆xi i=1 The Definition and Existence of the Integral Definition of Riemann Integrals Definition (6.1) We put Z b fdx = inf U(P; f ) a and we call this the upper Riemann integral of f . We also put Z b fdx = sup L(P; f ) a and we call this the lower Riemann integral of f . The Definition and Existence of the Integral Definition of Riemann Integrals Definition (6.1) If the upper and lower Riemann integrals are equal then we say f is Riemann-integrable on [a; b] and we write f 2 R. We denote the common value, which we call the Riemann integral of f on [a; b] as Z b Z b fdx or f (x)dx a a The Definition and Existence of the Integral Left and Right Riemann Integrals If f is bounded then there exists two numbers m and M such that m ≤ f (x) ≤ M if (a ≤ x ≤ b) Hence for every partition P we have m(b − a) ≤ L(P; f ) ≤ U(P; f ) ≤ M(b − a) and so L(P; f ) and U(P; f ) both form bounded sets (as P ranges over partitions).
    [Show full text]
  • Generalizations of the Riemann Integral: an Investigation of the Henstock Integral
    Generalizations of the Riemann Integral: An Investigation of the Henstock Integral Jonathan Wells May 15, 2011 Abstract The Henstock integral, a generalization of the Riemann integral that makes use of the δ-fine tagged partition, is studied. We first consider Lebesgue’s Criterion for Riemann Integrability, which states that a func- tion is Riemann integrable if and only if it is bounded and continuous almost everywhere, before investigating several theoretical shortcomings of the Riemann integral. Despite the inverse relationship between integra- tion and differentiation given by the Fundamental Theorem of Calculus, we find that not every derivative is Riemann integrable. We also find that the strong condition of uniform convergence must be applied to guarantee that the limit of a sequence of Riemann integrable functions remains in- tegrable. However, by slightly altering the way that tagged partitions are formed, we are able to construct a definition for the integral that allows for the integration of a much wider class of functions. We investigate sev- eral properties of this generalized Riemann integral. We also demonstrate that every derivative is Henstock integrable, and that the much looser requirements of the Monotone Convergence Theorem guarantee that the limit of a sequence of Henstock integrable functions is integrable. This paper is written without the use of Lebesgue measure theory. Acknowledgements I would like to thank Professor Patrick Keef and Professor Russell Gordon for their advice and guidance through this project. I would also like to acknowledge Kathryn Barich and Kailey Bolles for their assistance in the editing process. Introduction As the workhorse of modern analysis, the integral is without question one of the most familiar pieces of the calculus sequence.
    [Show full text]
  • Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration Over a Polygonal Element Sulaiman Abo Diab
    Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration over a Polygonal Element Sulaiman Abo Diab To cite this version: Sulaiman Abo Diab. Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration over a Polygonal Element. 2019. hal-02306578v1 HAL Id: hal-02306578 https://hal.archives-ouvertes.fr/hal-02306578v1 Preprint submitted on 6 Oct 2019 (v1), last revised 5 Nov 2019 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration over a Polygonal Element Sulaiman Abo Diab Faculty of Civil Engineering, Tishreen University, Lattakia, Syria [email protected] Abstracts: In this paper, the Riemann integral and the fundamental of calculus will be used to perform double integrals on polygonal domain surrounded by closed curves. In this context, the double integral with two variables over the domain is transformed into sequences of single integrals with one variable of its primitive. The sequence is arranged anti clockwise starting from the minimum value of the variable of integration. Finally, the integration over the closed curve of the domain is performed using only one variable.
    [Show full text]
  • Lecture 15-16 : Riemann Integration Integration Is Concerned with the Problem of finding the Area of a Region Under a Curve
    1 Lecture 15-16 : Riemann Integration Integration is concerned with the problem of ¯nding the area of a region under a curve. Let us start with a simple problem : Find the area A of the region enclosed by a circle of radius r. For an arbitrary n, consider the n equal inscribed and superscibed triangles as shown in Figure 1. f(x) f(x) π 2 n O a b O a b Figure 1 Figure 2 Since A is between the total areas of the inscribed and superscribed triangles, we have nr2sin(¼=n)cos(¼=n) · A · nr2tan(¼=n): By sandwich theorem, A = ¼r2: We will use this idea to de¯ne and evaluate the area of the region under a graph of a function. Suppose f is a non-negative function de¯ned on the interval [a; b]: We ¯rst subdivide the interval into a ¯nite number of subintervals. Then we squeeze the area of the region under the graph of f between the areas of the inscribed and superscribed rectangles constructed over the subintervals as shown in Figure 2. If the total areas of the inscribed and superscribed rectangles converge to the same limit as we make the partition of [a; b] ¯ner and ¯ner then the area of the region under the graph of f can be de¯ned as this limit and f is said to be integrable. Let us de¯ne whatever has been explained above formally. The Riemann Integral Let [a; b] be a given interval. A partition P of [a; b] is a ¯nite set of points x0; x1; x2; : : : ; xn such that a = x0 · x1 · ¢ ¢ ¢ · xn¡1 · xn = b.
