DOCSLIB.ORG

Chapter 9 One Dimensional Integrals in Several Variables

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 9 One Dimensional Integrals in Several Variables Chapter 9 One dimensional integrals in several variables 9.1 Differentiation under the integral Note: less than 1 lecture Letf(x,y) be a function of two variables and define b g(y):= f(x,y)dx �a Suppose that f is differentiable in y. The question we ask is when can we simply “differentiate under the integral”, that is, when is it true thatg is differentiable and its derivative b ? ∂f g�(y) = (x,y)dx. y �a ∂ Differentiation is a limit and therefore we are really asking when do the two limitting operations of integration and differentiation commute. As we have seen, this is not always possible. In particular, ∂f thefirst question we would face is the integrability of ∂y . Let us prove a simple, but the most useful version of this theorem. Theorem 9.1.1 (Leibniz integral rule). Suppose f:[a,b] [c,d] R is a continuous function, such × → that ∂f exists for all(x,y) [a,b] [c,d] and is continuous. Define ∂y ∈ × b g(y):= f(x,y)dx. �a Then g:[c,d] R is differentiable and → b ∂f g�(y) = (x,y)dx. y �a ∂ 53 54 CHAPTER 9. ONE DIMENSIONAL INTEGRALS IN SEVERAL VARIABLES ∂f Note that the continuity requirements for f and ∂y can be weakened but not dropped outright. ∂f The main point is for ∂y to exist and be continuous for a small interval in the y direction. In applications, the [c,d] can be made a very small interval around the point where you need to differentiate. Proof. Fix y [c,d] and let ε>0 be given. As ∂f is continuous on [a,b] [c,d] it is uniformly ∈ ∂y × continuous. In particular, there exists δ>0 such that whenever y [c,d] with y y <δ and all 1 ∈ | 1 − | x [a,b] we have ∈ ∂f ∂f (x,y ) (x,y) <ε. ∂y 1 − ∂y � � � � Now suppose h is such that y+h� [c,d] and h <δ .� Fix x for a moment and apply mean value �∈ | | � theorem tofind ay 1 betweeny andy+h such that f(x,y+h) f(x,y) ∂f − = (x,y ). h ∂y 1 If h <δ then | | f(x,y+h) f(x,y) ∂f ∂f ∂f − (x,y) = (x,y ) (x,y) <ε. h − ∂y ∂y 1 − ∂y � � � � � � � � This argument worked� for everyx [a,b]. Therefore,� as� a function ofx � � ∈ � � � f(x,y+h) f(x,y) ∂f x − converges uniformly tox (x,y) ash 0. �→ h �→ ∂y → We only defined uniform convergence for sequences although the idea is the same. If you wish you can replaceh with 1/n above and letn ∞. → Now consider the difference quotient g(y+h) g(y) b f(x,y+h)dx b f(x,y)dx b f(x,y+h) f(x,y) − = a − a = − dx. h h h � � �a Uniform convergence can be taken underneath the integral and therefore g(y+h) g(y) b f(x,y+h) f(x,y) b ∂f lim − = lim − dx= (x,y)dx. h 0 h a h 0 h a ∂y → � → � Example 9.1.2: Let 1 f(y) = sin(x2 y 2)dx. − �0 Then 1 2 2 f �(y) = 2ycos(x y )dx. − − �0 9.1. DIFFERENTIATION UNDER THE INTEGRAL 55 Example 9.1.3: Suppose we start with 1 x 1 − dx. (x) �0 ln The function under the integral can be checked to be continuous, and in fact extends to be continuous on [0,1], and hence the integral exists. Trouble isfinding it. Introduce a parameter y and define a function. 1 xy 1 f(y):= − dx. (x) �0 ln xy 1 Again it can be checked that ln(−x) is a continuous function (that is extends to be continuous) of x and y for (x,y) [0,1] [0,1] . Therefore f is a continuous function of on [0,1]. In particular f(0) =0 . ∈ × For any ε>0 , the y derivative of the integrand, that is xy is continuous on [0,1] [ε,1] . Therefore, × fory> 0 we can differentiate under the integral sign 1 y 1 ln(x)x y 1 f �(y) = dx= x dx= . (x) y+ �0 ln �0 1 1 We need tofigure out f(1) , knowing f �(y) = y+1 and f(0) =0 . By elementary calculus wefind 1 f(1) = 0 f �(y)dy= ln(2). Therefore � 1 x 1 − dx= ln(2). (x) �0 ln Exercise 9.1.1: Prove the two statements that were asserted in the example. x 1 a) Prove ln−(x) extends to a continuous function of[0,1]. xy 1 b) Prove − extends to be a continuous function on[0,1] [0,1]. ln(x) × 9.1.1 Exercises Exercise 9.1.2: Suppose h:R R is a continuous function. Suppose that g:R R is which is continuously → → differentiable and compactly supported. That is there exists some M>0 such that g(x)=0 whenever x M . | | ≥ Define ∞ f(x):= h(y)g(x y)dy. ∞ − �− Show that f is differentiable. Exercise 9.1.3: Suppose f is an infinitely differentiable function (all derivatives exist) such that f(0) =0 . Then show that there exists another infinitely differentiable function g(x) such that f(x)=xg(x) . Finally show that if f (0)=0 then g(0)=0 . Hint:first write f(x)= x f (s)ds and then rewrite the integral to go � � � 0 � from0 to 1. � 56 CHAPTER 9. ONE DIMENSIONAL INTEGRALS IN SEVERAL VARIABLES 1 tx 1 n x Exercise 9.1.4: Compute 0 e dx. Derive the formula for 0 x e dx not using itnegration by parts, but by differentiation underneath� the integral. � n Exercise 9.1.5: Let U R be an open set and suppose f(x,y 1,y 2,...,y n) is a continuous function defined ⊂ ∂f ∂f ∂f on [0,1] U R n+1. Suppose , ,..., exist and are continuous on [0,1] U . Then prove that × ⊂ ∂y 1 ∂y 2 ∂y n × F:U R defined by → 1 F(y1,y 2,...,y n):= f(x,y 1,y 2,...,y n)dx �0 is continuously differentiable. 9.2. PATH INTEGRALS 57 9.2 Path integrals Note: ??? lectures 9.2.1 Piecewise smooth paths Definition 9.2.1. A continuously differentiable function γ:[a,b] R n is called a smooth path or a → continuously differentiable path ifγ is continuously differentiable andγ (t)= 0 for allt [a,b]. ∗ � � ∈ The function γ is called a piecewise smooth path or a piecewise continuously differentiable path if there existfinitely many points t =a<t <t < <t =b such that the restriction of the 0 1 2 ··· k functionγ [tj 1,tj] is smooth path. | − We say γ is a simple path if γ is a one-to-one function. A γ is a closed path if γ(a) =γ(b) , |(a,b) that is if the path starts and ends in the same point. Since γ is a function of one variable, we have seen before that treating γ�(t) as a matrix is equivalent to treating it as a vector since it is an n 1 matrix, that is, a column vector. In fact, by an × exercise, even the operator norm of γ�(t) is equal to the euclidean norm. Therefore, we will write γ�(t) as a vector as is usual. Generally, it is the direct image γ [a,b] that is what we are interested in. We will informally talk about a curve, by which we will generally mean the setγ [a,b] . � � Example 9.2.2: Letγ:[0,4] R 2 be defined by � � → (t,0) ift [0,1], ∈ (1,t 1) ift (1,2], γ(t):= − ∈ (3 t,1) ift (2,3], − ∈ (0,4 t) ift (3,4]. − ∈ Then the reader can check that the path is the unit square traversed counterclockwise. We can check that for example γ (t)=(1,t 1) and therefore (γ ) (t)=(0,1)=0 . It is good to |[1,2] − |[1,2] � � notice at this point that (γ ) (1) = (0,1), (γ ) (1) = (1,0), and γ (1) does not exist. That is, |[1,2] � |[0,1] � � at the corners γ is of course not differentiable, even though the restrictions are differentiable and the derivative depends on which restriction you take. Example 9.2.3: The condition that γ (t)=0 means that the image of γ has no “corners” where γ is � � smooth. For example, the function (t2,0) ift< 0, γ(t):= (0,t 2) ift 0. � ≥ ∗Note that the word “smooth” is used sometimes of continuously differentiable, sometimes for infinitely differen- tiable in the literature. 