THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
|
|
- Jerome Moore
- 5 years ago
- Views:
Transcription
1 THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH This document is proof of the existence-uniqueness theorem for first-order differentil equtions, lso known s the Picrd-Lindelöf or Cuchy-Lipschitz theorem It ws written with specil ttention to both rigor nd clrity The proof is primrily bsed on the one given in the textbook I used in my Differentil Equtions clss (Differentil Equtions nd Their Applictions by Mrtin Brun, fourth edition)
2 2 RADON ROSBOROUGH 1 Gol Let f be function of two vribles nd let nd y 0 be rel numbers This defines n initil-vlue problem: (1) y (t) = f(t, y(t)) y( ) = y 0 Our gol is to prove tht under certin conditions, there is exctly one solution y to this differentil eqution Tht is, we would like to prove both the existence nd uniqueness of solutions to the eqution 2 Initil Steps We will strt our proof by trnsforming the differentil eqution (1) into more convenient form This is done by integrting both sides from to t: y (τ) dτ = f(τ, y(τ)) dτ Here, to void mbiguity, we re using the vrible τ s our vrible of integrtion insted of t The bove eqution reduces to y(t) y 0 = since y( ) = y 0, nd solving for y(t) gives the eqution (2) y(t) = y 0 + f(τ, y(τ)) dτ, f(τ, y(τ)) dτ By this resoning, ny function stisfying (1) must lso stisfy eqution (2) However, the converse sttement requires little more work Suppose tht function y stisfies eqution (2) Then it follows immeditely tht y( ) = y 0, becuse y( ) = y f(τ, y(τ)) dτ = y 0 Now, the fundmentl theorem of clculus tells us tht if function f is continuous, then b f(x) dx is differentible with respect to b, nd d db b f(x) dx = f(b) Hence, if we ssume tht if both f nd y re continuous, then f(τ, y(τ)) is continuous function of τ nd we hve y (t) = d dt f(τ, y(τ)) dτ = f(t, y(t)) Therefore, if y is continuous function stisfying (2) nd f is continuous, then y is solution of the originl differentil eqution (1)
3 THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS 3 3 Outline Our proof will consist of the following mjor steps: () Construct sequence of functions {y 0 (t), y 1 (t), }, clled Picrd itertes, which pproximte solution to the eqution (2) (b) Show tht the sequence of functions converges nd define y(t) = lim y n (t) (c) Show tht the function y(t) stisfies eqution (2) (d) Show tht the function y(t) is continuous (e) Show tht there cn only be one solution to the eqution (2) Hving completed these tsks, our resoning bove will imply tht the function y(t) is the unique solution to (1) 4 Construction of the Picrd itertes As our first pproximtion to solution to the differentil eqution (1), we will choose the simplest possible function tht stisfies the condition y( ) = y 0 ; tht is, y 0 (t) = y 0 Our procedure for generting better pproximtions is motivted by the reltion (2) which is stisfied by ny solution y to (1), reprinted here: In prticulr, we will define y(t) = y 0 + (3) y n (t) = y 0 + f(τ, y(τ)) dτ f(τ, y n 1 (τ)) dτ for every n 1 Observe tht we hve y n ( ) = y 0 for every n 0, so every Picrd iterte obeys the initil condition As concrete exmple of this itertion process, consider the differentil eqution y (t) = y(t) y(0) = 1, whose unique solution is y(t) = e t For this eqution, we hve f(t, y) = y, = 0, nd y 0 = 1 This trnsforms the recurrence reltion (3) to therefore, y 0 (t) = 1 y n (t) = 1 + y 1 (t) = 1 + y 2 (t) = y n 1 (τ) dτ; dτ = 1 + t 1 + τ dτ = 1 + t + t2 2 y n (t) = 1 + t + t tn n!
