MATH 174A: PROBLEM SET 5. Suggested Solution

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "MATH 174A: PROBLEM SET 5. Suggested Solution"

Transcription

1 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 of C(I; R) with respect to. 1. (1) Show tht the Riemnn integrl : C(I; R) R, f b f() d, etends to bounded liner mp : L 1 (I) R (which would be clled the Lebesgue integrl). () Suppose f L 1 (I), i.e. f is the equivlence clss of Cuchy sequence {f n } of continuous functions. We sy tht f if there eists Cuchy sequence {f n } representing f such tht f n or ll n. Show tht if f nd g then f + g, nd if c R, c then cf. Show lso tht if f then f. (Thus, it mkes sense to sy whether n element of L 1 (I) is non-negtive on I. We lso sy f g if f g, so it mkes sense to sy tht n element of L 1 (I) is greter thn nother element of L 1 (I).) (3) Show tht if f nd f then f. Use this to conclude tht is prtil order on L 1 (I), i.e. f g nd g h implies f h, nd f g nd g f implies f g. (4) Suppose tht {f (k) } k1 is sequence in L1 (I) nd f (k) f in L 1 (I). Show tht if f (k) for ll k then f. (Hint: Consider f (k) C(I; R) such tht f (k) nd f (k) is close to f (k).) (5) Suppose I. Show tht there is no continuous liner mp E : L 1 (I) R such tht E (f) f() for ll f C(I; R). Tht is, evluting continuous functions t cnnot be etended in resonble mnner to L 1 (I), i.e. elements of L 1 (I) do not hve vlues t ny given point. (Hint: If such mp eisted, nd f L 1 (I) were represented by Cuchy sequence {f n }, wht would E (f) be? Now find different Cuchy sequences representing the sme f.) Solution. (1) Recll the B.L.T. theorem: Suppose T is bounded liner trnsformtion from normed vector spce (V 1,. 1 ) to complete liner spce (V,. ). Then T cn be uniquely etended to bounded liner trnsformtion (with the sme bound), T, from the completion of V 1 to (V,. ). Note tht (C(I; R),. 1 ) is normed vector spce with completion L 1 (I) nd R is complete vector spce. We only need to show tht the Riemnn integrl : C(I; R) R is bounded liner trnsformtion, then we cn pply the B.L.T. theorem bove to etend it to bounded liner mp on L 1 (R). Linerity of the Riemnn integrl is well-known result from elementry nlysis. For

2 boundedness, simply note tht f f f 1. So f is bounded liner trnsformtion with 1 (ctully equlity holds). () Suppose f nd g. By definition, there re Cuchy sequences {f n } nd {g n } representing f nd g respectively with f n, g n for ll n. Then {f n + g n } is non-negtive Cuchy sequence representing f + g. Thus, f + g. On the other hnd, if c R nd c, then {cf n } is non-negtive Cuchy sequence representing cf, hence cf. Now, ssume tht f nd let {f n } be non-negtive Cuchy sequence representing f. By definition of the Lebesgue integrl in prt (1), we hve f lim f n. Since ech f n is non-negtive, f n. This implies tht f. (3) Assume tht f nd f. Then there eist two non-negtive Cuchy sequences {f n } nd {f n} for f nd f respectively. Note tht { f n} is nonpositive Cuchy sequence representing f, therefore, lim f n +f n 1. Since both f n nd f n re non-negtive, f n + f n 1 f n 1 + f n 1. Hence lim f n 1 nd therefore {f n } lso represents. We conclude tht f. Net, we show tht is prtil order on L 1 (I). For trnsitivity, suppose f g nd g h. Then f g nd g h. Since ddition preserves non-negtivity (see ()), f h (f g) + (g h), i.e. f h. For ntisymmetry, ssume f g nd g f, then f g nd g f. Apply the result we hve just proved, we hve f g, in other words, f g. (4) Let {f (k) } k1 be sequence in L1 (I) with f (k) f in L 1 (I). Suppose further tht f (k) for ll k. We wnt to prove tht f. For ech k, let {f n (k) } be non-negtive Cuchy sequence in C(I; R) representing f (k). Note tht f n (k) f (k) in L 1 (I) s n goes to infinity, we cn choose n(k) such tht f (k) n(k) f (k) 1 1. k Therefore, {f (k) n(k) } is non-negtive Cuchy sequence representing f nd f. (5) Suppose I. Assume the contrry tht there eists continuous liner mp E : L 1 (I) R such tht E (f) f() for ll f C(I; R). Suppose f L 1 (I) is represented by Cuchy sequence {f n }. By continuity of E, E (f) lim E (f n ) lim f n (). Consider the Cuchy sequence of functions f n C(I; R) defined by { n( ) + 1, [, + 1 f n () ] n., otherwise Then it is not hrd to see tht f n converges to in L 1 (I) but f n () 1 for ll n. Therefore, lim f n () 1, which is contrdiction. Problem. Suppose tht V is n inner product spce, D is liner subspce, nd A : D V is liner opertor (not necessrily continuous). We sy tht A is symmetric if (Au, v) (u, Av) for ll u, v D. We sy tht non-zero vector v D is n eigenvector of A with eigenvlue λ C if Av λv.

