Lecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Lecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature"

Transcription

1 Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics - A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the evlution of singulr integrls using so-clled open qudrture formule. We lso discuss vrious techniques to obtin more ccurte pproximtions to singulr integrls such s subtrcting out the singulrity nd trnsformtions to non singulr integrls. We next introduce Guss Integrtion, which exploits the orthogonlity properties of orthogonl polynomils in order to obtin integrtion rules tht cn integrte polynomil of degree N exctly using only N smple points. We lso discuss integrtion on infinite integrls nd dptive integrtion. Key Concepts: Singulr Integrls, Open Newton-Cotes Formule, Guss Integrtion. 6 Singulr Integrls, Open Qudrture rules, nd Guss Qudrture Consider evluting singulr integrls of the form I 6. Integrting functions with singulrities e x x/ We cnnot just use the trpezoidl rule in this cse s f. Insted we use wht re clled open integrtion formule tht do not use the endpoints in the numericl pproximtion of the integrls. 6.. Open Newton-Cotes formule The Midpoint rule x +h/ x h/ f(x) f(x ) + (x x )f (x ) + (x x ) f (x ) + x +h/ f(x) f + (x x )f + (x x ) f (ξ) x h/ hf + h/ h/ hf + s 6 sf + s f (ξ) dξ h/ f (ξ) hf + h 8 f (ξ) hf + h 4 f (ξ)

2 x x x N The Composite Midpoint rule For cell x k +h/ x k h/ I f(x) h h h h f(x) hf(x k ) + h 4 f (x k ) Open Newton-Cotes Formule: x x k +h/ x k h/ h/ h/ h/ h/ f(x) f(x k + s)ds f(x k ) + sf (x k ) + s f (x k ) + ds f(x k ) + f(x k ) + h 4 f (x k ) h f (x k ) f(x k ) + h f (x) 4 f(x k ) + h 4 {f (b) f ()} (h) f(x) hf + 4 f (ξ) MidpointRule ξ (x, x ) x x x f(x) x 4 x f(x) h (f + f ) + h 4 f () (ξ) ξ (x, x ) 4h (f f + f ) + 8h5 9 f (4) (ξ) ξ (x, x 4 )

3 Numericl Integrtion 6.. Chnge of vrible (Eg.) I (Eg. ) I x /n f(x) n f(t n ) t nt n dt let t x /n x t n nt n dt I n (Eg. ) I π π/ f(t n )t n dt which is proper integrl for n f(x) x cos t sin t dt ( x ) / f(cos t) dt proper f(x) [x( x)] / x sin t sin t cos t dt f(sin t) sin t cos t dt sin t cos t π/ f(sin t)dt. 6.. Subtrcting the singulrity Consider evluting the integrl I e x x/ We note tht close to the singulr point x in the integrnd, the numertor cn be expnded bout the singulr point in the Tylor series: e x + x + x! I We now choose to rerrnge the integrnd s follows e x (e x ) + x/ x/ x / e x x / Using this decomposition we cn thus evlute the singulr prt nlyticlly nd the non-singulr prt numericlly. We cn expect to obtin more ccurte result thn simply using n open integrtion formul nd ignoring the singulrity. Since the ccurcy of the midpoint rule, for exmple, depends on the second derivtive of the integrnd (e x ), we cnnot expect even the midpoint rule to chieve its theoreticl rte of convergence for this integrl. To x /

4 4 retrieve the O(h ) ccurcy of the Midpoint rule we need to subtrct t lest three terms s follows: I + e x x/ + x/ x + x/ + 5 x / x / e x x x / + x/ x / + e x x x / x / e x x x / x / In figure we plot the errors obtined when the midpoint rule is used directly s well s the errors when nd terms re subtrcted from the integrnd. The second order ccurcy only returns when terms re removed so tht g is bounded on [, ], where g(x) ex x x / x /. Error(h) Midpoint Errors O(h p / ) in evluting exp(x)/x 4 Direct p.5 term p. terms p 6 h Figure. Plots of the errors vs h when the midpoint rule is used directly, nd when nd terms re removed

