Consequently, the temperature must be the same at each point in the cross section at x. Let:


 Andrew Blankenship
 1 years ago
 Views:
Transcription
1 HW 2 Comments: L13. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the rod. Solution: The derivtion is virtully the sme s for the homogeneous cse: As in tht cse, the length of rod is L lterl surfce of rod is insulted rod hs constnt therml conductivity k = k (x) (k>0) rod hs constnt specific hetc = c (x) (c>0) rod hs constnt density ρ = ρ (x) (mss/volume) rod hs congruent cross sections perpendiculr to its lterl xis, the xxis; ech cross section hs re A H1. Het flows only prllel to the lterl xis of the rod. Consequently, the temperture must be the sme t ech point in the cross section t x. Let: u = u (x, t) be the temperture t cross section x t time t. (0 x L, t 0) q = q (x, t) be the het flux (the time rte of het trnsfer per unit re) cross the cross section t x t time t. Het flows from left to right when q (x, t) is positive nd flows from right to left when q (x, t) is negtive. e = e (x, t) be the specific internl energy (het energy per unit mss) t cross section x t time t. f = f (x, t), ssumed known, be the time rte per unit volume t which het is generted by processes within the rod. H2. Therml Energy is Conserved: Thetimerteofchngeoftherml energy in ny region V in het conducting solid is equl to the net rte t which het flows into V cross its boundry plus the net rte t which het is produced by sources nd/or sinks in V. Apply H2 to the region V of the rod with x b to obtin: e t ρa dx = for ll <bin [0,L] nd for ll t 0. ( q x + f) Adx (1) 1
2 Lemm 1 If r (x) nd s (x) re continuous functionsonnintervli nd if r (x) dx = then r (x) =s (x) for ll x in I. The Lemm nd (1) imply tht s (x) dx for ll <bin I, e t ρ = q x + f (2) Now, we need to relte the three unknowns e, q, nd u through properties chrcteristic of het flow. H3. (Fourier s Lw of Het Conduction) Het flows from regions of higher temperture to regions of lower temperture nd the rte of flow is greter the more rpidly temperture chnges with respect to distnce. Therefore we ssume tht the rte of het flow is proportionl to the temperture grdient: q (x, t) = ku x with k>0. The therml conductivity k is defined by this reltion. Use of Fourier s lw in (2) give e t ρ =(ku x ) x + f (3) becuse now k is function of x. (You cn expnd this using the product rule but tht turns out not to be n dvntge for mny purposes.) H4. For mny mterils nd over wide temperture rnges the specific internl energy is liner function of the temperture: e = cu + d The specific hetc = c (x) > 0 is defined by this reltion. The units of specific het re [energy]/[mss][deg]. From H4 e t = cu t. So (3) cn be expressed s ρcu t =(ku x ) x + f, the inhomogeneous het eqution (in one sptil dimension). Of course, you cn write this eqution in other equivlent wys. 2
3 L17. Consider homogenous lterlly insulted rod with het flow in one sptil dimension s in clss. Assume tht initilly the temperture in the rod is f (x) for 0 x L, the left end of the rod is lwys held t constnt temperture 0, nd the right hnd end of the rod is insulted. () Formulte n initil boundry vlue problem (IBVP) for the temperture u = u (x, t)in the rod. (b) Find ll nontrivil seprted solutions to the homogeneous equtions in the IBVP in (). Solution: () Since not sources or sinks re specified, ssume there re none. Then the temperture chnge is governed by the bsic het eqution u t = α 2 u xx. The initil temperture in the rod is u (x, 0) which is given s f (x). Sou (x, 0) = f (x). The temperture t the left end of the rod is u (0,t) nd it is mintined t temperture 0; so u (0,t)=0. Therightendoftherodis insulted, which mens there is no het flux there; tht is, q (L, t) =0. By Fourier s lw q (x, t) = ku x (x, t). Set x = L to get u x (L, t) =0t the right end of the rod. The relevnt IBVP is u t = α 2 u xx for 0 <x<l, t>0, (HE) u (x, 0) = f (x) for 0 x L, (IC) u (0,t)=0,u x (L, t) =0 for t 0. (BC) (b) The function u = T (t) X (x) with stisfy (HE) iff (TX) t = α 2 (TX) xx, TX = α 2 TX 00, T α 2 T = X00 X = λ2 where λ 2 is seprtion constnt. (Any letter cn be used in plce of λ 2 ; however,thechoiceof λ 2 is convenient becuse we expect exponentil decy in time.) Thus, T nd X must stisfy T + λ 2 α 2 T =0nd X 00 + λ 2 X =0. The homogeneous boundry conditions will be stisfied if u (0,t) = T (t) X (0) = 0 u x (L, t) = T (t) X 0 (L) =0 If either X (0) 6= 0or X 0 (L) 6= 0, the forgoing equtions would imply T (t) =0 for ll t nd the corresponding seprted solution would be trivil. So, ny nontrivil seprted solutions must stisfy T + λ 2 α 2 T =0nd X 00 + λ 2 X =0 X (0) = 0 X 0 (L) =0. 3