3 (1) Show tht if A : D V is symmetric, then ll eigenvlues of A re rel, nd ll eigenvectors corresponding to different eigenvlues re orthogonl, i.e. Av λv, Aw µw, λ µ implies (v, w). () Let A 1 d on D {f i d C1 ([, π]) : f() f(π)}, with V C([, π]). Show tht A is symmetric, nd the functions e in re orthogonl to ech other on [, π]. (3) Let V C([, b]), nd let D be subspce of C ([, b]). Under wht conditions on D is A, given by Af f, symmetric? (Hint: Clculte (Af, g) (f, Ag).) (4) Show tht the functions sin n, n 1 integer, re orthogonl to ech other on [, π]. (5) Show tht the functions sin(n + 1 ), n, 1,,... re orthogonl to ech other on [, π]. 3 Solution. Note tht we dopt the convention tht n inner product is conjugte liner in the second slot. (1) Suppose A : D V is symmetric nd λ is n eigenvlue of A with n eigenvector v D, i.e. Av λv nd v. By symmetry of A, we hve λ(v, v) (λv, v) (Av, v) (v, Av) (v, λv) λ(v, v). Since (v, v), dividing through gives λ λ. Hence λ is rel. All eigenvlues of symmetric opertor is rel. Suppose λ, µ re two distinct eigenvlues of A nd Av λv, Aw µw. We hve to show tht v nd w re orthogonl, i.e. (v, w). We hve lredy proved tht λ nd µ re rel. Note tht by symmetry gin, λ(v, w) (λv, w) (Av, w) (v, Aw) (v, µw) µ(v, w) µ(v, w). Therefore, (λ µ)(v, w) nd λ µ, hence (v, w). () Note tht for ny f, g V, we define (f, g) f()g() d to be the inner product on V. Note tht D is liner subspce of V nd for ll f, g D, (Af, g) ( 1 df i d, g) 1 df i d gd 1 i fg π 1 f dg d (f, Ag), i d where the first term drops out becuse f, g D. Thus, A is symmetric. Since A(e in ) 1 d i d (ein ) ne in, e in is n eigenvector corresponding to the eigenvlue n. By the result of prt (1), e in re orthogonl to ech other on [, π].