5 Numericl Integrtion 5 6. Guss Qudrture 6.. Orthogonl polynomils There exist fmilies of polynomil functions {ϕ n (x)} n over n intervl [, b] with weight w(x) : i.e.: ech of which re orthogonl with respect to integrtion ϕ m (x)ϕ n (x)w(x) δ mn C n. Eg. () Legendre Polynomils: {P n (x)} ; [, b] [, ]; w(x). In generl P n (x) cn be constructed by the recursion: P (x), P (x) x, P (x) (x ),... P n (x) n xp n (x) n (n ) P n (x). n ODE: ( x )y xy + (n + )ny ; y P n (x) Eg. () Lguerre Polynomils: {L n (x)}; [, b) [, ); w(x) e x L (x) ; L (x) x, L (x) 4x + x,... Recursion reltion: L n (x) (n x )L n (x) (n ) L n (x). ODE: xy + ( x)y + ny ; y L n (x). Eg. () Chebyshev Polynomils: {T n (x)}, [, b] [, ], w(x) / x Definition: T n (x) cos nθ where θ cos x. T (x), T (x) x, T (x) cos θ cos θ x,... The recursion reltion follows from the identity: cos nθ cos θ cos(n )θ cos(n ) T n (x) xt n (x) T n (x) ODE: ( x )y xy + n y y T n (x)

6 6 Hermite Polynomils: {H n (x)} (, b) (, ) w(x) e x H (x), H (x) x, H (x) 4x,... Recursion: H n (x) xh n (x) (n )H n (x) ODE: y xy + ny y H n (x). 6.. Expnsion of n rbitrry polynomil in terms of orthogonl polynomils Let q k (x) α + α x + + α k x k be ny polynomil of degree k. Then since the orthogonl polynomils {ϕ j (x)} re linerly independent, we cn lso express q k (x) s liner combintion of {ϕ j (x)}, j,..., k s follows: q k (x) α + α x + + α k x k β ϕ + β ϕ + + β k ϕ k. ( ) Exmple: Expnd q (x) x + x in terms of Legendre Polynomils P k (x) q (x) x + x in terms of Legendre polynomils β + β x + β (x ) ( β β ) + β x + β x β β 4 β β + β 5 q (x) 5 P (x) + P (x) 4 P (x) 6.. ϕ n (x) is orthogonl w.r.t the weight w(x) to ll lower degree polynomils q k (x), k,..., n The fct tht ny polynomil q k (x) cn be expnded s liner combintion of orthogonl polynomils {ϕ j (x)} k j up to degree k, s ws shown in the expnsion ( ), implies tht n orthogonl polynomil ϕ n (x) is orthogonl with respect to the weight w(x) to ny polynomil of lower degree thn n. In other words, if {q k (x)} n k polynomils of degrees k,..., n, then w(x)ϕ n (x)q k (x) for k,..., n re ny

7 Numericl Integrtion 7 To see this, consider ny kth degree polynomil q k (x) nd use use the expnsion ( ) to write w(x)ϕ n (x)q k (x) k m w(x)ϕ n (x) β k m k β k ϕ m (x) w(x)ϕ n (x)ϕ m (x) The ltter integrls vnish becuse of the orthogonlity of polynomils of distinct degree with respect to the weight w(x). Ide behind Guss Qudrture: We ssume tht the pproximtion of 6. Guss-Legendre qudrture f(x) is given by: f(x) w i f(x i ) where the w i re weights given to the function vlues f(x i ). If we regrd the x i s free then cn we do better by choosing these x i ppropritely? Shift to the intervl [, ] : There is no loss of generlity in ssuming tht [, b] [, ] since the chnge of vribles x [, b] to t [, ]: will reduce the integrl to x t(b ) i + F (t)dt where F (t) (b ) f(x(t)) ( + b) Let us pproximte f on [, ] by polynomil of degree M nd integrte the resulting polynomil. The error involved is of the form: f(x) M p M (x) + f (M) (ξ) M! f k M f k w k + l k (x) + (x x )... (x x M ) f[x,..., x M, x](x x )... (x x M ) f[x,..., x M, x](x x )... (x x M ) where l k (x) nd w k This formul will be exct if f is polynomil of degree M since then P M (x) f(x). M (x x j )/x k x j ) j j k l k (x).

