First Order Linear Differential Equations

 Clarence Pearson
 11 months ago
 Views:
Transcription
1 LECTURE 8 First Orer Linear Differential Equations We now turn our attention to the problem of constructing analytic solutions of ifferential equations; that is to say,solutions that can be epresse in terms of elementary functions (or formulae. We consier first the case of first orer linear ifferential equations. 1. Linear vs NonLinear Differential Equations An orinary or partial ifferential equation is sai to be linear if it is linear in the unknowns f, f etc.. Thus,a general,linear,orinary,n th orer,ifferential equation woul be one of the form a n ( n f ( +a n 1( n 1 f n n 1 ( + +a 1( f ( +f(=g(. It is important to note that the functions a n (,...,a1(,g( nee not be linear functions of. following two eamples shoul convey the general iea., 2 f 2, The Eample f isa2 n orer,linear,partial,ifferential equation. Eample f + z y 2 3 f 3 + f 2 = ezy + f 2 =1 is a nonlinear,orinary,ifferential equation of orer 3. The equation is nonlinear arises because of the presence of the term f 2 which is a quaratic function of the unknown function f. 2. Solving First Orer Linear ODEs A linear first orer orinary ifferential equation is a ifferential equation of the form (8.1 a(y + b(y = c(. Here y represents the unknown function,y its erivative with respect to the variable,an a(, b(c an c( are certain prescribe functions of (the precise functional form of a(,b(,an c( will fie in any specific eample. So long as a( 0,this equation is equivalent to a ifferential equation of the form (8.2 y + p(y = g( where p( = b( a(, g( = c( a(. 23
2 2. SOLVING FIRST ORDER LINEAR ODES 24 We shall refer to a ifferential equation (8.2 as the stanar form of ifferential equation (8.1. (In general,we shall say that an orinary linear ifferential equation is in stanar form when the coefficient of the highest erivative is 1. Our goal now is to evelop a formula for the general solution of (8.2. To accomplish this goal,we shall first construct solutions for several special cases. Then with the knowlege gaine from these simpler eamples, we will evelop a general formula for the solution of any ifferential equation of the form (8.2. Case (i p( =0,g( = arbtrary function. In this case,we have y = g(, an so we are looking for a function whose erivative is g(. Calculus (integral antierivative,we have y( = y = where C is an arbitrary constant of integration. Eample 8.3. y g( = g( + C, = 3 cos(4 Applying the Funamental Theorem of y( = 3cos(4 + C = 3 sin(4 +C 4 Case (ii: g( =0,p( = arbitrary function. In this case we are trying to solve a ifferential equation of the form (8.3 y + p(y =0. To construct a solution,we first rewrite (8.3 as an equation involving ifferentials y + p(y = 0 y = y p( Integrating both sies of the latter equation (the left han sie with respect to y an the right han sie with respect to yiels or,eponentiating both sies ln(y = p( + C [ y = ep ] p( + C