4 4 (3) Following the hint: (Af, g) (f, Ag) ( f, g) (f, g ) (f, g) + (f, g ) b f gd + b b b f g f g b fg d f g d + fg b fg f g b Hence, in order to gurntee tht A is symmetric, we need f()g () f ()g() f(b)g (b) f (b)g(b) for ll f, g D. (4) Note tht sin n re smooth nd A(sin n) (sin n) n sin n. Let f() sin n nd g() sin m, where n, m re positive integers. Then f()g () f ()g() f(π)g (π) f (π)g(π) since f() g() f(π) g(π). Therefore, sin n re eigenvectors of the symmetric opertor A with respect to the eigenvlue n. By prt (1), they re orthogonl on [, π]. (5) Similrly sin(n + 1) re smooth nd A(sin(n + 1 (n + 1 ) ) sin(n + 1 ). Let f() sin(n + 1) nd g() sin(m + 1). Then f () (n + 1) cos(n + 1) nd g () (m + 1) cos(m + 1). Moreover, f()g () f ()g() since f() g(), nd f(π)g (π) f (π)g(π) since f (π) g (π). Therefore, they re orthogonl to ech other on [, π] by prt (1). Problem 3. (cf. Tylor 3.1.5, 3.1.6) Suppose f is π-periodic C 1 function on R, i.e. f C 1 p([, π]). Let c n (f, e in ), b n (f, e in ), where (f, g) (π) 1 f() g() d. (1) Prove tht b n < nd conclude tht n c n <. () Prove tht c n <. (3) Prove tht M n M c ne in is uniformly convergent s M. (4) Let S N (f)(θ) N c ne inθ. Prove tht (S N f)(θ) 1 π The π-periodic function D N () sin(n + 1 ) sin(/) is known s the Dirichlet kernel. f(θ + ) sin(n + 1 ) sin(/) e in d.

5 (5) Using tht 1 π D π N () d 1, show tht for f Cp([, 1 π]) ( ) 1 f(θ) S N f(θ) sin(n + ), g θ, 5 g θ () f(θ+) f(θ) sin(/), g θ C ([, π]). (6) Using tht sin(n + 1 ), N, 1,,... is n orthonorml set in L ([, π]), show tht S N f(θ) f(θ), nd conclude tht the Fourier series of f C 1 p([, π]) converges uniformly to f. Solution. (1) We know tht e in / π is n orthonorml set in L ([, π]), nd we know tht f C ([, π]), so f L ([, π]), so we hve tht > f (e in / π, f ) b n. Moreover, b n 1 π e in f ()d in π e in f()d inc n, so > b n inc n n c n. () Write c n 1 n nc n, then Cuchy-Schwrz gives: ( ) 1/ 1 ( ) 1/ cn n c n n <. (3) C ([, π]) is Bnch spce, so it is enough to show tht c n e in / π is bsolutely summble, but cn e in / π c n e in / π 1 π cn <, so the sequence is summble nd thus converges in C, i.e. converges uniformly. (4) First observe tht e in e in n e in ei(n+1) e in e i 1 ei(n+ 1 ) e i(n+ 1 ) sin(n + 1). e i/ e i/ sin(/)

6 6 Then, using the bove, nd the substitution + θ, S N (f)(θ) c n e inθ / π ( 1 π 1 π 1 π ) e in f()d e inθ / π π e in( θ) f()d f() sin(n + 1 )( θ) d sin( θ)/ f(θ + ) sin(n + 1) d. sin / (5) First we show tht the function g θ C ([, π]). This is clerly C, ecept possibly t the endpoints. Using the reltion tht lim 1, we see tht g θ () lim f(θ + ) f(θ) sin(/) lim f( + θ) f(θ) sin sin(/) f (θ), nd then we use the periodicity of f, nd the fct tht sin( + π) sin to see g θ (π) lim π f( + θ) f(θ) sin(/) lim f( + π + θ) f(θ) sin(/ + π) f( + θ) f(θ) lim f (θ), / sin(/) so g C ([, π]). We hve tht f(θ) 1 π f(θ)d π N ()d, so f(θ) S N f(θ) 1 π 1 π (f(θ) f(θ + ))D N ()d f(θ) f(θ + ) sin(n + 1/) d sin(/) ( sin(n + 1/), gθ ), s desired. (6) Note tht the functions g θ C ([, π]), so g θ L ([, π]), so using the ssumption tht sin(n + 1/) is n orthonorml set (we cn proved this by

