# Integrals along Curves.

Size: px
Start display at page:

Download "Integrals along Curves."

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 spce-filling 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 one-sided 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. Riemnn-Stieltjes integrls. There is simple generliztion of the Riemnn integrl, clled the Riemnn-Stieltjes integrl. The Riemnn-Stieltjes integrl of rel-vlued 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 Riemnn-Stieltjes 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 Riemnn-Stieltjes 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 Riemnn-Stieltjes 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 two-dimensionl 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 Riemnn-Stieltjes 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 Cuchy-Riemnn 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

### Math Advanced Calculus II

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

More information

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

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

More information

### u(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.

Lecture 4 Complex Integrtion MATH-GA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex

More information

### 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 information

### The Regulated and Riemann Integrals

Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

More information

### II. Integration and Cauchy s Theorem

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

More information

### Complex variables lecture 5: Complex integration

omplex vribles lecture 5: omplex integrtion Hyo-Sung Ahn School of Mechtronics Gwngju Institute of Science nd Technology (GIST) 1 Oryong-dong, Buk-gu, Gwngju, Kore Advnced Engineering Mthemtics omplex

More information

### p(t) dt + i 1 re it ireit dt =

Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)

More information

### Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004

Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when

More information

### INDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Analysis Autumn 2012

Lecture 6: Line Integrls INDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Anlysis Autumn 2012 August 8, 2012 Lecture 6: Line Integrls Lecture 6: Line Integrls Lecture 6: Line Integrls Integrls of complex

More information

### Review of Calculus, cont d

Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

### Notes on length and conformal metrics

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

More information

### 7.2 Riemann Integrable Functions

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

More information

### Section 17.2 Line Integrals

Section 7. Line Integrls Integrting Vector Fields nd Functions long urve In this section we consider the problem of integrting functions, both sclr nd vector (vector fields) long curve in the plne. We

More information

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

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

More information

### Review on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.

Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5. - 5.3) Remrks on the course. Slide Review: Sec. 5.-5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description

More information

### We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.

Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn

More information

### MAA 4212 Improper Integrals

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

More information

### g i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f

1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where

More information

### Mapping the delta function and other Radon measures

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

More information

### Week 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 information

### Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

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

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

More information

### Week 10: Line Integrals

Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.

More information

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

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

More information

### Review of Riemann Integral

1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.

More information

### MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL

MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL DR. RITU AGARWAL MALVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR, INDIA-302017 Tble of Contents Contents Tble of Contents 1 1. Introduction 1 2. Prtition

More information

### 10 Vector Integral Calculus

Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve

More information

### Riemann is the Mann! (But Lebesgue may besgue to differ.)

Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >

More information

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

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

More information

### 11 An introduction to Riemann Integration

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

More information

### 1 The Riemann Integral

The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re

More information

### One dimensional integrals in several variables

Chpter 9 One dimensionl integrls in severl vribles 9.1 Differentition under the integrl Note: less thn 1 lecture Let f (x,y be function of two vribles nd define g(y : b f (x,y dx Suppose tht f is differentible

More information

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

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

More information

### A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

### Definite integral. Mathematics FRDIS MENDELU

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

More information

### Lecture 1. Functional series. Pointwise and uniform convergence.

1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

More information

### The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

### Chapter 6. Riemann Integral

Introduction to Riemnn integrl Chpter 6. Riemnn Integrl Won-Kwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl

More information

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

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

More information

### df dt f () b f () a dt

Vector lculus 16.7 tokes Theorem Nme: toke's Theorem is higher dimensionl nlogue to Green's Theorem nd the Fundmentl Theorem of clculus. Why, you sk? Well, let us revisit these theorems. Fundmentl Theorem

More information

### The Riemann Integral

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

More information

### Integrals - Motivation

Integrls - Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is non-liner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but

More information

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

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

More information

### ODE: Existence and Uniqueness of a Solution

Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =

More information

### MATH 311: COMPLEX ANALYSIS INTEGRATION LECTURE

MATH 311: COMPLEX ANALYSIS INTEGRATION LECTURE Contents 1. Introduction 1 2. A fr-reching little integrl 4 3. Invrince of the complex integrl 5 4. The bsic complex integrl estimte 6 5. Comptibility 8 6.

More information

### Anonymous 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 information

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

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

More information

### Calculus II: Integrations and Series

Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]

More information

### arxiv: v1 [math.ca] 11 Jul 2011

rxiv:1107.1996v1 [mth.ca] 11 Jul 2011 Existence nd computtion of Riemnn Stieltjes integrls through Riemnn integrls July, 2011 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde

More information

### Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

### Line Integrals. Chapter Definition

hpter 2 Line Integrls 2.1 Definition When we re integrting function of one vrible, we integrte long n intervl on one of the xes. We now generlize this ide by integrting long ny curve in the xy-plne. It

More information

### MA123, 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. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

### 1. On some properties of definite integrals. We prove

This short collection of notes is intended to complement the textbook Anlisi Mtemtic 2 by Crl Mdern, published by Città Studi Editore, [M]. We refer to [M] for nottion nd the logicl stremline of the rguments.

