PDE Notes. Paul Carnig. January ODE s vs PDE s 1

 Ashlee Edwards
 11 months ago
 Views:
Transcription
1 PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution Fourier s w of Het Conduction Energy Conservtion Idelistions Infinitesiml Approch: Integrl Approch: Higher Dimensions of the Het Eqution: Section Wve Phenomenon: Section Potentil Eqution Section Clssifiction of PDE s Generl Solution PDE Clssifictions Section 3.3 Introduction to Fourier Series 7 7 Section 3.4 Determining Fourier Coefficients Period ODE s vs PDE s ODE s ODE s re Ordinry Differentil Equtions. Exmple 1
2 d 2 u dt 2 = t2 + cost here u = ut which only hs one independent vrible, t. PDE s PDE s re Prtil Differentil Equtions. Exmple 2 u t 2 = 2 u x 2 here u = ut, x which hs two independent vribles t nd x. PDE s must hve t lest two independent vribles. 2 Section 1.2 Het diffusion Eqution Flux: Consider the flow of physicl quntity mss, het, energy. The flux q x, t of this flow is vector in the direction of the flow t x, t whose mgnitude is the mount of quntity crossing unit re t x norml i.e. orthogonl to the norml flow in unit time. So t = time nd x = So q x, t = lim s 0 t 0 x y z Quntity pssing through s in time[t, t] s t Here s is smll surfce re t x tht is norml to the flow. So, the quntity pssing through s in time t is Q x, t, s, t = q x, t s t Suppose wter is flowing in river with velocity v x, t et s be smll surfce re norml to v x, t Then the mount of the wter flowing through s in time [t, t + ] is: Mss = Q x, t, s, t = ρ v s t where ρ = density so q x, t = ρ v A chnge Q in the mount of het in body of mss is ccompnied by chnge u of temperture. The reltionship is: Q = cm u where m = mss nd c is the specific het of the mteril. Normlize to Qcmu so Q nd u re proportionl. Het is trnsported in the direction opposite to the temperture grdient t rte proportionl to it. So the het flux t q x, t is q x, t = κ u x, t where κ is the therml conductivity. 2
3 2.1 Fourier s w of Het Conduction q x, t = κ u x u y u z 2.2 Energy Conservtion The rte of chnge of het = rte het inrte het out 2.3 Idelistions First ssume we re working with thin rod. 1 Rod is homogeneous i.e. c, κ nd ρ re ll constnt 2 ength of the rod is constnt i.e. no expnding or contrcting 3 Rod is perfectly insulted so het flows only horizontlly 4 Het flows from left to right, so the left side is wrmer thn the right side. Now tht we hve the proper frmework we cn derive the het eqution with 2 pproches. 2.4 Infinitesiml Approch: First let use strt with the following: qx, ta qx + xa u = temperture = ux, t volume = A x mss = ρa x Now the mount of het is Qx, t, x = cρa xu dq dt = cρa xu t By our conservtion lw we get dq dt cρa xu t = A[qx + x, t qx, t] cρ xu t = [qx + x, t qx, t] = qx, ta qx + x, ta [qx + x, t qx, t] cρu t = here let x 0 x cρu t = q x 3
4 q = κu x so q x = κu xx cρu t = κu xx Now let the 1 k = cρ κ which is the therml diffusivity This give us 1 k u t = u xx which is the 1D het eqution. 2.5 Integrl Approch: Consider portion of the rod between x = nd x = b. So the totl mount of het in this section of the rod cn be given by: Qt,, b = dq dt = cρaux, tdx cρa du dt dx now by the conservtion lw we hve : dq dt = dq = Aq, t Aqb, t dt cρa du dt dx = A dq dx dx Now recll Fourier s lw tht sttes q = κ du dx dq dx = κu xx cρau t dx = A κu xx dx cρau t dx Aκu xx dx = 0 cρau t Aκu xx dx = 0 A dq dx dx cρau t Aκu xx = 0 which is wht we used the Infinitesiml Approch. 2.6 Higher Dimensions of the Het Eqution: 1D: 1 k u t = u xx 2D: 1 k u t = u xx + u yy 3D: 1 k u t = u xx + u yy + u zz ny D: 1 k u t = 2 u where 2 u is the plcin of u i.e. u x0 x u xn x n 4