7 problem (), since they re eigenfunctions of Af f with boundry condition f() f(π) ), we hve g θ ( sin(n + 1 ), g θ), n so in prticulr, ( sin(n + 1 ), g θ), so f(θ) S N f(θ). We know, then, tht S N f converges to f pointwise, nd, from prt (3), tht S N converges to some limit h uniformly. Uniform convergence implies pointwise convergence, nd pointwise limits re unique, so f h, nd so S N converges to f uniformly. Problem 4. (cf. Tylor ) Suppose now tht f is piecewise C 1 nd π-periodic on R, i.e. there eist finite number of points j [, π), j 1,,..., n, such tht f( j ±) lim j ± f() nd lim j ± f () eist, but re not necessrily equl to ech other, nd wy from f is C 1. S { j + πk : j 1,..., n, k Z}, Show tht t ech / S, the Fourier series converges to f(), while t ech j, the Fourier series converges to 1 (f( j+) + f( j )). Hint: use tht sin(n + 1 ), N, 1,,... is n orthonorml set in L ([, π]) with inner product (f, g) L ([,π]) π 1 f() g() d. 7 Solution. Recll the useful Riemnn-Lebesgue Lemm: if f L 1 ([, π]), then c n (f, e in ) s n. A proof of Riemnn-Lebesgue lemm is s follows: note tht if f C 1 ([, π]), then we hve lredy proved in problem 3 (). Net, we know tht C 1 -functions re dense in L 1 ([, π]) (continuous functions re clerly dense in L 1 by completion, nd ny continuous function on [, π] cn indeed be pproimted by smooth functions, sy polynomils by Stone-Weierstrss theorem), so for ny ɛ >, there eists g C 1 such tht f g L 1 < ɛ/. Since g C 1, there eists N such tht (g, e in ) < ɛ/ for ll n N. Hence for ny n N, we hve (f, e in ) (f g, e in ) + (g, e in ) f g L 1 + (g, e in ) < ɛ. So the lemm is proved.

8 8 Let θ / S, then f is continuously differentible in neighborhood of θ. Therefore, g θ () f(+θ) f(θ) sin(/) is in L 1 ([, π]) s f L 1 nd g θ is continuous t. Apply Riemnn- Lebesgue lemm to g θ, we get 1 π g θ ()e in d π s n goes to infinity. By the sme clcultions we did in the previous problem, f(θ) S N f(θ) 1 π sin(n + 1 ) g θ()d. We wnt to show tht the right hnd side converges to s N. Observe tht sin(n + 1/) sin N cos / cos N sin /. Hence sin(n + 1 ) g θ()d sin N (cos g θ())d cos N (sin g θ())d. Apply Riemnn-Lebesgue lemm to the L 1 functions in brckets, we see tht they converge to s N. Therefore, the Fourier series converges to f(θ) when θ / S. Now, consider those singulr points j, by doing trnsltion if necessry, we cn ssume tht j (note tht f is π-periodic.) Note tht S N f() N (f, ein ) nd Using periodicity, f() cos nd (f, e in ) + (f, e in ) π (f, cos n) 1 π f() cos nd+ f() cos nd f() cos nd. (f()+f( )) cos nd. Hence S N f() only depends on the even prt of f: f even () f()+f( ). More precisely, S N f() 1 π f even ()D N ()d sin(n+1/) where D N is the Dirichlet kernel. Note tht we cn re-define f sin(/) even () f(+)+f( ) so tht it is continuous t. Consider S N f() f even () 1 (f even () f even ())D N () d π ( f even () f even (), ) sin(n + 1 sin ) L ([,π]) which converges to since sin(n + 1/), N, 1,,..., is n orthonorml bsis in L ([, π]) nd f even () f even () L ([, π]). sin

9 (Note tht f even () f even () lim f (+) f ( ) + eists, so the function bove is bounded round.) 9 Alterntive Solution: sin(n + 1/) is n orthonorml set in L ([, π]) since they re eigenfunctions of the opertor Af f with f() f (π) (see Problem ). If f is C 1 t, then the sme rgument s in the previous problem shows tht g θ is L, hence the Fourier series converges to f(). If f is singulr t, then S N f() 1 f(θ) π D N(θ)dθ + 1 π by periodicity of f nd chnge of vrible. Therefore, f(+) + f( ) S N f() ( f(θ) f(+) sin θ/, sin(n + 1 )θ f( θ) D N (θ)dθ ) ( f( θ) f( ) + sin θ/, sin(n + 1 ) )θ. Apply the rgument in the previous problem with g θ replced by the functions f(θ) f(+) sin θ/ respectively on [, π]. This yields our desired conclusion. nd f( θ) f( ) sin θ/