58 CHAPTER 9. ONE DIMENSIONAL INTEGRALS IN SEVERAL VARIABLES It is left for the reader to check that γ is continuously differentiable, yet the image γ(R) = (x,y) { ∈ R2 :(x,y) = ( s,0) or(x,y) = (0,s) for somes 0 has a “corner” at the origin. − ≥ } Example 9.2.4: A graph of a continuously differentiable function f:[a,b] R is a smooth path. → That is, defineγ:[a,b] R 2 by → γ(t):= t,f(t) . Thenγ �(t)= 1,f �(t) , which is never zero. � � There are other ways of parametrizing the path. That is, having a different path with the same � � image. For example, the function that takes t to (1 t)a+tb , takes the interval [0,1] to [a,b]. So let − α:[0,1] R 2 be defined by → α(t):= (1 t)a+tb,f((1 t)a+tb) .
Load more

Recommended publications
  • Leibniz's Rule and Fubini's Theorem Associated with Hahn Difference
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE I S S N 2provided 3 4 7 -by1921 KHALSA PUBLICATIONS Volume 12 Number 06 J o u r n a l of Advances in Mathematics Leibniz’s rule and Fubini’s theorem associated with Hahn difference operators å Alaa E. Hamza † , S. D. Makharesh † † Jeddah University, Department of Mathematics, Saudi, Arabia † Cairo University, Department of Mathematics, Giza, Egypt E-mail: [email protected] å † Cairo University, Department of Mathematics, Giza, Egypt E-mail: [email protected] ABSTRACT In 1945 , Wolfgang Hahn introduced his difference operator Dq, , which is defined by f (qt ) f (t) D f (t) = , t , q, (qt ) t 0 where = with 0 < q <1, > 0. In this paper, we establish Leibniz’s rule and Fubini’s theorem associated with 0 1 q this Hahn difference operator. Keywords. q, -difference operator; q, –Integral; q, –Leibniz Rule; q, –Fubini’s Theorem. 1Introduction The Hahn difference operator is defined by f (qt ) f (t) D f (t) = , t , (1) q, (qt ) t 0 where q(0,1) and > 0 are fixed, see [2]. This operator unifies and generalizes two well known difference operators. The first is the Jackson q -difference operator which is defined by f (qt) f (t) D f (t) = , t 0, (2) q qt t see [3, 4, 5, 6]. The second difference operator which Hahn’s operator generalizes is the forward difference operator f (t ) f (t) f (t) = , t R, (3) (t ) t where is a fixed positive number, see [9, 10, 13, 14].
    [Show full text]
  • Elementary Calculus
    Elementary Calculus 2 v0 2g 2 0 v0 g Michael Corral Elementary Calculus Michael Corral Schoolcraft College About the author: Michael Corral is an Adjunct Faculty member of the Department of Mathematics at School- craft College. He received a B.A. in Mathematics from the University of California, Berkeley, and received an M.A. in Mathematics and an M.S. in Industrial & Operations Engineering from the University of Michigan. This text was typeset in LATEXwith the KOMA-Script bundle, using the GNU Emacs text editor on a Fedora Linux system. The graphics were created using TikZ and Gnuplot. Copyright © 2020 Michael Corral. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. Preface This book covers calculus of a single variable. It is suitable for a year-long (or two-semester) course, normally known as Calculus I and II in the United States. The prerequisites are high school or college algebra, geometry and trigonometry. The book is designed for students in engineering, physics, mathematics, chemistry and other sciences. One reason for writing this text was because I had already written its sequel, Vector Cal- culus. More importantly, I was dissatisfied with the current crop of calculus textbooks, which I feel are bloated and keep moving further away from the subject’s roots in physics. In addi- tion, many of the intuitive approaches and techniques from the early days of calculus—which I think often yield more insights for students—seem to have been lost.
    [Show full text]
  • Lp-Solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term
    mathematics Article Lp-solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term Juan Carlos Cortés * and Marc Jornet Instituto Universitario de Matemática Multidisciplinar, Building 8G, access C, 2nd floor, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain; [email protected] * Correspondence: [email protected] Received: 25 May 2020; Accepted: 18 June 2020; Published: 20 June 2020 Abstract: This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay t > 0, by adding a random forcing term f (t) that varies with time: x0(t) = ax(t) + bx(t − t) + f (t), t ≥ 0, with initial condition x(t) = g(t), −t ≤ t ≤ 0. The coefficients a and b are assumed to be random variables, while the forcing term f (t) and the initial condition g(t) are stochastic processes on their respective time domains. The equation is regarded in the Lebesgue space Lp of random variables with finite p-th moment. The deterministic solution constructed with the method of steps and the method of variation of constants, which involves the delayed exponential function, is proved to be an Lp-solution, under certain assumptions on the random data. This proof requires the extension of the deterministic Leibniz’s integral rule for differentiation to the random scenario. Finally, we also prove that, when the delay t tends to 0, the random delay equation tends in Lp to a random equation with no delay. Numerical experiments illustrate how our methodology permits determining the main statistics of the solution process, thereby allowing for uncertainty quantification.
    [Show full text]
  • Integral and Differential Structure on the Free C∞-Ring Modality
    INTEGRAL AND DIFFERENTIAL STRUCTURE ON THE FREE C1-RING MODALITY Geoffrey CRUTTWELL Jean-Simon Pacaud LEMAY Rory B. B. LUCYSHYN-WRIGHT Resum´ e.´ Les categories´ integrales´ ont et´ e´ recemment´ developp´ ees´ comme homologues aux categories´ differentielles.´ En particulier, les categories´ inte-´ grales sont equip´ ees´ d’un operateur´ d’integration,´ appele´ la transformation integrale,´ dont les axiomes gen´ eralisent´ les identites´ d’integration´ de base du calcul comme l’integration´ par parties. Cependant, la litterature´ sur les categories´ integrales´ ne contient aucun exemple decrivant´ l’integration´ de fonctions lisses arbitraires : les exemples les plus proches impliquent l’inte-´ gration de fonctions polynomiales. Cet article comble cette lacune en develo-´ ppant un exemple de categorie´ integrale´ dont la transformation integrale´ agit sur des 1-formes differentielles´ lisses. De plus, nous fournissons un autre point de vue sur la structure differentielle´ de cet exemple cle,´ nous etudions´ les derivations´ et les coder´ elictions´ dans ce contexte et nous prouvons que les anneaux C1 libres sont des algebres` de Rota-Baxter. Abstract. Integral categories were recently developed as a counterpart to differential categories. In particular, integral categories come equipped with an integration operator, known as an integral transformation, whose axioms generalize the basic integration identities from calculus such as integration by parts. However, the literature on integral categories contains no example that captures integration of arbitrary smooth functions: the closest are exam- ples involving integration of polynomial functions. This paper fills in this gap G.C,J-S.P.L,R.L-W INT.& DIFF. STRUCT. ON C1-RING MOD. by developing an example of an integral category whose integral transforma- tion operates on smooth 1-forms.
    [Show full text]
  • Why Do Fingers Cool Before Than the Face When It Is Cold?