8 8 Now let M N nd choose x,..., x N to be the zeros of the Legendre polynomil P N (x) of degree N. In this cse, ll the weights w k for k N + s cn be seen from the clcultion or w k f(x) C k C k l k (x) P N (x)q k,n (x) P N (x) ( N {}}{{}}{ (x x )... (x x N ) (x x N+ )... (x x k )(x x k+ )... (x x M ) k N + (x k x )... (x k x N+ )... (x k x M ) S β S P S (x) no mtter where we choose the x N+,..., x N. f(x) f k w k + f (N) (ξ) (N)! f k w k + f (N) (ξ) (N)! ) (x x )... (x x N ) C N [P N (x)] f k w k + N+ (N!) 4 (N + )[(N)!] f (N) (ξ). Thus for only N points we cn integrte polynomil of degree N exctly. For rbitrrily chosen smple points {x k }, we would hve required N points to chieve the sme ccurcy. Expressions for the bscisse nd the weights The {x k } N re the zeros of the Legendre polynomil of degree N. The weights w k ( x k ) (N + ) [P N+ (x k )] m x k w k ± /.8 8 8/9. ± /9

9 Numericl Integrtion Generting the coefficients nd weights using the method of undetermined coefficients N: This qudrture rule must integrte polynomil of degree exctly + x + x + x + w f(x ) + w f(x ) w w x x w ( + x ) w x x N: This qudrture rule must integrte polynomil of degree 5 exctly. + x + x + x + 4 x x w ( + x + 4 x 4 ) + w w + w w x w x 4 5 x 5 / 5 x 5 w 5 w w w 8 9

10 Exmple: Evlute I sin πx We mke use of the trnsformtion of vribles x t( ) + t+ t x I sin πx ( + t) sin π dt ( ) sin π π cos sin π ( + ) Compre this result with the trpezium rule using two function evlutions, which yields.5. Now using three point Guss-Legendre formul: I N 5 9. sin π ) π cos ( sin π sin Other Guss-Qudrture formule ) Hermite-Guss: w(x) e x (, b) (, ) e x f(x) w k f(x k ) + N! π N (N)! f (N) (ξ) w k N+ N! π [H N+ (x k )] m x k w k ± ±

11 ) Chebyshev-Guss Qudrture: w(x) ( x ) [, b] [, ]. Numericl Integrtion f(x) N π w k f(x k ) + x N (N)! f (N) (ξ) w k π T N (x k)t N+ (x k ) π N (weights re ll equl). 6.4 Integrting Functions on Infinite Intervls Consider evluting integrls of the form If f(x) x p s x then exists only if p >. I f(x) x p x p p 6.4. Truncte the Infinite Intervl c I f(x) + f(x) c I + I Use the symptotic behviour of f to determine how lrge c should be for I < ϵ/ I c Eg. cos xe x cos xe x c ln(ϵ/) 8.4 OR use n symptotic pproximtion for I. Evlute I using the stndrd integrtion rules. c e x e c ϵ 8

12 6.4. Mp to Finite Intervl I Choose the mp such tht x p dt f(x) where f(x) x x p Eg. p : x p x t t dt I x t f(x) Now s t f ( ) ( ) t t t so integrnd is finite OR t e x x ln t f f(x) ( ) dt t t f( ln t) t OR [, ) [, S] [S, ) nd on [, S] set t x/s on [S, ) set t S/x dt 6.4. Specilized Guss integrtion rules for infinite intervls () Guss-Lguerre Integrtion: (, ) w e x e x f(x) g(x) (b) Guss-Hermite integrtion: (, ) w e x e x f(x) w k f(ξ k ) e x ( e x g(x) ) }{{} f(x) w k f(ξ k )