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

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS Throughout, we let [, b] be bounded intervl in R. C 2 ([, b]) denotes the spce of functions with derivtives of second order continuous up to the endpoints. Cc 2

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

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

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

Variational Techniques for Sturm-Liouville Eigenvalue Problems

Variational Techniques for Sturm-Liouville Eigenvalue Problems Vritionl Techniques for Sturm-Liouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment

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

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

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 270A: Numerical Linear Algebra

Math 270A: Numerical Linear Algebra Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner

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

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

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

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

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below . Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

More information

Lectures On Fourier Series. S. Kesavan Institute of Mathematical Sciences Chennai , INDIA

Lectures On Fourier Series. S. Kesavan Institute of Mathematical Sciences Chennai , INDIA Lectures On Fourier Series S. Kesvn Institute of Mthemticl Sciences Chenni-6 113, INDIA Third Annul Foundtionl School - Prt I December 4 3, 6 Contents 1 Introduction 3 Orthonorml Sets 6 3 Vritions on the

More information

Sturm-Liouville Theory

Sturm-Liouville Theory LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory

More information

Inner-product spaces

Inner-product spaces Inner-product spces Definition: Let V be rel or complex liner spce over F (here R or C). An inner product is n opertion between two elements of V which results in sclr. It is denoted by u, v nd stisfies:

More information

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,

More 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

Week 7 Riemann Stieltjes Integration: Lectures 19-21

Week 7 Riemann Stieltjes Integration: Lectures 19-21 Week 7 Riemnn Stieltjes Integrtion: Lectures 19-21 Lecture 19 Throughout this section α will denote monotoniclly incresing function on n intervl [, b]. Let f be bounded function on [, b]. Let P = { = 0

More information

NOTES AND PROBLEMS: INTEGRATION THEORY

NOTES AND PROBLEMS: INTEGRATION THEORY NOTES AND PROBLEMS: INTEGRATION THEORY SAMEER CHAVAN Abstrct. These re the lecture notes prepred for prticipnts of AFS-I to be conducted t Kumun University, Almor from 1st to 27th December, 2014. Contents

More information

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests. ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

More information

Euler-Maclaurin Summation Formula 1

Euler-Maclaurin Summation Formula 1 Jnury 9, Euler-Mclurin Summtion Formul Suppose tht f nd its derivtive re continuous functions on the closed intervl [, b]. Let ψ(x) {x}, where {x} x [x] is the frctionl prt of x. Lemm : If < b nd, b Z,

More information

Mapping the delta function and other Radon measures

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

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

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

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

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

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.

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

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

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

A Convergence Theorem for the Improper Riemann Integral of Banach Space-valued Functions

A Convergence Theorem for the Improper Riemann Integral of Banach Space-valued Functions Interntionl Journl of Mthemticl Anlysis Vol. 8, 2014, no. 50, 2451-2460 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2014.49294 A Convergence Theorem for the Improper Riemnn Integrl of Bnch

More information

M597K: Solution to Homework Assignment 7

M597K: Solution to Homework Assignment 7 M597K: Solution to Homework Assignment 7 The following problems re on the specified pges of the text book by Keener (2nd Edition, i.e., revised nd updted version) Problems 3 nd 4 of Section 2.1 on p.94;

More information

1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q.

1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q. Chpter 6 Integrtion In this chpter we define the integrl. Intuitively, it should be the re under curve. Not surprisingly, fter mny exmples, counter exmples, exceptions, generliztions, the concept of the

More information

MATH 573 FINAL EXAM. May 30, 2007

MATH 573 FINAL EXAM. May 30, 2007 MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.

More 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

Chapter 4. Lebesgue Integration

Chapter 4. Lebesgue Integration 4.2. Lebesgue Integrtion 1 Chpter 4. Lebesgue Integrtion Section 4.2. Lebesgue Integrtion Note. Simple functions ply the sme role to Lebesgue integrls s step functions ply to Riemnn integrtion. Definition.