5 3 Section Wve Phenomenon: On guitr string See pge 11 for criteri. On curve let T 1 be the tngent force 1 nd α 1 be the ngle of T 1 nd the sme with T 2 nd β let b nd let u 1 so we get tht s x nd the ccelertion of the string given by u tt Newton s second lw sttes tht F = m Now the sum of the verticl forces cting on the segment of the string is TsinβTsinα = F where the mss is the liner density given by; mss = ρ s So ρ su tt = TsinβTsinα with smll slopes sinα tnα nd the sme for β. tnα = u t x, t tnβ = u t x + x, t so ρ su tt = Tu x x + x, t u x x, t ρu tt = T u xx + x, t u x x, t note tht s x s ρu tt = T u xx + x, t u x x, t let x 0 x ρu tt =Tu xx Now let c 2 = T ρ 1 c 2 u tt = u xx Which is the eqution for the 1D sitution. 4 Section Potentil Eqution Recll tht the 2D Het Eqution is: 1 u k t = 2 u x u y 2 If we hve stedy stte temperture distribution then the temperture does not chnge with respect to time. So in the cse of the 2D Het Eqution we hve tht u t = 0. From which we get: Which is the 2D Potentil Eqution. 2 u x u y 2 = 0 5 Section Clssifiction of PDE s 1. Order The order of the PDE is the order of its highest derivtive. 5
6 Exmple 1. u t u xx = 0 is second order PDE. Exmple 2. u t 2 u = 0 is first order PDE. 2. inerity A PDE in u is liner iff it nd its derivtives re liner in u. Exmple 1. u t u xx = cosx is liner. Exmple 2. sinu+u t =0 is non liner Exmple 3. uu t = 0 is non liner. 3. Homogeneous A PDE is homogeneous homog iff the terms not contining u or its derivtives sum to 0. Exmple 1. u t u xx = 0 is homogeneous. Exmple 2. u t u xx = tx is nonhomogeneous. Exmple 3. u t u xx = 1 is nonhomogeneous. Theorem 1. If u 1 nd u 2 re both solution of liner homogeneous PDE then so is: w = c 1 u 1 + c 2 u 2 where c 1 nd c 2 re both constnts. Exmple et u 1 nd u 2 both stisfy u t u xx = 0 We see here tht both u 1 nd u 2 re liner nd homogeneous nd becuse they re both solution to the PDE we get: u 1 t u 1 xx = 0 nd u 2 t u 2 xx = 0 et w = c 1 u 1 + c 2 u 2 Now tking its derivtives we get: w t = c 1 u 1 t + c 2 u 2 t nd w xx = c 1 u 1 xx + c 2 u 2 xx Now consider w t w xx. We will be done when we show tht w t w xx = 0 6
7 w t w xx = c 1 u 1 t + c 2 u 2 t [c 1 u 1 xx + c 2 u 2 xx ] w t w xx = c 1 [u 1 t c 1 u 1 xx ] + c 2 [u 2 t c 2 u 2 xx ] w t w xx = c c 2 0 w t w xx = 0 Theorem 2. If u 1 nd u 2 re both solution of liner nonhomogeneous PDE then u 1 u 2 is solution of the corresponding liner homogeneous PDE. Exmple et the nonhomogeneous liner PDE u t u xx = f x, t cll it 1. Now let u 1 nd u 2 be solutions to 1. so we hve: u 1 t u 1 xx = f x, t nd u 2 t u 2 xx = f x, t We will be done when we show tht u 1 u 2 t u 1 u 2 xx = 0 now tke u 1 u 2 t u 1 u 2 xx = [u 1 t u 1 xx ] [u 2 t u 2 xx ] [u 1 t u 1 xx ] [u 2 t u 2 xx ] = f x, t f x, t = Generl Solution Now to find the generl solution to PDE you dd the homogeneous solution to the nonhomogeneous solution i.e. if w = u 1 u 2 then u 1 = w + u PDE Clssifictions Consider the PDE 11 u xx + 12 u xy + 22 u yy +[OT]= f x, y note tht [OT]= lower order terms. et = if < 0 then we sy the PDE is Elliptic. if = 0 then we sy the PDE is Prbolic if > 0 then we sy the PDE is Hyperbolic 7
8 Exmple 1. 1D Het Eqution 1 k u t = u xx 1 k u t u xx = 0 so we hve 11 = 1, 12 = 0 nd 22 = 0 so we hve =0 so the 1D Het Eqution is Prbolic Exmple 2. 1D Wve Eqution 1 c 2 u tt = u xx 1 c 2 u tt u xx = 0 so we hve 11 = 1, 12 = 0 nd 22 = 1 c 2 so we hve =0 = 0 1 c 2 = 1 so the 1D Wve Eqution is Hyperbolic c2 Exmple 3. 2D Potentil Eqution u xx + u yy = 0 so 11 = 1, 12 = 0 nd 22 = 1 so = 1. So the 2D Potentil Eqution is Elliptic. 6 Section 3.3 Introduction to Fourier Series Given function f x we wnt to express in the form: f x = n cos nπx So the gol is to find the n nd b n terms, given. Definition: Orthogonlity + b nsin nπx which is the Fourier Series. et f x nd gx both be not identicl 0 nd f x = gx. Now let f x, gx, nd wx be integrble on the intervl, b nd wx>0. Now if we hve: f xgxwx dx = 0 then we sy tht f x nd gx re orthogonl with respect to w on the intervl [, b]. In Fourier Series we hve wx = 1 nd the intervl [, b] is [, ] Fct 8