    Why do fingers cool before than the face when it is cold? Julio Ben´ıtez L´opez, Universidad Polit´ecnica de Valencia, [email protected] Abstract The purpose of this note is to show a physical application of vector calculus by using the heat equation. In this note we will answer the question of the title under reasonable physical hypotheses and without considering medical reasons as blood pressure or sweating. Vector calculus is used in many fields of physical sciences: electromagnetism, gravitation, thermodynamics, fluid dynamics, ... (see, for example, [1]). Here, we will apply vector calculus to the heat equation in order to show a simple and intuitive physical fact. Throughout this note some scattered mathematical concepts shall appear, as the gradient, the chain rule, the space curves, the divergence theorem, the Leibniz integral rule, and the isoperimetric inequality. We start with a brief introduction, the interested reader can consult [1, 2]. Let us consider a body which occupies a region Ω ⊂ R3 with a temperature distribution. Let T (x, y, z, t) be the temperature of the point (x, y, z) ∈ Ω at the time t. Thus, we can 1 model the temperature as a mapping T :Ω × R → R differentiable enough. The colder regions warm up and the warmer regions cool, and therefore we can imagine that there is a heat flux from warmer areas to cooler ones. It is natural to assume that the magnitude of this flux is proportional to the spatial change of the temperature T . Fourier’s Law relates the heat flux, J, and the gradient of T : ∂T ∂T ∂T J = −k∇T = −k , , , (1) ∂x ∂y ∂z where k is a positive constant which depends on the material.
    [Show full text]
  • Lecture 3: the Fundamental Theorem of Calculus
    Lecture 3: The Fundamental Theorem of Calculus Today: The Fundamental Theorem of Calculus, Leibniz Integral Rule, Mean Value Theo- rem We defined the definite integral and introduced the Evaluation Theorem to help us evaluate definite integrals. In today’s class, we will explore the relation between two central ideas of calculus: differentiation and integration. The Fundamental Theorem of Calculus The fundamental theorem of calculus deals with functions of the form x F (x) = f(t) dt, Za where f(x) is continuous on the interval [a, b] and x is taken on [a, b]. Recall that the definite integral is always a number that depends on the integrand, the upper and lower limits, but does not depend on the variable t that we integrate over. If x is fixed, the value of the function F (x) is also fixed, and if we let x vary, the function F (x), defined as the definite integral above, would vary with x. Example. Let f(x) = x + 1. Define x F (x) = f(t) dt. Z0 (1) Sketch the graph of f(x) over the interval [0, 3]. (2) Evaluate F (0), F (1), F (2), and F (3). (3) Find a formula for F (x). (4) Calculate F 0(x). The graph of f(x) is a straight line. 0 1 2 3 4 4 3 3 2 2 1 1 0 0 0 1 2 3 19 0 First, F (0) = 0 f(t) dt = 0. The values of F (x) at x = 1, 2, 3 could be computed by area of trapezoids: R 1 (1 + 2) 1 3 F (1) = f(t) dt = · = , 2 2 Z0 2 (1 + 3) 2 F (2) = f(t) dt = · = 4, 2 Z0 3 (1 + 4) 3 15 F (3) = f(t) dt = · = .
    [Show full text]
  • Math 346 Lecture #17 8.6 Fubini's Theorem and Leibniz's Integral Rule
    Math 346 Lecture #17 8.6 Fubini's Theorem and Leibniz's Integral Rule Fubini's Theorem { the switching of the order of the iterated integrals for the multivariate integral { is a consequence of passing the switching of the order of iterated integrals on step functions (which is easily shown) to L1 functions by means of the Monotone Convergence Theorem. A consequence of Fubini's Theorem is Leibniz's integral rule which gives conditions by which a derivative of a partial integral is the partial integral of a derivative, which is a useful tool in computation of multivariate integrals. 8.6.1 Fubini's Theorem We fix some notation to aid in stating Fubini's Theorem. Let X = [a; b] ⊂ Rn and Y = [c; d] ⊂ Rm. For g 2 L1(X; R) we write the integral of g as Z g(x) dx: X For h 2 L1(Y; R) we write the integral of h as Z h(y) dy: Y For f 2 L1(X × Y; R) we write the integral of f as Z f(x; y) dxdy: X×Y n+m Note. The measure dxdy on R is not quite the \product" of the measures λn = dx n n on R and λm = dy on R . The measure dxdy = λn+m is the \completion" of the product of the measures dx and dy, that is, the missing subsets of sets of measure zero are added and the product measure is extended. For f : X × Y ! R we define for each x 2 X the function fx : Y ! R by fx(y) = f(x; y): Theorem 8.6.1 (Fubini's Theorem).