More information

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

Lecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics - A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the

More 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

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

Partial Differential Equations

Partial Differential Equations Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for One-Dimensionl Eqution The reen s function provides complete solution to boundry

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

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

Necessary and Sufficient Conditions for Differentiating Under the Integral Sign

Necessary and Sufficient Conditions for Differentiating Under the Integral Sign Necessry nd Sufficient Conditions for Differentiting Under the Integrl Sign Erik Tlvil 1. INTRODUCTION. When we hve n integrl tht depends on prmeter, sy F(x f (x, y dy, it is often importnt to know when

More 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

Math 211A Homework. Edward Burkard. = tan (2x + z)

Math 211A Homework. Edward Burkard. = tan (2x + z) Mth A Homework Ewr Burkr Eercises 5-C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z =

More information

The Dirac distribution

The Dirac distribution A DIRAC DISTRIBUTION A The Dirc distribution A Definition of the Dirc distribution The Dirc distribution δx cn be introduced by three equivlent wys Dirc [] defined it by reltions δx dx, δx if x The distribution

More information

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

7 - Continuous random variables

7 - Continuous random variables 7-1 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7 - Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin

More 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

5.5 The Substitution Rule

5.5 The Substitution Rule 5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in

More information

Practice final exam solutions

Practice final exam solutions University of Pennsylvni Deprtment of Mthemtics Mth 26 Honors Clculus II Spring Semester 29 Prof. Grssi, T.A. Asher Auel Prctice finl exm solutions 1. Let F : 2 2 be defined by F (x, y (x + y, x y. If

More information

We know that if f is a continuous nonnegative function on the interval [a, b], then b

We know that if f is a continuous nonnegative function on the interval [a, b], then b 1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going

More 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

2 Definitions and Basic Properties of Extended Riemann Stieltjes Integrals

2 Definitions and Basic Properties of Extended Riemann Stieltjes Integrals 2 Definitions nd Bsic Properties of Extended Riemnn Stieltjes Integrls 2.1 Regulted nd Intervl Functions Regulted functions Let X be Bnch spce, nd let J be nonempty intervl in R, which my be bounded or

More information

PARTIAL FRACTION DECOMPOSITION

PARTIAL FRACTION DECOMPOSITION PARTIAL FRACTION DECOMPOSITION LARRY SUSANKA 1. Fcts bout Polynomils nd Nottion We must ssemble some tools nd nottion to prove the existence of the stndrd prtil frction decomposition, used s n integrtion

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

Lecture Notes on Functional Analysis. Kai-Seng Chou Department of Mathematics The Chinese University of Hong Kong Hong Kong

Lecture Notes on Functional Analysis. Kai-Seng Chou Department of Mathematics The Chinese University of Hong Kong Hong Kong Lecture Notes on Functionl Anlysis Ki-Seng Chou Deprtment of Mthemtics The Chinese University of Hong Kong Hong Kong My 29, 2014 2 Contents 1 Normed Spce: Exmples 5 1.1 Vector Spces of Functions...................................

More information

Math 324 Course Notes: Brief description

Math 324 Course Notes: Brief description Brief description These re notes for Mth 324, n introductory course in Mesure nd Integrtion. Students re dvised to go through ll sections in detil nd ttempt ll problems. These notes will be modified nd

More information

Math 4200: Homework Problems

Math 4200: Homework Problems Mth 4200: Homework Problems Gregor Kovčič 1. Prove the following properties of the binomil coefficients ( n ) ( n ) (i) 1 + + + + 1 2 ( n ) (ii) 1 ( n ) ( n ) + 2 + 3 + + n 2 3 ( ) n ( n + = 2 n 1 n) n,

More information

Homework Solution - Set 5 Due: Friday 10/03/08

Homework Solution - Set 5 Due: Friday 10/03/08 CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

More information

Introduction to Real Analysis (Math 315) Martin Bohner

Introduction to Real Analysis (Math 315) Martin Bohner ntroduction to Rel Anlysis (Mth 315) Spring 2005 Lecture Notes Mrtin Bohner Author ddress: Version from April 20, 2005 Deprtment of Mthemtics nd Sttistics, University of Missouri Roll, Roll, Missouri 65409-0020

More information

5.2 Volumes: Disks and Washers

5.2 Volumes: Disks and Washers 4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of cross-section or slice. In this section, we restrict

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

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

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

STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH

STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH XIAO-BIAO LIN. Qudrtic functionl nd the Euler-Jcobi Eqution The purpose of this note is to study the Sturm-Liouville problem. We use the vritionl problem s

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

MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL

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

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

More information

Summary of Elementary Calculus

Summary of Elementary Calculus Summry of Elementry Clculus Notes by Wlter Noll (1971) 1 The rel numbers The set of rel numbers is denoted by R. The set R is often visulized geometriclly s number-line nd its elements re often referred

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

Integrals along Curves.