4 4 RADON ROSBOROUGH The stute reder will recognize y n (t) s the nth prtil sum of the Mclurin series for e t It follows esily, then, tht the sequence {y 0 (t), y 1 (t), } converges nd tht the limiting function y(t) = lim y n (t) = e t is continuous nd stisfies (2) We now show tht this conclusion is true in generl under suitble ssumptions 5 Bounding the Picrd itertes In generl, even if the differentil eqution (1) hs unique solution, the solution my only be vlid on specified intervl typiclly becuse f(t, y(t)) is not defined for one or more vlues of t Therefore, we will hve to restrict our resoning to limited intervl contining However, it is difficult to reson bout the lrgest possible intervl tht is, the lrgest intervl over which the differentil eqution hs solution Insted, we will pick smller intervl in such wy tht the behvior of the Picrd itertes is esy to nlyze over the intervl To construct this intervl, we will strt by picking two rbitrry positive rel numbers nd b These numbers define rectngle in the t-y plne tht hs vertices t (, y 0 b), (, y 0 + b), ( +, y 0 b), nd ( +, y 0 + b) This rectngle is illustrted in Figure 1 y y 0 + b (, y 0 + b) ( +, y b + b) y 0 (, y 0 ) y = y(t) y 0 b (, y 0 b) + ( +, y 0 b) t Figure 1 Let R denote the rectngle nd its interior, ie the set of ll points (t, y) such tht t + nd y 0 b y y 0 + b Since we re ssuming tht f is continuous, it follows tht f is continuous nd hs mximum vlue on R We let M denote this mximum vlue, ie M = mx f(t, y) (t,y) R Next, we consider the lines through the point (, y 0 ) tht hve slope M nd M, respectively These lines hve equtions y = y 0 ± M(t ), nd re shown in Figure 2 From the figure, it is esy to see tht depending on the vlue of M, the lines will leve the rectngle t either t = or t = b/m, whichever is smller We will denote this t-vlue by α, ie α = min (, We will now prove tht for t + α, every Picrd iterte lies between the two lines Tht is, until the lines leve the rectngle R, every y n (t) lies within the shded regions in Figure 2 We cn reformulte this hypothesis s follows: b M ) y 0 M(t ) y n (t) y 0 + M(t ) M(t ) y n (t) y 0 M(t )
5 THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS 5 (, y 0 + M) (y 0 + b/m, b) (, y 0 ) (, y 0 ) (, y 0 M) (y 0 b/m, b) Figure 2 y n (t) y 0 M(t ) Becuse M 0 nd t, we need no bsolute vlue brs on the right-hnd side To prove the hypothesis, we use induction on n The cse of n = 0 follows immeditely, s y 0 (t) y 0 = y 0 y 0 = 0 M(t ) For the inductive cse, we ssume for some n 0 tht y n (t) y 0 M(t ) for t α nd seek to prove tht y n+1 (t) y 0 M(t ) on the sme intervl We now use the definition (3) of the Picrd itertes, reprinted here: y n+1 (t) = y 0 + f(τ, y n (τ)) dτ In prticulr, we note tht y n+1 (t) y 0 = f(τ, y n (τ)) dτ Next, we use the following two elementry properties of definite integrls: b b f(x) dx f(x) dx b ( ) b f(x) g(x) dx mx f(x) g(x) dx x b Note tht we hve τ t + α Consequently, it follows from the inductive hypothesis tht y n (τ) lies between the lines nd hence within R on this intervl Thus, (τ, y n (τ)) lies within R for ll τ from to t, nd f(τ, y n (τ)) M over this intervl From these properties, we hve: f(τ, y n (τ)) dτ f(τ, y n (τ)) dτ M(t ) We hve therefore shown tht y n+1 (t) y 0 M(t ), which completes the proof tht y n (t) y 0 M(t ) for every n 0 6 Proof tht the Picrd itertes converge Now tht we hve obtined bound on the size of y n (t) on suitble intervl, we cn show tht the sequence {y 0 (t), y 1 (t), } converges on tht intervl We do this by rewriting y n (t) s telescoping series: y n (t) = y 0 (t) + y 1 (t) y 0 (t) + y 2 (t) y 1 (t) + + y n (t) y n 1 (t)