    [Show full text]
  • Integral Calculus of One Variable Functions Northwestern University, Summer 2019
    Math 224: Integral Calculus of One Variable Functions Northwestern University, Summer 2019 Shuyi Weng Last Update: August 12, 2019 Contents Lecture 1: Antiderivatives and the Area Problem 1 Lecture 2: Definite Integrals 11 Lecture 3: The Fundamental Theorem of Calculus 19 Lecture 4: Substitution Rule and Integration by Parts 26 Lecture 5: Trigonometric Integrals 32 Lecture 6: Partial Fractions 39 Lecture 7: Improper Integrals 47 Lecture 8: Areas, Volumes, and Arc Lengths 54 Lecture 9: Sequences and Limits 62 Lecture 10: Series 71 Lecture 11: Convergence Tests 79 Lecture 12: Power Series 85 Lecture 13: Taylor and Maclaurin Series 92 Lecture 1: Antiderivatives and the Area Problem Today: Introduction, Review of Math 220, Antiderivatives, The Area Problem Welcome to Math 224! You have studied the idea of differentiation in your previous calculus course(s). We will steer our focus to integration in this course. Let's start by reviewing some important ideas of differential calculus. Review: Differential Calculus The central idea in differential calculus is, of course, differentiation. Definition. Given a function f(x), we define the derivative of f(x) by d f(x + h) − f(x) f 0(x) = f(x) = lim ; dx h!0 h given any value of x for which this limit exists. If the limit exists for all x in the domain of definition of f(x), we say that f(x) is differentiable. The following theorem includes some of the elementary differentiation rules. Theorem 1. Some elementary differentiation rules: i. (Constant Function) If c is a constant, then d (c) = 0: dx ii.
    [Show full text]
  • Fractional Calculus: Definitions and Applications Joseph M
    Western Kentucky University TopSCHOLAR® Masters Theses & Specialist Projects Graduate School 4-2009 Fractional Calculus: Definitions and Applications Joseph M. Kimeu Western Kentucky University, [email protected] Follow this and additional works at: http://digitalcommons.wku.edu/theses Part of the Algebraic Geometry Commons, and the Numerical Analysis and Computation Commons Recommended Citation Kimeu, Joseph M., "Fractional Calculus: Definitions and Applications" (2009). Masters Theses & Specialist Projects. Paper 115. http://digitalcommons.wku.edu/theses/115 This Thesis is brought to you for free and open access by TopSCHOLAR®. It has been accepted for inclusion in Masters Theses & Specialist Projects by an authorized administrator of TopSCHOLAR®. For more information, please contact [email protected]. FRACTIONAL CALCULUS: DEFINITIONS AND APPLICATIONS A Thesis Presented to The Faculty of the Department of Mathematics Western Kentucky University Bowling Green, Kentucky In Partial Fulfillment Of the Requirements for the Degree Master of Science By Joseph M. Kimeu May 2009 FRACTIONAL CALCULUS: DEFINITIONS AND APPLICATIONS Date Recommended 04/30/2009 Dr. Ferhan Atici, Director of Thesis Dr. Di Wu Dr. Dominic Lanphier _________________________________________ Dean, Graduate Studies and Research Date A C K N O W L E D G E M E N TS I would like to express my deepest gratitude to my advisor, Dr. Ferhan Atici, for her thoughtful suggestions and excellent guidance, without which the completion of this thesis would not have been possible. I would also like to extend my sincere appreciation to Dr. Wu and Dr. Lanphier for their serving as members of my thesis committee; and to Dr. Chen and Dr. Magin for their insightful discussions on the Mittag-Leffler function.