9 The set of functions {1, sin nπx nπx, cos } is orthogonl. To prove this we need to show the following Proof of 1 sin nπx dx = 0 2 sin nπx cos nπx mπx cos dx = 0 4 mπx cos dx = 0 cos nπx dx = 0 sin nπx mπx sin dx = 0 sin nπx dx = nπx cos nπ = cosnπ cos nπ note here tht the cos nπ is even so we get: nπ cosnπ cosnπ = nπ nπ 0 = 0 Definition: Even Function f x is n even function provided f x = f x x domx Even functions re symmetric to the yxis for y = f x Even functions lso hve the property tht if f x is n even function then: f xdx = 2 f xdx 0 Definition: Odd Function f x is n odd function provided f x = f x x domx Odd functions re symmetric bout the origin. Odd functions lso hve the property tht if f x is n odd function then: f xdx = 0 7 Section 3.4 Determining Fourier Coefficients Recll tht: 9
10 f x = n cos nπx + b nsin nπx is the Fourier Series. First find 0. 0 = = 0 f xdx = 2 dx + 0 n = 0 2 dx + Here we should note tht n 2 + n cos nπx + b nsin nπx n cos nπx dx + cos nπx dx + b n b n sin nπx dx sin nπx dx cos nπx dx nd b n sin nπx dx re equl to zero due to them being orthogonl on [, ] with respect to wx where wx = 1. Now we hve 0 = = 0 2 Thus 0 = 0 2 dx + 0 dx = = 0 f x dx Now tht we hve 0 to find the other terms consider: f xcos mπx dx = n cos nπx 0 mπx cos 2 + n cos nπx cos mπx dx + b nsin nπx + b nsin nπx cos mπx dx mπx cos dx Here note tht just like in the cse where we found 0 ll of the terms will go to zero due to orthogonlity except for when n = m. Which gives us: m cos 2 mπx dx = m so m = f xcos mπx nlogously we get b m = dx f xsin mπx dx 7.1 Period The Fourier series is helpful when f is periodic with period 2. 10
11 Exmple: sinx sinx is periodic with its period equl to 2π i.e. sinx = sinx 2π = sinx 4π nd so on. so we cn sy tht sinx = sinx p cll it 1 where p is the smllest positive number tht mkes 1 true. So here we hve 2 = 2π giving us = π in this exmple. The point is tht it is importnt to identify correctly Exmple: Find the Fourier Series for f x given its grph Clerly f is periodic with period 10 which gives us =5 nd our f x is defined s: f x = { 3 if n < x < 10n 3 if 10n < x < n for n Z 11
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 informationPartial 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 righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationLecture 1  Introduction and Basic Facts about PDEs
* 18.15  Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1  Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV
More informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
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 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 informationBest 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 informationTopic 1 Notes Jeremy Orloff
Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble
More informationMath 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 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 information7.6 The Use of Definite Integrals in Physics and Engineering
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems
More informationKinematic Waves. These are waves which result from the conservation equation. t + I = 0. (2)