    [Show full text]
  • Intuitive Mathematical Discourse About the Complex Path Integral Erik Hanke
    Intuitive mathematical discourse about the complex path integral Erik Hanke To cite this version: Erik Hanke. Intuitive mathematical discourse about the complex path integral. INDRUM 2020, Université de Carthage, Université de Montpellier, Sep 2020, Cyberspace (virtually from Bizerte), Tunisia. hal-03113845 HAL Id: hal-03113845 https://hal.archives-ouvertes.fr/hal-03113845 Submitted on 18 Jan 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Intuitive mathematical discourse about the complex path integral Erik Hanke1 1University of Bremen, Faculty of Mathematics and Computer Science, Germany, [email protected] Interpretations of the complex path integral are presented as a result from a multi-case study on mathematicians’ intuitive understanding of basic notions in complex analysis. The first case shows difficulties of transferring the image of the integral in real analysis as an oriented area to the complex setting, and the second highlights the complex path integral as a tool in complex analysis with formal analogies to path integrals in multivariable calculus. These interpretations are characterised as a type of intuitive mathematical discourse and the examples are analysed from the point of view of substantiation of narratives within the commognitive framework.
    [Show full text]
  • More Information About Line and Contour Integrals
    More information about line and contour integrals. Real line integrals. Let P (x, y)andQ(x, y) be continuous real valued functions on the plane and let γ be a smooth directed curve. We wish to define the line integral Z P (x, y) dx + Q(x, y) dy. γ This can be done in several ways. One of which is to partition the curve γ by a very large number of points, define Riemann sums for the partition, and then take a limit as the “mesh” of the partition gets finer and finer. However an argument like the one we did in class for complex line integrals shows that if c(t)=(x(t),y(t)) with a ≤ t ≤ b is a smooth parameterization of γ then Z Z b P (x, y) dx + Q(x, y) dy = P (x(t),y(t))x0(t)+Q(x(t),y(t)) dt. γ a R This reduces computing γ P (x, y) dx + Q(x, y) dy to first finding a parameterization of γ and then computing a standard Riemann integral such as we have all seen in calculus. Here is an example. Let γ be the upper half of the circle |z| =1 transversed counter clockwise. Then compute Z (1) xy2 dx − (x + y + x3y) dy. γ This half circle is parameterized by x(t)=cos(t),y(t)=sin(t) with 0 ≤ t ≤ π. Now the calculation proceeds just as if we were doing a change of variables. That is dx = − sin(t) dt, dy = cos(t) dt 1 2 Using these formulas in (1) gives Z xy2 dx − (x + y + x3y) dy γ Z π = cos(t)sin2(t)(− sin(t)) dt − (cos(t)+sin(t)+cos3(t)sin(t)) cos(t) dt Z0 π = cos(t)sin2(t)(− sin(t)) − (cos(t)+sin(t)+cos3(t)sin(t)) cos(t) dt 0 With a little work1 this can be shown to have the value −(2/5+π/2).
    [Show full text]
  • Fractional Integrals and Derivatives • Yuri Luchko Fractional Integrals and Derivatives: “True” Versus “False”
    Fractional Integrals and Derivatives • Yuri Luchko Yuri • Fractional Integrals and Derivatives “True” versus “False” Edited by Yuri Luchko Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Fractional Integrals and Derivatives: “True” versus “False” Fractional Integrals and Derivatives: “True” versus “False” Editor Yuri Luchko MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editor Yuri Luchko Beuth Technical University of Applied Sciences Berlin Germany Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Mathematics (ISSN 2227-7390) (available at: https://www.mdpi.com/journal/mathematics/special issues/Fractional Integrals Derivatives). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Volume Number, Page Range. ISBN 978-3-0365-0494-0 (Hbk) ISBN 978-3-0365-0494-0 (PDF) © 2021 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Editor .............................................. vii Preface to ”Fractional Integrals and Derivatives: “True” versus “False”” ............. ix Yuri Luchko and Masahiro Yamamoto The General Fractional Derivative and Related Fractional Differential Equations Reprinted from: Mathematics 2020, 8, 2115, doi:10.3390/math8122115 ...............