3 [ ] 2. SOLVING FIRST ORDER LINEAR ODES 25 y = ep p( + C [ ] = e C ep p( ] p( = A ep [ In the last step we have simply replace the constant e C,which is arbitrary since C is arbitrary,by another arbitrary constant A. There is nothing tricky here; the point is that in the general solution the numerical factor in front of the eponential function is arbitrary an so rather than writing this factor as e C we use the simpler form A. Thus,the general solution of y + p( =0 is y = A ep [ p( ]. We note that in both Cases (i an (ii,we constructe a solution by carrying out a single integration,an in oing so an arbitrary parameter (ue to a constant of integration was introuce. This is typical of first orer ifferential equations. Inee,a general solution to an n th orer ifferential equation will involve n arbitrary parameters. We shall see latter that in physical applications these arbitrary constants correspon to initial conitions. Case (iii: g( 0, p(= a,a constant. In this case we have y + ay = g(. To solve this equation we employ a trick. (This will not be the last trick you see in this course. Let s multiply both sies of this equation by e a : e a y + ae a y = e a g(. Noticing that the right han sie is y (via the prouct rule for ifferentiation we have,equivalently, (ea e a y = We now take antierivatives of both sies to get (e a y=e a g(. e a g( + C or Eample 8.4. y( = 1 e a e a g( + Ce a. y 2y = 2 e 2 This equation is of type (iii with So we multiply both sies by e 2 to get p = 2 g( = 2 e 2. ( e 2 y = 2 e (y 2y =e 2( 2 e 2 = 2
4 3. THE GENERAL CASE 26 Integrating both sies with respect to,an employing the Funamental Theorem of Calculus on the left yiels e 2 y = C or y = e 2 + Ce 2. Let us now confirm that this is a solution y = 2 e e 2 +2Ce 2 2y = e 2 2Ce 2 so y 2y = 2 e 2 3. The General Case We are now prepare to hanle the case of a general first orer linear ifferential equation; i.e.,ifferential equations of the form (8.4 y + p(y = g( with p( an g( are arbitrary functions of. Note: This case inclues all the preceing cases. We shall construct a solution of this equation in a manner similar to case when p( is a constant; that is, we will try fin a function µ( satisfying (8.5 µ((y +p(y= (µ(y Multiplying (8.4 by µ(,we coul then obtain which when integrate yiels or (µ(y =µ(g( µ(y = µ( g( + C y = 1 (8.6 µ( g( + C µ( µ( It thus remains to fin a suitable function µ(; i.e.,we nee to fin a function µ( sothat This will certainly be true if (8.7 (µ(y +µ(yp( =µ(y µ( =p(µ(.
5 3. THE GENERAL CASE 27 Thus,we have to solve another first orer,linear,ifferential equation of type (iii. As before we rewrite (8.7 in terms of ifferentials to get an then integrate both sies; yieling µ µ = p(, Eponentiating both sies of this relation yiels So a suitable function µ( is ln(µ = p( + A. µ =ep( p( + A µ = ep( p( + A = A ep( p( µ(=a ep( p( Inserting this epression for µ( into our formula (14 for y yiels ( y( = A ep p( ( 1 [ A ep p( g( + C ] It is easily see that the constant A in the enominator is irrelevant to the final answer. This is because it can be cancele out by the A within the integral over,an it can be absorbe into the arbitrary constant C in the secon term. Thus,the general solution to a first orer linear equation y + p(y = g( is given by (8.8 y( = 1 µ( g( + C µ( µ( µ( =ep ( p( Eample 8.5. (8.9 y +2y= sin( Putting this equation in stanar form requires we set Now p( g( p( = 2 = 2 = sin( = 2 ln( =ln ( 2,
6 3. THE GENERAL CASE 28 so Therefore, [ ] µ( = ep [ p( ( =ep ln 2] = 2 1 y( = µ( g( + C µ( µ( = 1 2 ( 2 sin( + C 2 = 1 2 sin( + C 2 Now can be integrate by parts. Set sin( u =, v = sin( Then u =, v = v = cos( an the integration by parts formula, uv = uv vu, tells us that sin( = cos( + cos( Therefore,we have as a general solution of (8.9, = cos( + sin(. y( = 1 2 ( cos( + sin( + C 2 = 1 2 sin( 1 cos( + C 2.