Integrals along Curves. 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

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

20 MATHEMATICS POLYNOMIALS

20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

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

38 Riemann sums and existence of the definite integral.

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

1 Sequences. 2 Series. 2 SERIES Analysis Study Guide

1 Sequences. 2 Series. 2 SERIES Analysis Study Guide 2 SERIES Anlysis Study Guide 1 Sequences Def: An ordered field is field F nd totl order < (for ll x, y, z F ): (i) x < y, y < x or x = y, (ii) x < y, y < z x < z (iii) x < y x + z < y + z (iv) 0 < y, x

More information

1. Review: Eigenvalue problem for a symmetric matrix

1. Review: Eigenvalue problem for a symmetric matrix 1. Review: Eigenvlue problem for symmetric mtrix 1.1. Inner products For vectors u,v R n we define the inner product by u,v := nd the norm by u := u,u. We cll two vectors u,v orthogonl if u,v =. Recll

More information

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS 3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive

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

The solutions of the single electron Hamiltonian were shown to be Bloch wave of the form: ( ) ( ) ikr

The solutions of the single electron Hamiltonian were shown to be Bloch wave of the form: ( ) ( ) ikr Lecture #1 Progrm 1. Bloch solutions. Reciprocl spce 3. Alternte derivtion of Bloch s theorem 4. Trnsforming the serch for egenfunctions nd eigenvlues from solving PDE to finding the e-vectors nd e-vlues

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

NWI: Mathematics. Various books in the library with the title Linear Algebra I, or Analysis I. (And also Linear Algebra II, or Analysis II.

NWI: Mathematics. Various books in the library with the title Linear Algebra I, or Analysis I. (And also Linear Algebra II, or Analysis II. NWI: Mthemtics Literture These lecture notes! Vrious books in the librry with the title Liner Algebr I, or Anlysis I (And lso Liner Algebr II, or Anlysis II) The lecture notes of some of the people who

More information

MA Handout 2: Notation and Background Concepts from Analysis

MA Handout 2: Notation and Background Concepts from Analysis MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,

More information

MATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals.

MATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals. MATH 409 Advned Clulus I Leture 22: Improper Riemnn integrls. Improper Riemnn integrl If funtion f : [,b] R is integrble on [,b], then the funtion F(x) = x f(t)dt is well defined nd ontinuous on [,b].

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

The Wave Equation I. MA 436 Kurt Bryan

The Wave Equation I. MA 436 Kurt Bryan 1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

More information

x ) dx dx x sec x over the interval (, ).

x ) dx dx x sec x over the interval (, ). Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor

More information

Continuous Random Variables

Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

More information

MATH 423 Linear Algebra II Lecture 28: Inner product spaces.

MATH 423 Linear Algebra II Lecture 28: Inner product spaces. MATH 423 Liner Algebr II Lecture 28: Inner product spces. Norm The notion of norm generlizes the notion of length of vector in R 3. Definition. Let V be vector spce over F, where F = R or C. A function

More information

k and v = v 1 j + u 3 i + v 2

k and v = v 1 j + u 3 i + v 2 ORTHOGONAL FUNCTIONS AND FOURIER SERIES Orthogonl functions A function cn e considered to e generliztion of vector. Thus the vector concets like the inner roduct nd orthogonlity of vectors cn e extended

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

Section 7.1 Area of a Region Between Two Curves

Section 7.1 Area of a Region Between Two Curves Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

More information

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS 1 TRANSFORMING BIVARIATE DENSITY FUNCTIONS Hving seen how to trnsform the probbility density functions ssocited with single rndom vrible, the next logicl step is to see how to trnsform bivrite probbility

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

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

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

(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