4 You cn solve the differentil eqution for T using n integrting fctor becuse it is first order liner. You cn lso solve it just like you solve second order constnt coefficients by finding one linerly independent solution vi the Euler exponentil guess pproch. Which ever route you follow you find (check this) T (t) =e λ2 α 2 t nd ll its constnt multiples. Likewise if λ 6= 0, the differentil eqution in the BVP for X hs generl solution (check this) X = A cos λx + B sin λx where A nd B re rbitrry constnts. When λ =0, the differentil eqution hs generl solution X = A + Bx nd cn only stisfy the boundry conditions if X (0) = A =0nd X 0 (L) = B =0. Tht is X (x) =0for ll x nd we get the trivil solution for u = TX. So ny nontrivil solutions occur when λ 6= 0. Then X = A cos λx + B sin λx nd this solution to X 00 + λ 2 X =0will stisfy the boundry conditions iff X (0) = A =0, X 0 (L) = Bλcos λl =0. To get n nontrivil solution for X we must tke B 6= 0nd choose λ so tht cos λl =0. The only solutions to this eqution re the odd multiples of π/2 : λl = (2n +1) π for n ny integer, 2 (2n +1)π λ = λ n = for n ny integer. For these vlues of λ the differentil eqution X 00 + λ 2 X =0hs the nontrivil solutions (2n +1)πx X = X n (x) =sinλ n x =sin nd ll its nonzero multiples. Since the T differentil eqution must be solved with the sme λ, the corresponding T fctors re T = T n (t) =e λ2 n α2t = e (2n+1)2 π 2 α 2 t/. Finlly, the seprted solutions to the homogeneous equtions in the IBVP re u = u n (x, t) =T n (t) X n (x) =e (2n+1)2 π 2 α 2 t/ sin nd ll their nonzero multiples. (2n +1)πx 4
5 L21. Let f (s) be twice differentible function of the rel vrible s. In wht follows t is time nd x is the sptil vrible with <x<. Imgine string with infinite extent. () Show tht u = f (x ct) nd v = f (x + ct) re both solutions to the wve eqution in one sptil dimension, which in this cse re the trnsverse deflections of the string. (b) Show tht u = f (x ct) represents trveling wve tht trvels to the right t speed c. This tells you wht c mens in the wve eqution. Hint. At time t =0the wve profile is u = f (x). Drw simple possibility for this profile. Sketch the wve profile t lter time t on the sme set of xes. Now think. (c) Wht is the interprettion of the other solution in ()? Solution: () Let u = f (x ct) nd use the chin rule to obtin Consequently, u t = f 0 (x ct)( c), u tt = f 00 (x ct)( c) 2, u x = f 0 (x ct), u xx = f 00 (x ct). u tt = c 2 f 00 (x ct) =c 2 u xx nd u = f (x ct) is solution of the wve eqution. Likewise for v = f (x + ct). (b) (Remember tht dimensionl nlysis of the wve eqution revels tht the constnt c hs the units of velocity. We wnt to understnd wht it is the velocity of.) At time 0 thewveprofile is u (x, 0) = f (x). This mens tht t time 0 the trnsverse deflection of the string t ny position x is f (x). At time t thewveprofile is u (x, t) =f (x ct). So the trnsverse deflection of the wve t x + ct t time t is u (x + ct, t) =f (x + ct ct) =f (x). Consequently, the trnsverse deflection tht occurred t position x t time 0 occurs t position x + ct t time t. This mens the entire wve is moving to the right with speed c. (c) Likewise, v = f (x + ct) is trvelling wve solution to the wve eqution tht trvels to the left with speed c. 5
Math 5440 Problem Set 3 Solutions
Mth 544 Mth 544 Problem Set 3 Solutions Aron Fogelson Fll, 213 1: (Logn, 1.5 # 2) Repet the derivtion for the eqution of motion of vibrting string when, in ddition, the verticl motion is retrded by dmping
More informationMath 5440 Problem Set 3 Solutions
Mth 544 Mth 544 Problem Set 3 Solutions Aron Fogelson Fll, 25 1: Logn, 1.5 # 2) Repet the derivtion for the eqution of motion of vibrting string when, in ddition, the verticl motion is retrded by dmping
More informationHeat flux and total heat
Het flux nd totl het John McCun Mrch 14, 2017 1 Introduction Yesterdy (if I remember correctly) Ms. Prsd sked me question bout the condition of insulted boundry for the 1D het eqution, nd (bsed on glnce
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 information1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation
1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview
More informationPDE Notes. Paul Carnig. January ODE s vs PDE s 1
PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................
