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

Size: px
Start display at page:

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

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 in-rte 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 Tsinβ-Tsinα = F where the mss is the liner density given by; mss = ρ s So ρ su tt = Tsinβ-Tsinα 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 non-homogeneous. Exmple 3. u t u xx = 1 is non-homogeneous. 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 non-homogeneous PDE then u 1 -u 2 is solution of the corresponding liner homogeneous PDE. Exmple et the non-homogeneous 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 y-xis 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

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

More information

Math 5440 Problem Set 3 Solutions

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

More information

Math 5440 Problem Set 3 Solutions

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 information

1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation

1.1. 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 information

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

Consequently, the temperature must be the same at each point in the cross section at x. Let: HW 2 Comments: L1-3. 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

More information

Math 124A October 04, 2011

Math 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

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

Partial Derivatives. Limits. For a single variable function f (x), the limit lim Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

Conservation Law. Chapter Goal. 5.2 Theory

Conservation 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 information

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 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 information

Heat flux and total heat

Heat 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 information

2. THE HEAT EQUATION (Joseph FOURIER ( ) in 1807; Théorie analytique de la chaleur, 1822).

2. 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 (1768-1830) in 1807; Théorie nlytique de l chleur, 1822). One dimension. Consider uniform br (of some mteril, sy metl, tht conducts het),

More information

Lecture 1 - Introduction and Basic Facts about PDEs

Lecture 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 information

13.4 Work done by Constant Forces

13.4 Work done by Constant Forces 13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

More information

Math 8 Winter 2015 Applications of Integration

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

More information

Math Calculus with Analytic Geometry II

Math 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 x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

More information

1 2-D Second Order Equations: Separation of Variables

1 2-D Second Order Equations: Separation of Variables Chpter 12 PDEs in Rectngles 1 2-D 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 information

Week 10: Line Integrals

Week 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 information

c n φ n (x), 0 < x < L, (1) n=1

c n φ n (x), 0 < x < L, (1) n=1 SECTION : Fourier Series. MATH4. In section 4, we will study method clled Seprtion of Vribles for finding exct solutions to certin clss of prtil differentil equtions (PDEs. To do this, it will be necessry

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

Improper Integrals, and Differential Equations

Improper 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 information

Overview of Calculus I

Overview 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 information

Conservation Law. Chapter Goal. 6.2 Theory

Conservation 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 information

Green function and Eigenfunctions

Green function and Eigenfunctions Green function nd Eigenfunctions Let L e regulr Sturm-Liouville opertor on n intervl (, ) together with regulr oundry conditions. We denote y, φ ( n, x ) the eigenvlues nd corresponding normlized eigenfunctions

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties 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 information

Line Integrals. Partitioning the Curve. Estimating the Mass

Line Integrals. Partitioning the Curve. Estimating the Mass Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to

More information

MA 201: Partial Differential Equations Lecture - 12

MA 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 information

df dt f () b f () a dt

df dt f () b f () a dt Vector lculus 16.7 tokes Theorem Nme: toke's Theorem is higher dimensionl nlogue to Green's Theorem nd the Fundmentl Theorem of clculus. Why, you sk? Well, let us revisit these theorems. Fundmentl Theorem

More information

Convergence of Fourier Series and Fejer s Theorem. Lee Ricketson

Convergence 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 information

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1 Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the

More information

1 The Riemann Integral

1 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 information

PHYSICS 116C Homework 4 Solutions

PHYSICS 116C Homework 4 Solutions PHYSICS 116C Homework 4 Solutions 1. ( Simple hrmonic oscilltor. Clerly the eqution is of the Sturm-Liouville (SL form with λ = n 2, A(x = 1, B(x =, w(x = 1. Legendre s eqution. Clerly the eqution is of