Introduction Kinemtic Wves These re wves which result from the conservtion eqution E t + I = 0 (1) where E represents sclr density field nd I, its outer flux. The onedimensionl form of (1) is E t + I
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 informationSTEP 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 informationSection 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 information38 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 xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 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 informationLecture 24: Laplace s Equation
Introductory lecture notes on Prtil Differentil Equtions  c Anthony Peirce. Not to e copied, used, or revised without explicit written permission from the copyright owner. 1 Lecture 24: Lplce s Eqution
More informationMATH34032: 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 informationMASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS
MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK WRITTEN EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/MCUPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION
More informationModule 9: The Method of Green s Functions
Module 9: The Method of Green s Functions The method of Green s functions is n importnt technique for solving oundry vlue nd, initil nd oundry vlue prolems for prtil differentil equtions. In this module,
More informationChapter H1: Introduction, Heat Equation
Nme Due Dte: Problems re collected on Wednesdy. Mth 3150 Problems Hbermn Chpter H1 Submitted work. Plese submit one stpled pckge per problem set. Lbel ech problem with its corresponding problem number,
More informationBig 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 informationMA123, 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. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationContinuous 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 relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationl 2 p2 n 4n 2, the total surface area of the
Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n nsided regulr polygon of perimeter p n with vertices on C. Form cone
More informationOrthogonal Polynomials and LeastSquares Approximations to Functions
Chpter Orthogonl Polynomils nd LestSqures Approximtions to Functions **4/5/3 ET. Discrete LestSqures Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
More informationEquations of Motion. Figure 1.1.1: a differential element under the action of surface and body forces
Equtions of Motion In Prt I, lnce of forces nd moments cting on n component ws enforced in order to ensure tht the component ws in equilirium. Here, llownce is mde for stresses which vr continuousl throughout
More informationCalculus and linear algebra for biomedical engineering Week 11: The Riemann integral and its properties
Clculus nd liner lgebr for biomedicl engineering Week 11: The Riemnn integrl nd its properties Hrtmut Führ fuehr@mth.rwthchen.de Lehrstuhl A für Mthemtik, RWTH Achen Jnury 9, 2009 Overview 1 Motivtion:
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 informationSection 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 informationHomework 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 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 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 informationMath 113 Exam 1Review
Mth 113 Exm 1Review September 26, 2016 Exm 1 covers 6.17.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
More informationIf 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 ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
More informationMA 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 informationTest 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).