    [Show full text]
  • On a New Generalized Integral Operator and Certain Operating Properties
    axioms Article On a New Generalized Integral Operator and Certain Operating Properties Paulo M. Guzman 1,2,†, Luciano M. Lugo 1,†, Juan E. Nápoles Valdés 1,† and Miguel Vivas-Cortez 3,*,† 1 FaCENA, UNNE, Av. Libertad 5450, Corrientes 3400, Argentina; [email protected] (P.M.G.); [email protected] (L.M.L.); [email protected] (J.E.N.V.) 2 Facultad de Ingeniería, UNNE, Resistencia, Chaco 3500, Argentina 3 Facultad de Ciencias Exactas y Naturales, Escuela de Ciencias Físicas y Matemática, Pontificia Universidad Católica del Ecuador, Quito 170143, Ecuador * Correspondence: [email protected] † These authors contributed equally to this work. Received: 20 April 2020; Accepted: 22 May 2020; Published: 20 June 2020 Abstract: In this paper, we present a general definition of a generalized integral operator which contains as particular cases, many of the well-known, fractional and integer order integrals. Keywords: integral operator; fractional calculus 1. Preliminars Integral Calculus is a mathematical area with so many ramifications and applications, that the sole intention of enumerating them makes the task practically impossible. Suffice it to say that the simple procedure of calculating the area of an elementary figure is a simple case of this topic. If we refer only to the case of integral inequalities present in the literature, there are different types of these, which involve certain properties of the functions involved, from generalizations of the known Mean Value Theorem of classical Integral Calculus, to varied inequalities in norm. Let (U, ∑ U, u) and (V, ∑ V, m) be s-fìnite measure spaces, and let (W, ∑ W, l) be the product of these spaces, thus W = UxV and l = u x m.
    [Show full text]
  • The Generalized Riemann Integral Robert M
    AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 20 20 The Generalized Riemann Integral Robert M. McLeod The Generalized Riemann Integral The Generalized The Generalized Riemann Integral is addressed to persons who already have an acquaintance with integrals they wish to extend and to the teachers of generations of students to come. The organization of the Riemann Integral work will make it possible for the rst group to extract the principal results without struggling through technical details which they may nd formidable or extraneous to their purposes. The technical level starts low at the opening of each chapter. Thus, readers may follow Robert M. McLeod each chapter as far as they wish and then skip to the beginning of the next. To readers who do wish to see all the details of the arguments, they are given. The generalized Riemann integral can be used to bring the full power of the integral within the reach of many who, up to now, haven't gotten a glimpse of such results as monotone and dominated conver- gence theorems. As its name hints, the generalized Riemann integral is de ned in terms of Riemann sums. The path from the de nition to theorems exhibiting the full power of the integral is direct and short. AMS / MAA PRESS 4-color Process 290 pages spine: 9/16" finish size: 5.5" X 8.5" 50 lb stock 10.1090/car/020 THE GENERALIZED RIEMANN INTEGRAL By ROBERT M. McLEOD THE CARUS MATHEMATICAL MONOGRAPHS Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications E.
    [Show full text]
  • Study of New Class of Q-Fractional Integral Operator
    Study of new class of q−fractional integral operator M. Momenzadeh, N. I. Mahmudov Near East University Eastern mediterranean university Lefkosa, TRNC, Mersin 10, Turkey Famagusta, North Cyprusvia Mersin 10 Turkey Email: [email protected] [email protected] April 29, 2019 Abstract We study the new class of q−fractional integral operator. In the aid of iterated Cauchy integral approach to fractional integral operator, we applied tpf(t) instead of f(t) in these integrals and with parameter p a new class of q-fractional integral operator is introduced. Recently the q-analogue of fractional differential integral operator is studied.[8][4][9][10][13][14]All of these operators are q−analogue of Riemann fractional differential operator. The new class of introduced operatorgeneralize all these defined operator and can be cover the q-analogue of Hadamard fractional differential operator. Some properties of this operator is investigated. Keyword: q-fractional differential integral operator, fractional calculus, Hadamard fractional differ- ential operator 1 Introduction Fractional calculus has a long history and has recently gone through a period of rapid development. When Jackson (1908)[16] defined q-differential operator, q−calculus became a bridge between mathematics and physics. It has a lot of applications in different mathematical areas such as combinatorics, number theory, basic hypergeometric functions and other sciences: quantum theory, mechanics, theory of relativity, capacitor theory, electrical circuits, particle physics, viscoelastic, electro analytical chemistry, neurology, diffusion systems, control theory and statistics; The q−Riemann-Liouville fractional integral operator was introduced by Al-Salam [12], from that time few q-analogues of Riemann operator were studied.