6 6 RADON ROSBOROUGH so tht = y 0 (t) + n y k (t) y k 1 (t), lim y n(t) = y 0 (t) + y k (t) y k 1 (t) If the infinite series y k(t) y k 1 (t) converges, then so does the sequence {y 0 (t), y 1 (t), } Now, if we replce every term of series with its bsolute vlue nd it still converges, then certinly the originl series must lso converge Thus, it suffices to show the convergence of y k(t) y k 1 (t) To do this, we will use series of pproximtions involving the quntity y k (t) y k 1 Firstly, we will use the definition (3) of the Picrd iterte, gin reprinted here: y n (t) = y 0 + f(τ, y n 1 (τ)) dτ In prticulr, we find tht ( ) ( ) y k (t) y k 1 (t) = y 0 + f(τ, y k 1 (τ)) dτ y 0 + f(τ, y k 2 (τ)) dτ = f(τ, y k 1 (τ)) f(τ, y k 2 (τ)) dτ f(τ, y k 1 (τ)) f(τ, y k 2 (τ)) dτ, provided tht k 2 Next we invoke the men vlue theorem, which sttes tht if function g is continuous on, b nd differentible on (, b) then there exists number ξ (, b) such tht Now, for ny given τ we cn define g (ξ) = g(b) g() b g(y) = f(τ, y) = y k 2 (τ) b = y k 1 (τ) If we ssume tht f is continuous nd the prtil derivtive f y = f/ y exists, ie tht g is differentible, then the men vlue theorem tells us tht there exists number ξ between y k 2 (τ) nd y k 1 (τ) such tht f y (τ, ξ) = f(τ, y k 1(τ)) f(τ, y k 2 (τ)) y k 1 (τ) y k 2 (τ) Rerrnging this eqution, we find the useful reltion f(τ, y k 1 (τ)) f(τ, y k 2 (τ)) = f y (τ, ξ) y k 1 (τ) y k 2 (τ) If we mke this rgument for every τ t, we my obtin different number ξ for ech τ Tht is, we must replce the number ξ with function ξ(τ) We then find tht f(τ, y k 1 (τ)) f(τ, y k 2 (τ)) dτ = f y (τ, ξ(τ)) y k 1 (τ) y k 2 (τ) dτ = f y (τ, ξ(τ)) y k 1 (τ) y k 2 (τ) dτ
7 THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS 7 Now, let us ssume tht f/ y not only exists over the rectngle R, but it is lso continuous 1 Then f/ y is lso continuous, nd therefore hs mximum vlue on R We will denote this mximum vlue by L, ie L = mx (t,y) R f y(t, y) Since ξ(τ) lies between the Picrd itertes y k 2 (τ) nd y k 1 (τ), our work in the previous section proves tht it lies between the lines y = y 0 ± M(t ) for τ α Hence, ll points (τ, ξ(τ)) lie within R for τ t, nd so f y (τ, ξ(τ)) L The sme elementry properties of definite integrls we used erlier pply gin, so tht f y (τ, ξ(τ)) y k 1 (τ) y k 2 (τ) dτ L y k 1 (τ) y k 2 (τ) dτ In summry, y k (t) y k 1 (t) L y k 1 (τ) y k 2 (τ) dτ for every k 2 We now switch to n inductive rgument on k For k = 1, recll we proved in the previous section tht y 1 (t) y 0 (t) M(t ) for t + α For k = 2, we hve y 2 (t) y 1 (t) L y 1 (τ) y 0 (τ) dτ For k = 3, we hve Inductively, we find tht L y 3 (t) y 2 (t) L M(t ) dτ = ML(t ) 2 2 L y 2 (τ) y 1 (τ) dτ ML(t ) 2 2 = ML2 (t ) 3 3! (4) y k (t) y k 1 (t) MLk 1 (t ) k for t + α But now we cn esily show tht the series y k(t) y k 1 converges, becuse ML k 1 (t ) k y k (t) y k 1 (t) = M L(t ) k L = M L(t t0 ) k 1 L 1 This is not strictly necessry All we need is tht f/ y is bounded k=0 dτ
8 8 RADON ROSBOROUGH = M L ( e L(t ) 1 ) As t + α, we hve t α nd e L(t t0) e Lα This shows tht y k (t) y k 1 (t) M ( e Lα 1 ), L which implies the series converges This completes our proof tht the sequence of Picrd itertes {y 0 (t), y 1 (t), } converges We thus cn define y(t) = lim y n (t) 7 Proof tht y(t) stisfies eqution (2) We will now show tht the function y(t) = lim y n (t) stisfies eqution (2), reprinted here: y(t) = y 0 + f(τ, y(τ)) dτ To do so, we strt with the definition (3) of the Picrd itertes, reprinted here: y n+1 (t) = y 0 + f(τ, y n (τ)) dτ Tking the limits of both sides s n gives us y(t) = y 0 + lim f(τ, y n (τ)) dτ; to show tht y(t) stisfies (2), we must demonstrte tht f(τ, y(τ)) dτ = lim f(τ, y n (τ)) dτ, t 0 or equivlently tht lim f(τ, y(τ)) dτ f(τ, y n (τ)) dτ = 0 We my now use roughly the sme procedure tht we used to show the convergence of the sequence {y 0 (t), y 1 (t), } In prticulr: t f(τ, y(τ)) dτ f(τ, y n (τ)) dτ = f(τ, y(τ)) f(τ, y n (τ)) dτ f(τ, y(τ)) dτ f(τ, y n (τ)) dτ Next, pplying the men vlue theorem shows tht for every τ between nd t, there is number ξ(τ) between y n (τ) nd y(τ) such tht or equivlently We then find tht f y (τ, ξ(τ)) = f(τ, y(τ)) f(τ, y n(τ)), y(τ) y n (τ) f(τ, y(τ)) f(τ, y n (τ)) = f y (τ, ξ(τ)) y(τ) y n (τ) f(τ, y(τ)) f(τ, y n (τ)) dτ = f y (τ, ξ(τ)) y(τ) y n (τ) dτ As y(t) is the limit of sequence of functions y n (t) which ll lie within the closed rectngle R for τ t, it follows tht y(t) lso lies within R on tht intervl Becuse ξ(τ) is between y n (τ) nd
9 THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS 9 y(τ), ll points (τ, ξ(τ)) lie within R for τ t, nd f y (τ, ξ(τ)) L We cn thus conclude tht f y (τ, ξ(τ)) y(τ) y n (τ) dτ L y(τ) y n (τ) dτ nd Now observe tht the reltions my be combined to obtin Also, reltion (4) tells us tht so y(τ) y n (τ) = y n (τ) = y 0 (τ) + y(τ) = y 0 (τ) + y(τ) y n (τ) = n y k (τ) y k 1 (τ) y k (τ) y k 1 (τ) y k (τ) y k 1 (τ) y k (τ) y k 1 (τ) MLk 1 (τ ) k, y k (τ) y k 1 (τ) nd using the fct tht τ t α gives (5) y(τ) y n (τ) y k (τ) y k 1 (τ) Substituting (5) yields: f y (τ, ξ(τ)) y(τ) y n (τ) dτ L ML k 1 α k ML k 1 α k dτ ML k 1 (τ ) k, Since every term of this series is nonnegtive, Tonelli s theorem gurntees tht we my swp the integrl nd summtion: ML k 1 α k ML k 1 α k L dτ = L dτ; this llows us to simplify s follows: ML k 1 α k L dτ M L k α k (t ) Mα (Lα) k Since the ltter summtion is the til end of series expnsion for e Lα, it pproches zero s n To prove this formlly, observe tht (Lα) k (Lα) k n (Lα) k n = = e Lα (Lα) k 1 1 1
10 10 RADON ROSBOROUGH nd Since nd it follows tht lim (Lα) k f(τ, y(τ)) dτ lim lim = e Lα lim n (Lα) k = 0 f(τ, y n (τ)) dτ Mα f(τ, y(τ)) dτ (Lα) k 1 = 0, f(τ, y n (τ)) dτ = 0, which completes the proof tht y(t) = lim y n (t) stisfies (2) (Lα) k 8 Proof tht y(t) is continuous To show tht y(t) is continuous, we must show tht for every ɛ > 0 there exists δ > 0 such tht h < δ implies y(t + h) y(t) < ɛ To do so, we observe tht nd consequently y(t + h) y(t) = y(t + h) y n (t + h) + y n (t + h) y n (t) + y n (t) y(t) y(t + h) y(t) y(t + h) y n (t + h) + y n (t + h) y n (t) + y n (t) y(t) for every n 0 By picking lrge enough n, we cn reduce the mgnitude of this sum to ɛ Since the summtion ML k 1 α k is the til end of convergent Mclurin series, we cn mke it s smll s we wish by selecting sufficiently lrge n In prticulr, we will choose n n such tht ML k 1 α k < ɛ 3 nd reltion (5) implies y(τ) y n (τ) < ɛ 3, for both τ = t nd τ = t + h This tkes cre of two out of the three terms For the third, y n (t + h) y n (t), note tht y n (t) is continuous for every n 0, nd so for every ɛ > 0 there exists δ > 0 such tht h < δ implies y n (t + h) y n (t) < ɛ We let ɛ = ɛ/3 nd define δ = δ Thus, ech of the three terms is strictly less thn ɛ/3, nd This completes the proof tht y(t) is continuous y(t + h) y(t) < ɛ 3 + ɛ 3 + ɛ 3 = ɛ
11 THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS 11 9 Proof tht the solution to eqution (2) is unique Hving shown tht solution to (1) exists, we now show tht it is unique Supposing tht two solutions of (1) re given by y(t) nd z(t), we define w(t) = y(t) z(t) ; it is sufficient, then, to show tht w(t) = 0 for ll t If y(t) nd z(t) re solutions of (1), then they re lso solutions of (2); tht is, Subtrcting, we find tht y(t) z(t) = w(t) y(t) = y 0 + z(t) = y 0 + f(τ, y(τ)) dτ f(τ, z(τ)) dτ f(τ, y(τ)) dτ f(τ, z(τ)) dτ f(τ, y(τ)) f(τ, z(τ)) dτ, nd by the sme resoning with the men vlue theorem tht we used twice before, We now define w(t) L y(τ) z(τ) dτ = L w(τ) dτ U(t) = w(τ) dτ, so tht U (t) = w(t) L w(τ) dτ = LU(t), or equivlently U (t) LU(t) 0 Multiplying both sides by the strictly positive integrting fctor e L(t t0) gives U (t)e L(t t0) LU(t)e L(t t0) 0 d U(t)e L(t ) 0 dt Now, U( ) = 0 w(τ) dτ = 0, so the function U(t)e L(t t0) is zero t t = Furthermore, w(t) is nonnegtive nd t, so U(t) is lso nonnegtive Since the function U(t)e L(t t0) is zero t t =, is never less thn zero, nd is never incresing, it must be zero for ll t > s well We must then conclude tht U(t) = 0 nd therefore tht w(t) = 0 This completes the proof
The Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More information63. 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 informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution
More informationLecture 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 informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationMORE 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 information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationUnit #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 informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationA 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 informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More informationLine 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 informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
More informationMath 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 informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
More informationThe 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 informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationTaylor Polynomial Inequalities
Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationSolution to HW 4, Ma 1c Prac 2016
Solution to HW 4 M c Prc 6 Remrk: every function ppering in this homework set is sufficiently nice t lest C following the jrgon from the textbook we cn pply ll kinds of theorems from the textbook without
More informationPolynomial 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 informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationAP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
More informationSYDE 112, LECTURES 3 & 4: The Fundamental Theorem of Calculus
SYDE 112, LECTURES & 4: The Fundmentl Theorem of Clculus So fr we hve introduced two new concepts in this course: ntidifferentition nd Riemnn sums. It turns out tht these quntities re relted, but it is
More informationAnonymous 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 informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationMath 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 informationMath& 152 Section Integration by Parts
Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationBernoulli Numbers Jeff Morton
Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f
More informationThe Henstock-Kurzweil integral
fculteit Wiskunde en Ntuurwetenschppen The Henstock-Kurzweil integrl Bchelorthesis Mthemtics June 2014 Student: E. vn Dijk First supervisor: Dr. A.E. Sterk Second supervisor: Prof. dr. A. vn der Schft
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More informationIntegrals - Motivation
Integrls - Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is non-liner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but
More informationNote 16. Stokes theorem Differential Geometry, 2005
Note 16. Stokes theorem ifferentil Geometry, 2005 Stokes theorem is the centrl result in the theory of integrtion on mnifolds. It gives the reltion between exterior differentition (see Note 14) nd integrtion
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More information1. On some properties of definite integrals. We prove
This short collection of notes is intended to complement the textbook Anlisi Mtemtic 2 by Crl Mdern, published by Città Studi Editore, [M]. We refer to [M] for nottion nd the logicl stremline of the rguments.