More information

Best Approximation. Chapter The General Case

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

More information

Math& 152 Section Integration by Parts

Math& 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 information

The Regulated and Riemann Integrals

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 information

1 1D heat and wave equations on a finite interval

1 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 information

1 E3102: a study guide and review, Version 1.0

1 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 information

Review of Calculus, cont d

Review 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 information

Harman Outline 1A1 Integral Calculus CENG 5131

Harman Outline 1A1 Integral Calculus CENG 5131 Hrmn Outline 1A1 Integrl Clculus CENG 5131 September 5, 213 III. Review of Integrtion A.Bsic Definitions Hrmn Ch14,P642 Fundmentl Theorem of Clculus The fundmentl theorem of clculus shows the intimte reltionship

More information

Final Exam - Review MATH Spring 2017

Final Exam - Review MATH Spring 2017 Finl Exm - Review MATH 5 - Spring 7 Chpter, 3, nd Sections 5.-5.5, 5.7 Finl Exm: Tuesdy 5/9, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.

More information

Line Integrals. Chapter Definition

Line Integrals. Chapter Definition hpter 2 Line Integrls 2.1 Definition When we re integrting function of one vrible, we integrte long n intervl on one of the xes. We now generlize this ide by integrting long ny curve in the xy-plne. It

More information

Summary: Method of Separation of Variables

Summary: 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 information

Integrals - Motivation

Integrals - 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 information

THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES

THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if

More information

Chapters 4 & 5 Integrals & Applications

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

More information

Math 116 Calculus II

Math 116 Calculus II Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................

More information

Definite integral. Mathematics FRDIS MENDELU

Definite 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 information

Math 231E, Lecture 33. Parametric Calculus

Math 231E, Lecture 33. Parametric Calculus Mth 31E, Lecture 33. Prmetric Clculus 1 Derivtives 1.1 First derivtive Now, let us sy tht we wnt the slope t point on prmetric curve. Recll the chin rule: which exists s long s /. = / / Exmple 1.1. Reconsider

More information

INTRODUCTION TO INTEGRATION

INTRODUCTION 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 information

Describe in words how you interpret this quantity. Precisely what information do you get from x?

Describe in words how you interpret this quantity. Precisely what information do you get from x? WAVE FUNCTIONS AND PROBABILITY 1 I: Thinking out the wve function In quntum mechnics, the term wve function usully refers to solution to the Schrödinger eqution, Ψ(x, t) i = 2 2 Ψ(x, t) + V (x)ψ(x, t),

More information

Topic 1 Notes Jeremy Orloff

Topic 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 information

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

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

More information

We 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.

We 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 information

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30

Definite 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 information

Theoretical foundations of Gaussian quadrature

Theoretical foundations of Gaussian quadrature Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of

More information

Abstract inner product spaces

Abstract 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 information

Math 100 Review Sheet

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

More information

APM346H1 Differential Equations. = u x, u = u. y, and u x, y =?. = 2 u t and u xx= 2 u. x,t, where u t. x, y, z,t u zz. x, y, z,t u yy.

APM346H1 Differential Equations. = u x, u = u. y, and u x, y =?. = 2 u t and u xx= 2 u. x,t, where u t. x, y, z,t u zz. x, y, z,t u yy. INTRODUCTION Types of Prtil Differentil Equtions Trnsport eqution: u x x, yu y x, y=, where u x = u x, u = u y, nd u x, y=?. y Shockwve eqution: u x x, yu x, yu y x, y=. The virting string eqution: u tt

More information

Chapter 28. Fourier Series An Eigenvalue Problem.

Chapter 28. Fourier Series An Eigenvalue Problem. Chpter 28 Fourier Series Every time I close my eyes The noise inside me mplifies I cn t escpe I relive every moment of the dy Every misstep I hve mde Finds wy it cn invde My every thought And this is why

More information

A. 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 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 information

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Definition 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 information

7.6 The Use of Definite Integrals in Physics and Engineering

7.6 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 information

Stuff You Need to Know From Calculus

Stuff You Need to Know From Calculus Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you

More information

10 Vector Integral Calculus

10 Vector Integral Calculus Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve

More information

4.4 Areas, Integrals and Antiderivatives

4.4 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 information

221B Lecture Notes WKB Method

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

More information

Chapter 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

Chapter 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 information

f(a+h) f(a) x a h 0. This is the rate at which

f(a+h) f(a) x a h 0. This is the rate at which M408S Concept Inventory smple nswers These questions re open-ended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnk-out-n-nswer problems! (There re plenty of those in the

More information

Kinematic Waves. These are waves which result from the conservation equation. t + I = 0. (2)

Kinematic 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 one-dimensionl form of (1) is E t + I

More information

( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.

( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists. AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find

More information

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

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

More information

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

More information

g i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f

g 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 information

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

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

More information

1.3 The Lemma of DuBois-Reymond

1.3 The Lemma of DuBois-Reymond 28 CHAPTER 1. INDIRECT METHODS 1.3 The Lemm of DuBois-Reymond We needed extr regulrity to integrte by prts nd obtin the Euler- Lgrnge eqution. The following result shows tht, t lest sometimes, the extr

More information

Elliptic Equations. Laplace equation on bounded domains Circular Domains

Elliptic Equations. Laplace equation on bounded domains Circular Domains Elliptic Equtions Lplce eqution on bounded domins Sections 7.7.2, 7.7.3, An Introduction to Prtil Differentil Equtions, Pinchover nd Rubinstein 1.2 Circulr Domins We study the two-dimensionl Lplce eqution

More information

4. Calculus of Variations

4. 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 information

CHM Physical Chemistry I Chapter 1 - Supplementary Material

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

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC 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 information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019 ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil

More information

Section 6.1 Definite Integral

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

More information

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2. Mth 43 Section 6. Section 6.: 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

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry and basic calculus

ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry and basic calculus ES 111 Mthemticl Methods in the Erth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry nd bsic clculus Trigonometry When is it useful? Everywhere! Anything involving coordinte systems

More information

The Riemann Integral

The Riemann Integral Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function

More information

1 The fundamental theorems of calculus.

1 The fundamental theorems of calculus. The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous

More information

MASTER 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 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/MC-UPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION

More information

ENGI 9420 Lecture Notes 7 - Fourier Series Page 7.01

ENGI 9420 Lecture Notes 7 - Fourier Series Page 7.01 ENGI 940 ecture Notes 7 - Fourier Series Pge 7.0 7. Fourier Series nd Fourier Trnsforms Fourier series hve multiple purposes, including the provision of series solutions to some liner prtil differentil

More information

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) = Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos( - 1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin( - 1 ) = -π 2 6 2 6 Cn you do similr problems?

More information

Chapter 0. What is the Lebesgue integral about?

Chapter 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 information

potentials 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)

potentials 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 information

UNIFORM 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 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 information

AM1 Mathematical Analysis 1 Oct Feb Exercises Lecture 3. sin(x + h) sin x h cos(x + h) cos x h

AM1 Mathematical Analysis 1 Oct Feb Exercises Lecture 3. sin(x + h) sin x h cos(x + h) cos x h AM Mthemticl Anlysis Oct. Feb. Dte: October Exercises Lecture Exercise.. If h, prove the following identities hold for ll x: sin(x + h) sin x h cos(x + h) cos x h = sin γ γ = sin γ γ cos(x + γ) (.) sin(x

More information

Calculus I-II Review Sheet

Calculus I-II Review Sheet Clculus I-II Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing

More information

5.7 Improper Integrals

5.7 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 information

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

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

More information

38 Riemann sums and existence of the definite integral.

38 Riemann sums and existence of the definite integral. 38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

More information

f(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

f(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 information

x = b a N. (13-1) The set of points used to subdivide the range [a, b] (see Fig. 13.1) is

x = b a N. (13-1) The set of points used to subdivide the range [a, b] (see Fig. 13.1) is Jnury 28, 2002 13. The Integrl The concept of integrtion, nd the motivtion for developing this concept, were described in the previous chpter. Now we must define the integrl, crefully nd completely. According

More information

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description

More information

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

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further

More information