More information1 2D Second Order Equations: Separation of Variables
Chpter 12 PDEs in Rectngles 1 2D Second Order Equtions: Seprtion of Vribles 1. A second order liner prtil differentil eqution in two vribles x nd y is A 2 u x + B 2 u 2 x y + C 2 u y + D u 2 x + E u +
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 informationThe 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 information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a nonconstant 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 information1 1D heat and wave equations on a finite interval
1 1D het nd wve equtions on finite intervl In this section we consider generl method of seprtion of vribles nd its pplictions to solving het eqution nd wve eqution on finite intervl ( 1, 2. Since by trnsltion
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 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 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 informationMath 124A October 04, 2011
Mth 4A October 04, 0 Viktor Grigoryn 4 Vibrtions nd het flow In this lecture we will derive the wve nd het equtions from physicl principles. These re second order constnt coefficient liner PEs, which model
More information(4.1) D r v(t) ω(t, v(t))
1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution
More 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 informationpotentials A z, F z TE z Modes We use the e j z z =0 we can simply say that the x dependence of E y (1)
3e. Introduction Lecture 3e Rectngulr wveguide So fr in rectngulr coordintes we hve delt with plne wves propgting in simple nd inhomogeneous medi. The power density of plne wve extends over ll spce. Therefore
More informationTravelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing
Applied Mthemtics ENotes 8(8)  c IN 675 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ Trvelling Profile olutions For Nonliner Degenerte Prbolic Eqution And Contour Enhncement In Imge
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 informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rellife exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
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 informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
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 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 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 informationConservation Law. Chapter Goal. 6.2 Theory
Chpter 6 Conservtion Lw 6.1 Gol Our long term gol is to unerstn how mthemticl moels re erive. Here, we will stuy how certin quntity chnges with time in given region (sptil omin). We then first erive the
More informationName Solutions to Test 3 November 8, 2017
Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier
More information3 Conservation Laws, Constitutive Relations, and Some Classical PDEs
3 Conservtion Lws, Constitutive Reltions, nd Some Clssicl PDEs As topic between the introduction of PDEs nd strting to consider wys to solve them, this section introduces conservtion of mss nd its differentil
More information2. THE HEAT EQUATION (Joseph FOURIER ( ) in 1807; Théorie analytique de la chaleur, 1822).
mpc2w4.tex Week 4. 2.11.2011 2. THE HEAT EQUATION (Joseph FOURIER (17681830) in 1807; Théorie nlytique de l chleur, 1822). One dimension. Consider uniform br (of some mteril, sy metl, tht conducts het),
More information1 E3102: a study guide and review, Version 1.0
1 E3102: study guide nd review, Version 1.0 Here is list of subjects tht I think we ve covered in clss (your milege my vry). If you understnd nd cn do the bsic problems in this guide you should be in very
More informationProblem set 1: Solutions Math 207B, Winter 2016
Problem set 1: Solutions Mth 27B, Winter 216 1. Define f : R 2 R by f(,) = nd f(x,y) = xy3 x 2 +y 6 if (x,y) (,). ()Show tht thedirectionl derivtives of f t (,)exist inevery direction. Wht is its Gâteux
More information(PDE) u t k(u xx + u yy ) = 0 (x, y) in Ω, t > 0, (BC) u(x, y, t) = 0 (x, y) on Γ, t > 0, (IC) u(x, y, 0) = f(x, y) (x, y) in Ω.