13 Numericl Integrtion 6.5 Adptive Integrtion 6.5. Adptive Simpson Integrtion I() h [f + 4f + f 5 ] h5 }{{} 9 f (4) (ξ) S (h) f f f f 4 f 5 h I() (h/) {[f + 4f + f + 4f 4 + f 5 ]} (h/)5 } {{} 9 S 4 (h) Assume f (4) (ξ) f (4) (ξ ) f (4) (ξ) pproximtely constnt. Substrct { } f (4) (ξ ) + f (4) (ξ ) S S 4 h5 9 f (4) (ξ) [ ] 5 5 ( ) h f (4) (ξ) h 5 9 f (4) (ξ) 6 5 (S S 4 ) I() S 4 h5 9 f (4) (ξ) ( 4 ) 5 S S 4 I, I 4 is 5 S S 4 < TOL S 4 YES DONE NO is ( ) ( ) h h 5 S S 4 < TOL 6.5. The Best of Both Worlds Guss-Ptterson Integrtion Guss Qudrture Rules obtin the highest ccurcy for the lest number of function evlutions. x x x x Newton-Cotes Formule llow for utomtic nd dptive integrtion rules becuse the regulr grid llows one to use ll previous function evlutions towrd subsequent refinements - the dptive Trpezium rule is n exmple of this. The Guss-Ptterson integrtion rules llow one to build higher order integrtion schemes which mke use of previous function evlutions in subsequent clcultions. These rules hve the ttrctive high order ccurcy typicl of Guss qudrture rules. This is idel for dptive integrtion. Ptterson, T.N.L. 968, The Optimum Addition of Points T Qudrture Formuls, Mth. Comp.,, p

Lecture 14: Quadrature

Lecture 14: Quadrature Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

More information

Interpolation. Gaussian Quadrature. September 25, 2011

Interpolation. Gaussian Quadrature. September 25, 2011 Gussin Qudrture September 25, 2011 Approximtion of integrls Approximtion of integrls by qudrture Mny definite integrls cnnot be computed in closed form, nd must be pproximted numericlly. Bsic building

More information

Discrete Least-squares Approximations

Discrete Least-squares Approximations Discrete Lest-squres Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve

More information

Lecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at Urbana-Champaign. March 20, 2014

Lecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at Urbana-Champaign. March 20, 2014 Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t Urbn-Chmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method

More information

Orthogonal Polynomials and Least-Squares Approximations to Functions

Orthogonal Polynomials and Least-Squares Approximations to Functions Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny

More information

Advanced Computational Fluid Dynamics AA215A Lecture 3 Polynomial Interpolation: Numerical Differentiation and Integration.

Advanced Computational Fluid Dynamics AA215A Lecture 3 Polynomial Interpolation: Numerical Differentiation and Integration. Advnced Computtionl Fluid Dynmics AA215A Lecture 3 Polynomil Interpoltion: Numericl Differentition nd Integrtion Antony Jmeson Winter Qurter, 2016, Stnford, CA Lst revised on Jnury 7, 2016 Contents 3 Polynomil

More information

COT4501 Spring Homework VII

COT4501 Spring Homework VII COT451 Spring 1 Homework VII The ssignment is due in clss on Thursdy, April 19, 1. There re five regulr problems nd one computer problem (using MATLAB). For written problems, you need to show your work

More information

Numerical integration

Numerical integration 2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter

More information

Numerical quadrature based on interpolating functions: A MATLAB implementation

Numerical quadrature based on interpolating functions: A MATLAB implementation SEMINAR REPORT Numericl qudrture bsed on interpolting functions: A MATLAB implementtion by Venkt Ayylsomyjul A seminr report submitted in prtil fulfillment for the degree of Mster of Science (M.Sc) in

More information

Review of Gaussian Quadrature method

Review of Gaussian Quadrature method Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

More information

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

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

More information

Best Approximation. Chapter The General Case

Best Approximation. Chapter The General Case Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given

More information

COSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III) - Gauss Quadrature and Adaptive Quadrature

COSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III) - Gauss Quadrature and Adaptive Quadrature COSC 336 Numericl Anlysis I Numericl Integrtion nd Dierentition III - Guss Qudrture nd Adptive Qudrture Edgr Griel Fll 5 COSC 336 Numericl Anlysis I Edgr Griel Summry o the lst lecture I For pproximting

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. 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

Lecture 14 Numerical integration: advanced topics

Lecture 14 Numerical integration: advanced topics Lecture 14 Numericl integrtion: dvnced topics Weinn E 1,2 nd Tiejun Li 2 1 Deprtment of Mthemtics, Princeton University, weinn@princeton.edu 2 School of Mthemticl Sciences, Peking University, tieli@pku.edu.cn

More information

New Expansion and Infinite Series

New Expansion and Infinite Series Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

More information

3.4 Numerical integration

3.4 Numerical integration 3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,

More information

Section 6.1 Definite Integral

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

More information

Lecture 19: Continuous Least Squares Approximation

Lecture 19: Continuous Least Squares Approximation Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for