More information

### Calculus in R. Chapter Di erentiation

Chpter 3 Clculus in R 3.1 Di erentition Definition 3.1. Suppose U R is open. A function f : U! R is di erentible t x 2 U if there exists number m such tht lim y!0 pple f(x + y) f(x) my y =0. If f is di

More information

### Complex integration. L3: Cauchy s Theory.

MM Vercelli. L3: Cuchy s Theory. Contents: Complex integrtion. The Cuchy s integrls theorems. Singulrities. The residue theorem. Evlution of definite integrls. Appendix: Fundmentl theorem of lgebr. Discussions

More information

### Calculus I-II Review Sheet

Clculus I-II Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing

More information

### Math 554 Integration

Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we

More information

### INTRODUCTION TO INTEGRATION

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

More information

### Riemann Sums and Riemann Integrals

Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties

More information

### Calculus and linear algebra for biomedical engineering Week 11: The Riemann integral and its properties

Clculus nd liner lgebr for biomedicl engineering Week 11: The Riemnn integrl nd its properties Hrtmut Führ fuehr@mth.rwth-chen.de Lehrstuhl A für Mthemtik, RWTH Achen Jnury 9, 2009 Overview 1 Motivtion:

More information

### Advanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015

Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n

More information

### Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

More information

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

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

More information

### Chapter 6. Infinite series

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

More information

### MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 5 SOLUTIONS. cos t cos at dt + i

MATH 85: COMPLEX ANALYSIS FALL 9/ PROBLEM SET 5 SOLUTIONS. Let R nd z C. () Evlute the following integrls Solution. Since e it cos t nd For the first integrl, we hve e it cos t cos t cos t + i t + i. sin

More information

### 38 Riemann sums and existence of the definite integral.

38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

More information

### Riemann Sums and Riemann Integrals

Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct

More information

### This is a short summary of Lebesgue integration theory, which will be used in the course.

3 Chpter 0 ntegrtion theory This is short summry of Lebesgue integrtion theory, which will be used in the course. Fct 0.1. Some subsets (= delmängder E R = (, re mesurble (= mätbr in the Lebesgue sense,

More information

### Main topics for the First Midterm

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

More information

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

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

More information

### MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

### LECTURE. INTEGRATION AND ANTIDERIVATIVE.

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

More information

### For a continuous function f : [a; b]! R we wish to define the Riemann integral

Supplementry Notes for MM509 Topology II 2. The Riemnn Integrl Andrew Swnn For continuous function f : [; b]! R we wish to define the Riemnn integrl R b f (x) dx nd estblish some of its properties. This

More information

### ON THE C-INTEGRAL BENEDETTO BONGIORNO

ON THE C-INTEGRAL BENEDETTO BONGIORNO Let F : [, b] R be differentible function nd let f be its derivtive. The problem of recovering F from f is clled problem of primitives. In 1912, the problem of primitives

More information

### The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

More information

### Section 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 information

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

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

More information

### 1 Line Integrals in Plane.

MA213 thye Brief Notes on hpter 16. 1 Line Integrls in Plne. 1.1 Introduction. 1.1.1 urves. A piece of smooth curve is ssumed to be given by vector vlued position function P (t) (or r(t) ) s the prmeter

More information

### MATH , Calculus 2, Fall 2018

MATH 36-2, 36-3 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly

More information

### Overview of Calculus I

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

More information

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

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

More information

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

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

More information

### Chapter One: Calculus Revisited

Chpter One: Clculus Revisited 1 Clculus of Single Vrible Question in your mind: How do you understnd the essentil concepts nd theorems in Clculus? Two bsic concepts in Clculus re differentition nd integrtion

More information

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

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

More information

### Introduction to the Calculus of Variations

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

More information

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

MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further

More information

### I. INTEGRAL THEOREMS. A. Introduction

1 U Deprtment of Physics 301A Mechnics - I. INTEGRAL THEOREM A. Introduction The integrl theorems of mthemticl physics ll hve their origin in the ordinry fundmentl theorem of clculus, i.e. xb x df dx dx

More information

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

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

More information

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

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

More information

### Review of basic calculus

Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

More information

### 1 The fundamental theorems of calculus.

The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection

More information

### arxiv:math/ v2 [math.ho] 16 Dec 2003

rxiv:mth/0312293v2 [mth.ho] 16 Dec 2003 Clssicl Lebesgue Integrtion Theorems for the Riemnn Integrl Josh Isrlowitz 244 Ridge Rd. Rutherford, NJ 07070 jbi2@njit.edu Februry 1, 2008 Abstrct In this pper,

More information

### Big idea in Calculus: approximation

Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:

More information

### x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b

CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick

More information

### (4.1) D r v(t) ω(t, v(t))

1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution

More information

### Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)

More information

### Calculus of Variations

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

More information

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

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

More information

### Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner

More information