Seprtion of Vriles for Higher Dimensionl Het Eqution 1. Het Eqution nd Eigenfunctions of the Lplcin: An 2D Exmple Ojective: Let Ω e plnr region with oundry curve Γ. Consider het conduction in Ω with fixed
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 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 information21.6 Green Functions for First Order Equations
21.6 Green Functions for First Order Equtions Consider the first order inhomogeneous eqution subject to homogeneous initil condition, B[y] y() = 0. The Green function G( ξ) is defined s the solution to
More informationRead section 3.3, 3.4 Announcements:
Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f
More informationMA 201: Partial Differential Equations Lecture  12
Two dimensionl Lplce Eqution MA 201: Prtil Differentil Equtions Lecture  12 The Lplce Eqution (the cnonicl elliptic eqution) Two dimensionl Lplce Eqution Two dimensionl Lplce Eqution 2 u = u xx + u yy
More informationu t = k 2 u x 2 (1) a n sin nπx sin 2 L e k(nπ/l) t f(x) = sin nπx f(x) sin nπx dx (6) 2 L f(x 0 ) sin nπx 0 2 L sin nπx 0 nπx
Chpter 9: Green s functions for timeindependent problems Introductory emples Onedimensionl het eqution Consider the onedimensionl het eqution with boundry conditions nd initil condition We lredy know
More information13: Diffusion in 2 Energy Groups
3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.Dec. 5 September Contents We study the diffusion eqution in two energy groups
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 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 information4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
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 information(9) P (x)u + Q(x)u + R(x)u =0
STURMLIOUVILLE THEORY 7 2. Second order liner ordinry differentil equtions 2.1. Recll some sic results. A second order liner ordinry differentil eqution (ODE) hs the form (9) P (x)u + Q(x)u + R(x)u =0
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 informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s oneminute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
More informationFunctions of Several Variables
Functions of Severl Vribles Sketching Level Curves Sections Prtil Derivtives on every open set on which f nd the prtils, 2 f y = 2 f y re continuous. Norml Vector x, y, 2 f y, 2 f y n = ± (x 0,y 0) (x
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 informationSeparation of Variables in Linear PDE
Seprtion of Vribles in Liner PDE Now we pply the theory of Hilbert spces to liner differentil equtions with prtil derivtives (PDE). We strt with prticulr exmple, the onedimensionl (1D) wve eqution 2 u
More informationJackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The twodimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationIn Section 5.3 we considered initial value problems for the linear second order equation. y.a/ C ˇy 0.a/ D k 1 (13.1.4)
678 Chpter 13 Boundry Vlue Problems for Second Order Ordinry Differentil Equtions 13.1 TWOPOINT BOUNDARY VALUE PROBLEMS In Section 5.3 we considered initil vlue problems for the liner second order eqution
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationReview SOLUTIONS: Exam 2
Review SOUTIONS: Exm. True or Flse? (And give short nswer ( If f(x is piecewise smooth on [, ], we cn find series representtion using either sine or cosine series. SOUTION: TRUE. If we use sine series,
More informationFamilies of Solutions to Bernoulli ODEs
In the fmily of solutions to the differentil eqution y ry dx + = it is shown tht vrition of the initil condition y( 0 = cuses horizontl shift in the solution curve y = f ( x, rther thn the verticl shift
More informationThe 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 informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
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 informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationProblems for HW X. C. Gwinn. November 30, 2009
Problems for HW X C. Gwinn November 30, 2009 These problems will not be grded. 1 HWX Problem 1 Suppose thn n object is composed of liner dielectric mteril, with constnt reltive permittivity ɛ r. The object
More informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
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 informationMath 116 Calculus II
Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................
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 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 informationg i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f
1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where
More informationIntroduction to the Calculus of Variations
Introduction to the Clculus of Vritions Jim Fischer Mrch 20, 1999 Abstrct This is selfcontined pper which introduces fundmentl problem in the clculus of vritions, the problem of finding extreme vlues
More information3 Mathematics of the Poisson Equation
3 Mthemtics of the Poisson Eqution 3. Green functions nd the Poisson eqution () The Dirichlet Green function stisfies the Poisson eqution with deltfunction chrge 2 G D (r, r o ) = δ 3 (r r o ) (3.) nd
More informationMath Fall 2006 Sample problems for the final exam: Solutions
Mth 425 Fll 26 Smple problems for the finl exm: Solutions Any problem my be ltered or replced by different one! Some possibly useful informtion Prsevl s equlity for the complex form of the Fourier series
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 informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More 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 informationLECTURE 1. Introduction. 1. Rough Classiæcation of Partial Diæerential Equations
LECTURE 1 Introduction 1. Rough Clssiction of Prtil Dierentil Equtions A prtil dierentil eqution is eqution relting function of n vribles x 1 ;::: ;x n, its prtil derivtives, nd the coordintes x =èx 1
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 information7.3 Problem 7.3. ~B(~x) = ~ k ~ E(~x)=! but we also have a reected wave. ~E(~x) = ~ E 2 e i~ k 2 ~x i!t. ~B R (~x) = ~ k R ~ E R (~x)=!