More information

Numerical Integration

Numerical Integration Numericl Integrtion Wouter J. Den Hn London School of Economics c 2011 by Wouter J. Den Hn June 3, 2011 Qudrture techniques I = f (x)dx n n w i f (x i ) = w i f i i=1 i=1 Nodes: x i Weights: w i Qudrture

More information

LECTURE 19. Numerical Integration. Z b. is generally thought of as representing the area under the graph of fèxè between the points x = a and

LECTURE 19. Numerical Integration. Z b. is generally thought of as representing the area under the graph of fèxè between the points x = a and LECTURE 9 Numericl Integrtion Recll from Clculus I tht denite integrl is generlly thought of s representing the re under the grph of fèxè between the points x = nd x = b, even though this is ctully only

More information

Sections 5.2: The Definite Integral

Sections 5.2: The Definite Integral Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)

More information

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

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

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx... Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

Best Approximation in the 2-norm

Best Approximation in the 2-norm Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion

More information

Lecture Notes: Orthogonal Polynomials, Gaussian Quadrature, and Integral Equations

Lecture Notes: Orthogonal Polynomials, Gaussian Quadrature, and Integral Equations 18330 Lecture Notes: Orthogonl Polynomils, Gussin Qudrture, nd Integrl Equtions Homer Reid My 1, 2014 In the previous set of notes we rrived t the definition of Chebyshev polynomils T n (x) vi the following

More information

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl

More information

Numerical Integration

Numerical Integration Chpter 1 Numericl Integrtion Numericl differentition methods compute pproximtions to the derivtive of function from known vlues of the function. Numericl integrtion uses the sme informtion to compute numericl

More information

III. Lecture on Numerical Integration. File faclib/dattab/lecture-notes/numerical-inter03.tex /by EC, 3/14/2008 at 15:11, version 9

III. Lecture on Numerical Integration. File faclib/dattab/lecture-notes/numerical-inter03.tex /by EC, 3/14/2008 at 15:11, version 9 III Lecture on Numericl Integrtion File fclib/dttb/lecture-notes/numerical-inter03.tex /by EC, 3/14/008 t 15:11, version 9 1 Sttement of the Numericl Integrtion Problem In this lecture we consider the

More information

MA 124 January 18, Derivatives are. Integrals are.

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

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

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

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s). Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different

More information

Math 100 Review Sheet

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

More information

2

2 1 Notes for Numericl Anlysis Mth 5466 by S. Adjerid Virgini Polytechnic Institute nd Stte University (A Rough Drft) 2 Contents 1 Differentition nd Integrtion 5 1.1 Introduction............................

More information

CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED NEWTON-COTES QUADRATURES

CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED NEWTON-COTES QUADRATURES Filomt 27:4 (2013) 649 658 DOI 10.2298/FIL1304649M Published by Fculty of Sciences nd Mthemtics University of Niš Serbi Avilble t: http://www.pmf.ni.c.rs/filomt CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED

More information

Lecture 1. Functional series. Pointwise and uniform convergence.

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

Big idea in Calculus: approximation

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

Midpoint Approximation

Midpoint Approximation Midpoint Approximtion Sometimes, we need to pproximte n integrl of the form R b f (x)dx nd we cnnot find n ntiderivtive in order to evlute the integrl. Also we my need to evlute R b f (x)dx where we do

More information

Harmonic Mean Derivative - Based Closed Newton Cotes Quadrature

Harmonic Mean Derivative - Based Closed Newton Cotes Quadrature IOSR Journl of Mthemtics (IOSR-JM) e-issn: - p-issn: 9-X. Volume Issue Ver. IV (My. - Jun. 0) PP - www.iosrjournls.org Hrmonic Men Derivtive - Bsed Closed Newton Cotes Qudrture T. Rmchndrn D.Udykumr nd

More information

Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivatives/Indefinite Integrals of Basic Functions Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second

More information

8 Laplace s Method and Local Limit Theorems

8 Laplace s Method and Local Limit Theorems 8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved

More information

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230 Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

More information

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ). AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

More information

221A Lecture Notes WKB Method

221A Lecture Notes WKB Method A Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using ψ x, t = e

More information

OPEN NEWTON - COTES QUADRATURE WITH MIDPOINT DERIVATIVE FOR INTEGRATION OF ALGEBRAIC FUNCTIONS

OPEN NEWTON - COTES QUADRATURE WITH MIDPOINT DERIVATIVE FOR INTEGRATION OF ALGEBRAIC FUNCTIONS IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 9-6 pissn: -78 OPEN NEWTON - COTES QUADRATURE WITH MIDPOINT DERIVATIVE FOR INTEGRATION OF ALGEBRAIC FUNCTIONS T. Rmchndrn R.Priml

More information

Keywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.

Keywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula. Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. Lmi-Athens Lmi 3500 Greece Abstrct Using

More information

Anonymous Math 361: Homework 5. x i = 1 (1 u i )

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

Math 360: A primitive integral and elementary functions

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

Section 6.1 INTRO to LAPLACE TRANSFORMS

Section 6.1 INTRO to LAPLACE TRANSFORMS Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform

More information

F (x) dx = F (x)+c = u + C = du,

F (x) dx = F (x)+c = u + C = du, 35. The Substitution Rule An indefinite integrl of the derivtive F (x) is the function F (x) itself. Let u = F (x), where u is new vrible defined s differentible function of x. Consider the differentil

More information

Math 554 Integration

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

Convex Sets and Functions

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

More information

Quantum Physics II (8.05) Fall 2013 Assignment 2

Quantum Physics II (8.05) Fall 2013 Assignment 2 Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.

More information

Riemann Integrals and the Fundamental Theorem of Calculus

Riemann Integrals and the Fundamental Theorem of Calculus Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums

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 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

MAT 168: Calculus II with Analytic Geometry. James V. Lambers

MAT 168: Calculus II with Analytic Geometry. James V. Lambers MAT 68: Clculus II with Anlytic Geometry Jmes V. Lmbers Februry 7, Contents Integrls 5. Introduction............................ 5.. Differentil Clculus nd Quotient Formuls...... 5.. Integrl Clculus nd

More information

1 Error Analysis of Simple Rules for Numerical Integration

1 Error Analysis of Simple Rules for Numerical Integration cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion

More information

Not for reproduction

Not for reproduction AREA OF A SURFACE OF REVOLUTION cut h FIGURE FIGURE πr r r l h FIGURE A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundry of solid of revolution of the type

More information

Unit 5. Integration techniques

Unit 5. Integration techniques 18.01 EXERCISES Unit 5. Integrtion techniques 5A. Inverse trigonometric functions; Hyperbolic functions 5A-1 Evlute ) tn 1 3 b) sin 1 ( 3/) c) If θ = tn 1 5, then evlute sin θ, cos θ, cot θ, csc θ, nd

More information

If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du

If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find nti-derivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible

More information

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

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

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

More information

We divide the interval [a, b] into subintervals of equal length x = b a n

We divide the interval [a, b] into subintervals of equal length x = b a n Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:

More information

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.-3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This

More information

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral. Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

More information

Chapters 4 & 5 Integrals & Applications

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

More information

MATH 174A: PROBLEM SET 5. Suggested Solution

MATH 174A: PROBLEM SET 5. Suggested Solution MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion

More information

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0. STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

More information

Week 10: Riemann integral and its properties

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

CS667 Lecture 6: Monte Carlo Integration 02/10/05

CS667 Lecture 6: Monte Carlo Integration 02/10/05 CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of

More information

Numerical Methods I. Olof Widlund Transcribed by Ian Tobasco

Numerical Methods I. Olof Widlund Transcribed by Ian Tobasco Numericl Methods I Olof Widlund Trnscribed by In Tobsco Abstrct. This is prt one of two semester course on numericl methods. The course ws offered in Fll 011 t the Cournt Institute for Mthemticl Sciences,

More information

Summer MTH142 College Calculus 2. Section J. Lecture Notes. Yin Su University at Buffalo

Summer MTH142 College Calculus 2. Section J. Lecture Notes. Yin Su University at Buffalo Summer 6 MTH4 College Clculus Section J Lecture Notes Yin Su University t Bufflo yinsu@bufflo.edu Contents Bsic techniques of integrtion 3. Antiderivtive nd indefinite integrls..............................................