    [Show full text]
  • Contents 1. the Riemann and Lebesgue Integrals. 1 2. the Theory of the Lebesgue Integral
    Contents 1. The Riemann and Lebesgue integrals. 1 2. The theory of the Lebesgue integral. 7 2.1. The Monotone Convergence Theorem. 7 2.2. Basic theory of Lebesgue integration. 9 2.3. Lebesgue measure. Sets of measure zero. 12 2.4. Nonmeasurable sets. 13 2.5. Lebesgue measurable sets and functions. 14 3. More on the Riemann integral. 18 3.1. The fundamental theorems of calculus. 21 3.2. Characterization of Riemann integrability. 22 1. The Riemann and Lebesgue integrals. Fix a positive integer n. Recall that Rn and Mn are the family of rectangles in Rn and the algebra of multirectangles in Rn, respec- tively. Definition 1.1. We let F + F B n ; n; n; be the set of [0; 1] valued functions on Rn; the vector space of real valued functions n on R ; the vector space of f 2 Fn such that ff =6 0g[rng f is bounded, respectively. We let S B \S M ⊂ F S+ S \F + ⊂ S+ M n = n ( n) n and we let n = n n ( n): n Thus s 2 Sn if and only if s : R ! R, rng s is finite, fs = yg is a multirectangle 2 R f 6 g 2 S+ 2 S ≥ for each y and s = 0 is bounded and s n if and only if s n and s 0. S+ S+ 2 S+ We let n;" be the set of nondecreasing sequences in n . For each s n;" we let 2 F + sup s n be such that n sup s(x) = supfsν (x): ν 2 Ng for x 2 R and we let n f + 2 N+g In;"(s) = sup In (sν ): ν : 2 S+ Remark 1.1.
    [Show full text]
  • MATH 311 Topics in Applied Mathematics I Lecture 32: Gradient, Divergence, and Curl
    MATH 311 Topics in Applied Mathematics I Lecture 32: Gradient, divergence, and curl. Review of integral calculus. Gradient, divergence, and curl Gradient of a scalar field f = f (x1, x2 ..., xn) is ∂f ∂f ∂f grad f = , ,..., . ∂x1 ∂x2 ∂xn Divergence of a vector field F =(F1, F2,..., Fn) is ∂F ∂F ∂F div F = 1 + 2 + + n . ∂x1 ∂x2 ··· ∂xn Curl of a vector field F =(F1, F2, F3) is ∂F ∂F ∂F ∂F ∂F ∂F curl F = 3 2 , 1 3 , 2 1 . ∂x2 − ∂x3 ∂x3 − ∂x1 ∂x1 − ∂x2 e1 e2 e3 Informally, curl F = ∂ ∂ ∂ . ∂x1 ∂x2 ∂x3 F1 F2 F3 Del notation Gradient, divergence, and curl can be denoted in a compact way using the del (a.k.a. nabla a.k.a. atled) “operator” ∂ ∂ ∂ = , ,..., . ∇ ∂x1 ∂x2 ∂xn Namely, grad f = f , div F = F, curl F = F. ∇ ∇· ∇ × Theorem 1 div(curl F) = 0 wherever the vector field F is twice continuously differentiable. Theorem 2 curl(grad f )= 0 wherever the scalar field f is twice continuously differentiable. In the del notation, ( F) = 0 and ( f )= 0. ∇· ∇ × ∇ × ∇ Note that f = ∆f (the Laplacian, also denoted 2f ). ∇·∇ ∇ Riemann sums and Riemann integral Definition. A Riemann sum of a function f : [a, b] R → with respect to a partition P = x , x ,..., xn of [a, b] { 0 1 } generated by samples tj [xj−1, xj ] is a sum ∈ n (f , P, tj )= f (tj )(xj xj−1). S Xj=1 − Remark. P = x , x ,..., xn is a partition of [a, b] if { 0 1 } a = x < x < < xn− < xn = b.
    [Show full text]