Test 3 Review Jiwen He Test 3 Test 3: Dec. 46 in CASA Mteril  Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 1417 in CASA You Might Be Interested
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 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 informationLecture 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 informationWeek 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 information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the xxis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More informationRiemann 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 informationEnergy 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 informationragsdale (zdr82) HW2 ditmire (58335) 1
rgsdle (zdr82) HW2 ditmire (58335) This printout should hve 22 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc
More informationLecture 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 informationNUMERICAL 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 informationVariational Techniques for SturmLiouville Eigenvalue Problems
Vritionl Techniques for SturmLiouville 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 informationu(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 MATHGA 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 informationLecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at UrbanaChampaign. March 20, 2014
Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t UrbnChmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method
More informationMTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008
MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul
More informationMath 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two
More informationMath 107H Topics for the first exam. csc 2 x dx = cot x + C csc x cotx dx = csc x + C tan x dx = ln secx + C cot x dx = ln sinx + C e x dx = e x + C
Integrtion Mth 07H Topics for the first exm Bsic list: x n dx = xn+ + C (provided n ) n + sin(kx) dx = cos(kx) + C k sec x dx = tnx + C sec x tnx dx = sec x + C /x dx = ln x + C cos(kx) dx = sin(kx) +
More informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
More informationNumerical 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 informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
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 information3.4 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 information5.5 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 ntiderivtive is not esily recognizble, then we re in
More information20 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 informationSection 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 informationMACsolutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements MACsolutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationLinear Systems with Constant Coefficients
Liner Systems with Constnt Coefficients 4305 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system
More informationSolutions to Assignment 1
MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove
More informationThe 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 evectors nd evlues
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationSections 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(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 informationCHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx
CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twicedifferentile function of x, then t
More informationChapter 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 informationTest , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes
Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl
More informationdy ky, dt where proportionality constant k may be positive or negative
Section 1.2 Autonomous DEs of the form 0 The DE y is mthemticl model for wide vriety of pplictions. Some of the pplictions re descried y sying the rte of chnge of y(t) is proportionl to the mount present.
More informationapproaches 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 informationB Veitch. Calculus I Study Guide
Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some
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 informationChapter 2. Constraints, Lagrange s equations
Chpter Constrints, Lgrnge s equtions Section Constrints The position of the prticle or system follows certin rules due to constrints: Holonomic constrint: f (r. r,... r n, t) = 0 Constrints tht re not
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 informationHomework Assignment 3 Solution Set
Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationImproper 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 informationPractice Problems Solution
Prctice Problems Solution Problem Consier D Simple Hrmonic Oscilltor escribe by the Hmiltonin Ĥ ˆp m + mwˆx Recll the rte of chnge of the expecttion of quntum mechnicl opertor t A ī A, H] + h A t. Let
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 informationSection 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40
Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since
More informationWave Equation on a Two Dimensional Rectangle
Wve Eqution on Two Dimensionl Rectngle In these notes we re concerned with ppliction of the method of seprtion of vriles pplied to the wve eqution in two dimensionl rectngle. Thus we consider u tt = c
More informationMathematics of Motion II Projectiles
Chmp+ Fll 2001 Dn Stump 1 Mthemtics of Motion II Projectiles Tble of vribles t time v velocity, v 0 initil velocity ccelertion D distnce x position coordinte, x 0 initil position x horizontl coordinte
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More informationPreSession Review. Part 1: Basic Algebra; Linear Functions and Graphs
PreSession Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationTheoretische 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 FriedrichAlexnderUniversität ErlngenNürnberg Theoretische
More informationu( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 218, pp 4448): Determine the equation of the following graph.
nlyzing Dmped Oscilltions Prolem (Medor, exmple 218, pp 4448): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $
More informationPART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.
PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic
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 informationMidpoint 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 informationThe 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 informationQuantum 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 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 informationApplied Physics Introduction to Vibrations and Waves (with a focus on elastic waves) Course Outline
Applied Physics Introduction to Vibrtions nd Wves (with focus on elstic wves) Course Outline Simple Hrmonic Motion && + ω 0 ω k /m k elstic property of the oscilltor Elstic properties of terils Stretching,
More informationOrdinary differential equations
Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview IntroductionModelling Bsic concepts to understnd n ODE. Description nd properties
More informationPhysics 9 Fall 2011 Homework 2  Solutions Friday September 2, 2011
Physics 9 Fll 0 Homework  s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you
More information