More information

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

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

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

Theoretische Physik 2: Elektrodynamik (Prof. A.-S. Smith) Home assignment 4

Theoretische Physik 2: Elektrodynamik (Prof. A.-S. Smith) Home assignment 4 WiSe 1 8.1.1 Prof. Dr. A.-S. Smith Dipl.-Phys. Ellen Fischermeier Dipl.-Phys. Mtthis Sb m Lehrstuhl für Theoretische Physik I Deprtment für Physik Friedrich-Alexnder-Universität Erlngen-Nürnberg Theoretische

More information

y = f(x) $ # Area & " % ! $ b f(x) g(x) dx = [F (x) G(x)] b a

y = f(x) $ # Area &  % ! $ b f(x) g(x) dx = [F (x) G(x)] b a MthsTrck (NOTE Feb 23: This is the old version of MthsTrck. New books will be creted during 23 nd 24) Topic 9 Module 9 Introduction Integrtion to Mtrices y = f(x) Income = Tickets! Price = =! 25 $! 25

More information

1B40 Practical Skills

1B40 Practical Skills B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need

More information

Chapter 6. Riemann Integral

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

Lecture 3. Limits of Functions and Continuity

Lecture 3. Limits of Functions and Continuity Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live

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[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

METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS

METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS Journl of Young Scientist Volume III 5 ISSN 44-8; ISSN CD-ROM 44-9; ISSN Online 44-5; ISSN-L 44 8 METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS An ALEXANDRU Scientific Coordintor: Assist

More information

CHM Physical Chemistry I Chapter 1 - Supplementary Material

CHM Physical Chemistry I Chapter 1 - Supplementary Material CHM 3410 - Physicl Chemistry I Chpter 1 - Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 59-6), nd "Mthemticl Bckground " (pp 109-111). 1. Derivtion

More information

Math 5440 Problem Set 3 Solutions

Math 5440 Problem Set 3 Solutions Mth 544 Mth 544 Problem Set 3 Solutions Aron Fogelson Fll, 25 1: Logn, 1.5 # 2) Repet the derivtion for the eqution of motion of vibrting string when, in ddition, the verticl motion is retrded by dmping

More information

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

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

Ordinary differential equations

Ordinary differential equations Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview Introduction-Modelling Bsic concepts to understnd n ODE. Description nd properties

More information

Efficient Computation of a Class of Singular Oscillatory Integrals by Steepest Descent Method

Efficient Computation of a Class of Singular Oscillatory Integrals by Steepest Descent Method Applied Mthemticl Sciences, Vol. 8, 214, no. 31, 1535-1542 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/1.12988/ms.214.43166 Efficient Computtion of Clss of Singulr Oscilltory Integrls by Steepest Descent

More information

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

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

Introduction to Numerical Analysis

Introduction to Numerical Analysis Introduction to Numericl Anlysis Doron Levy Deprtment of Mthemtics nd Center for Scientific Computtion nd Mthemticl Modeling (CSCAMM) University of Mrylnd June 14, 2012 D. Levy CONTENTS Contents 1 Introduction

More information

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

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

7. Numerical evaluation of definite integrals

7. Numerical evaluation of definite integrals 7. Numericl evlution of definite integrls Tento učení text yl podpořen z Operčního progrmu Prh - Adptilit Hn Hldíková Numericl pproximtion of definite integrl is clled numericl qudrture, the formuls re

More information

(0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35

(0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35 7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know

More information

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

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

Chapter 4. Additional Variational Concepts

Chapter 4. Additional Variational Concepts Chpter 4 Additionl Vritionl Concepts 137 In the previous chpter we considered clculus o vrition problems which hd ixed boundry conditions. Tht is, in one dimension the end point conditions were speciied.

More information

Czechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction

Czechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued

More information

440-2 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam

440-2 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam 440-2 Geometry/Topology: Differentible Mnifolds Northwestern University Solutions of Prctice Problems for Finl Exm 1) Using the cnonicl covering of RP n by {U α } 0 α n, where U α = {[x 0 : : x n ] RP

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

Line Integrals. Partitioning the Curve. Estimating the Mass

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

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

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

SOME PROPERTIES OF CHEBYSHEV SYSTEMS

SOME PROPERTIES OF CHEBYSHEV SYSTEMS SOME PROPERTIES OF CHEBYSHEV SYSTEMS RICHARD A. ZALIK Abstrct. We study Chebyshev systems defined on n intervl, whose constituent functions re either complex or rel vlued, nd focus on problems tht my hve

More information