More informationChapter 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 information8 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( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationEuler-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 informationConvex 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 informationMath 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 informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationUNIFORM 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 information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationNew 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 information7.2 Riemann Integrable Functions
7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous
More informationII. Integration and Cauchy s Theorem
MTH6111 Complex Anlysis 2009-10 Lecture Notes c Shun Bullett QMUL 2009 II. Integrtion nd Cuchy s Theorem 1. Pths nd integrtion Wrning Different uthors hve different definitions for terms like pth nd curve.
More informationVariational 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 informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd
More informationMath Advanced Calculus II
Mth 452 - Advnced Clculus II Line Integrls nd Green s Theorem The min gol of this chpter is to prove Stoke s theorem, which is the multivrible version of the fundmentl theorem of clculus. We will be focused
More informationThe 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 informationLecture 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 informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationNumerical Integration
Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the
More information4. Calculus of Variations
4. Clculus of Vritions Introduction - Typicl Problems The clculus of vritions generlises the theory of mxim nd minim. Exmple (): Shortest distnce between two points. On given surfce (e.g. plne), nd the
More informationA. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More informationLecture 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 informationPresentation Problems 5
Presenttion Problems 5 21-355 A For these problems, ssume ll sets re subsets of R unless otherwise specified. 1. Let P nd Q be prtitions of [, b] such tht P Q. Then U(f, P ) U(f, Q) nd L(f, P ) L(f, Q).
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationChapter 1. Basic Concepts
Socrtes Dilecticl Process: The Þrst step is the seprtion of subject into its elements. After this, by deþning nd discovering more bout its prts, one better comprehends the entire subject Socrtes (469-399)
More informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationLecture 1: Introduction to integration theory and bounded variation
Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. You
More informationProperties of the Riemann Integral
Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University Februry 15, 2018 Outline 1 Some Infimum nd Supremum Properties 2
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More information. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.
Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos( - 1 2 ) = rcsin( 1 2 ) = rcsin( - 1 2 ) = Cn you do similr problems? Review of Bsic Concepts
More informationChapters 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 informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationx = 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 informationp(t) dt + i 1 re it ireit dt =
Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)
More informationMath Solutions to homework 1
Mth 75 - Solutions to homework Cédric De Groote October 5, 07 Problem, prt : This problem explores the reltionship between norms nd inner products Let X be rel vector spce ) Suppose tht is norm on X tht
More informationMAT 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 informationIntegrals 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 informationConvergence of Fourier Series and Fejer s Theorem. Lee Ricketson
Convergence of Fourier Series nd Fejer s Theorem Lee Ricketson My, 006 Abstrct This pper will ddress the Fourier Series of functions with rbitrry period. We will derive forms of the Dirichlet nd Fejer
More informationAMATH 731: Applied Functional Analysis Fall Some basics of integral equations
AMATH 731: Applied Functionl Anlysis Fll 2009 1 Introduction Some bsics of integrl equtions An integrl eqution is n eqution in which the unknown function u(t) ppers under n integrl sign, e.g., K(t, s)u(s)
More information