7. Problem 7. We hve two semiinnite slbs of dielectric mteril with nd equl indices of refrction n >, with n ir g (n ) of thickness d between them. Let the surfces be in the x; y lne, with the g being
More informationCalculus of Variations
Clculus of Vritions Com S 477/577 Notes) YnBin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function
More informationThe Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5
The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 25pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationSturmLiouville Eigenvalue problem: Let p(x) > 0, q(x) 0, r(x) 0 in I = (a, b). Here we assume b > a. Let X C 2 1
Ch.4. INTEGRAL EQUATIONS AND GREEN S FUNCTIONS Ronld B Guenther nd John W Lee, Prtil Differentil Equtions of Mthemticl Physics nd Integrl Equtions. Hildebrnd, Methods of Applied Mthemtics, second edition
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationWaveguides Free Space. Modal Excitation. Daniel S. Weile. Department of Electrical and Computer Engineering University of Delaware
Modl Excittion Dniel S. Weile Deprtment of Electricl nd Computer Engineering University of Delwre ELEG 648 Modl Excittion in Crtesin Coordintes Outline 1 Aperture Excittion Current Excittion Outline 1
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 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 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 informationChapter 5. , r = r 1 r 2 (1) µ = m 1 m 2. r, r 2 = R µ m 2. R(m 1 + m 2 ) + m 2 r = r 1. m 2. r = r 1. R + µ m 1
Tor Kjellsson Stockholm University Chpter 5 5. Strting with the following informtion: R = m r + m r m + m, r = r r we wnt to derive: µ = m m m + m r = R + µ m r, r = R µ m r 3 = µ m R + r, = µ m R r. 4
More informationDifferential Equations 2 Homework 5 Solutions to the Assigned Exercises
Differentil Equtions Homework Solutions to the Assigned Exercises, # 3 Consider the dmped string prolem u tt + 3u t = u xx, < x , u, t = u, t =, t >, ux, = fx, u t x, = gx. In the exm you were supposed
More informationCBE 291b  Computation And Optimization For Engineers
The University of Western Ontrio Fculty of Engineering Science Deprtment of Chemicl nd Biochemicl Engineering CBE 9b  Computtion And Optimiztion For Engineers Mtlb Project Introduction Prof. A. Jutn Jn
More informationInClass Problems 2 and 3: Projectile Motion Solutions. InClass Problem 2: Throwing a Stone Down a Hill
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Deprtment of Physics Physics 8T Fll Term 4 InClss Problems nd 3: Projectile Motion Solutions We would like ech group to pply the problem solving strtegy with the
More informationPressure Wave Analysis of a Cylindrical Drum
Pressure Wve Anlysis of Cylindricl Drum Chris Clrk, Brin Anderson, Brin Thoms, nd Josh Symonds Deprtment of Mthemtics The University of Rochester, Rochester, NY 4627 (Dted: December, 24 In this pper, hypotheticl
More informationLecture 13  Linking E, ϕ, and ρ
Lecture 13  Linking E, ϕ, nd ρ A Puzzle... InnerSurfce Chrge Density A positive point chrge q is locted offcenter inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More informationPHYSICS 116C Homework 4 Solutions
PHYSICS 116C Homework 4 Solutions 1. ( Simple hrmonic oscilltor. Clerly the eqution is of the SturmLiouville (SL form with λ = n 2, A(x = 1, B(x =, w(x = 1. Legendre s eqution. Clerly the eqution is of
More informationPHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS
PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS To strt on tensor clculus, we need to define differentition on mnifold.a good question to sk is if the prtil derivtive of tensor tensor on mnifold?
More informationThe Basic Functional 2 1
2 The Bsic Functionl 2 1 Chpter 2: THE BASIC FUNCTIONAL TABLE OF CONTENTS Pge 2.1 Introduction..................... 2 3 2.2 The First Vrition.................. 2 3 2.3 The Euler Eqution..................
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 42 Chapter 7 Practice Problems Set B
Mth 42 Chpter 7 Prctice Problems Set B 1. Which of the following functions is solution of the differentil eqution dy dx = 4xy? () y = e 4x (c) y = e 2x2 (e) y = e 2x (g) y = 4e2x2 (b) y = 4x (d) y = 4x
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 informationR. I. Badran Solid State Physics
I Bdrn Solid Stte Physics Crystl vibrtions nd the clssicl theory: The ssmption will be mde to consider tht the men eqilibrim position of ech ion is t Brvis lttice site The ions oscillte bot this men position
More informationdx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.
Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd
More informationDIRECT CURRENT CIRCUITS
DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through
More information