    [Show full text]
  • Vector Calc Stuff - Integrals, Grad, Div, Curl
    VECTOR CALC STUFF - INTEGRALS, GRAD, DIV, CURL 1. Theory Note 1. The hardest part (for me at least) about this vector calculus stu is the notation. There are a few dierent ways of writing any one thing. You should nd a system of notation that YOU like and stick to it. You also have to be able to translate from other notations into your own so that you can understand the problem when it is given to you. I also did everything here in 3 dimensions for illustration purposes, but its not too hard to generalize to more dimensions if you are so inclined. Once you understand what is going on with these things you should do a lot of problems from old exams. There is a category called Multivariable Calculus lled with problems on the written exam wiki. Many are very routine once the machinery below is set up; the problems themselves usually are pretty simple. Note 2. WARNING! On functions I talk about here, I assume the functions are well behaved, usually meaning that they are dierentiable. You should watch out because the denition of a mutli-dimensional function being dierentiable can be a bit slippery. One thing that will give us what we want is if f(x; y; z) has continuous partial derivatives , @f , @f and @f . @x @y @z Fact 3. (Chain Rule and Gradients) Let x : R3 ! R,y : R3 ! R, and z : R3 ! R be three dierentiable functions. We will use u; v; w here as our coordinates so that we write x(u; v; w); y(u; v; w); z(u; v; w).
    [Show full text]
  • Download Lebesgue Integration Free Ebook
    LEBESGUE INTEGRATION DOWNLOAD FREE BOOK J. H. Williamson | 128 pages | 15 Oct 2014 | Dover Publications Inc. | 9780486789774 | English | New York, United States Riemann integral These properties can be shown to hold in many different cases. A general not necessarily positive measurable function Lebesgue Integration is Lebesgue integrable if the area between the graph of f and the x -axis is finite:. I have to pay a certain sum, which I have collected in my pocket. Since this is Lebesgue Integration for Lebesgue Integration partition, f is not Riemann integrable. September The problem with this definition becomes Lebesgue Integration when we try to split the integral into two pieces. It Lebesgue Integration that Cantor did not keep up with the work of the later generation of French analysts. Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem. The integral of a simple function is equal Lebesgue Integration the measure of a given layer, times the height of that layer. As we stated earlier, these two definitions are equivalent. In applications such as Fourier series it is important to be able to approximate the integral of Lebesgue Integration function using integrals of approximations to the function. Measure Theory and Integration. Because the Riemann integral of a function is a number, this makes the Riemann integral a linear functional on the vector space of Riemann-integrable functions. Namespaces Article Talk. Since Lebesgue Integration started from an arbitrary partition and ended up as close as we wanted to either zero or one, it is false to say that we are eventually trapped near some number sso this function is not Riemann integrable.
    [Show full text]
  • Econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics del Valle, Gerardo Hernández Working Paper On a new class of barrier options Working Papers, No. 2014-23 Provided in Cooperation with: Bank of Mexico, Mexico City Suggested Citation: del Valle, Gerardo Hernández (2014) : On a new class of barrier options, Working Papers, No. 2014-23, Banco de México, Ciudad de México This Version is available at: http://hdl.handle.net/10419/129941 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. www.econstor.eu Banco de México Documentos de Investigación Banco de México Working Papers N° 2014-23 On a new class of barrier options Gerardo Hernández del Valle Banco de Mexico November 2014 La serie de Documentos de Investigación del Banco de México divulga resultados preliminares de trabajos de investigación económica realizados en el Banco de México con la finalidad de propiciar el intercambio y debate de ideas.
    [Show full text]