Section 7.1: Integration by Parts
Section 7.1: Integration by Parts 1. Introuction to Integration Techniques Unlike ifferentiation where there are a large number of rules which allow you (in principle) to ifferentiate any function, the
More informationIntegration by Parts
Integration by Parts 63207 If u an v are functions of, the Prouct Rule says that (uv) = uv +vu Integrate both sies: (uv) = uv = uv + u v + uv = uv vu, vu v u, I ve written u an v as shorthan for u an
More informationLinear FirstOrder Equations
5 Linear FirstOrer Equations Linear firstorer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationx = c of N if the limit of f (x) = L and the righthanded limit lim f ( x)
Limit We say the limit of f () as approaches c equals L an write, lim L. OneSie Limits (Left an RightHane Limits) Suppose a function f is efine near but not necessarily at We say that f has a lefthane
More informationIntegration: Using the chain rule in reverse
Mathematics Learning Centre Integration: Using the chain rule in reverse Mary Barnes c 999 University of Syney Mathematics Learning Centre, University of Syney Using the Chain Rule in Reverse Recall that
More informationOrdinary Differential Equations
Orinary Differential Equations Example: Harmonic Oscillator For a perfect Hooke slaw spring,force as a function of isplacement is F = kx Combine with Newton s Secon Law: F = ma with v = a = v = 2 x 2
More informationImplicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
More informationCenter of Gravity and Center of Mass
Center of Gravity an Center of Mass 1 Introuction. Center of mass an center of gravity closely parallel each other: they both work the same way. Center of mass is the more important, but center of gravity
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.
More information2 ODEs Integrating Factors and Homogeneous Equations
2 ODEs Integrating Factors an Homogeneous Equations We begin with a slightly ifferent type of equation: 2.1 Exact Equations These are ODEs whose general solution can be obtaine by simply integrating both
More informationHyperbolic Functions. Notice: this material must not be used as a substitute for attending. the lectures
Hyperbolic Functions Notice: this material must not be use as a substitute for attening the lectures 0. Hyperbolic functions sinh an cosh The hyperbolic functions sinh (pronounce shine ) an cosh are efine
More informationd dx [xn ] = nx n 1. (1) dy dx = 4x4 1 = 4x 3. Theorem 1.3 (Derivative of a constant function). If f(x) = k and k is a constant, then f (x) = 0.
Calculus refresher Disclaimer: I claim no original content on this ocument, which is mostly a summaryrewrite of what any stanar college calculus book offers. (Here I ve use Calculus by Dennis Zill.) I
More informationChapter 6: Integration: partial fractions and improper integrals
Chapter 6: Integration: partial fractions an improper integrals Course S3, 006 07 April 5, 007 These are just summaries of the lecture notes, an few etails are inclue. Most of what we inclue here is to
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More informationDerivatives and the Product Rule
Derivatives an the Prouct Rule James K. Peterson Department of Biological Sciences an Department of Mathematical Sciences Clemson University January 28, 2014 Outline Differentiability Simple Derivatives
More information11.7. Implicit Differentiation. Introduction. Prerequisites. Learning Outcomes
Implicit Differentiation 11.7 Introuction This Section introuces implicit ifferentiation which is use to ifferentiate functions expresse in implicit form (where the variables are foun together). Examples
More informationSeparation of Variables
Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical
More information1 Lecture 20: Implicit differentiation
Lecture 20: Implicit ifferentiation. Outline The technique of implicit ifferentiation Tangent lines to a circle Derivatives of inverse functions by implicit ifferentiation Examples.2 Implicit ifferentiation
More informationQF101: Quantitative Finance September 5, Week 3: Derivatives. Facilitator: Christopher Ting AY 2017/2018. f ( x + ) f(x) f(x) = lim
QF101: Quantitative Finance September 5, 2017 Week 3: Derivatives Facilitator: Christopher Ting AY 2017/2018 I recoil with ismay an horror at this lamentable plague of functions which o not have erivatives.
More informationMA 2232 Lecture 08  Review of Log and Exponential Functions and Exponential Growth
MA 2232 Lecture 08  Review of Log an Exponential Functions an Exponential Growth Friay, February 2, 2018. Objectives: Review log an exponential functions, their erivative an integration formulas. Exponential
More informationLecture 6: Calculus. In Song Kim. September 7, 2011
Lecture 6: Calculus In Song Kim September 7, 20 Introuction to Differential Calculus In our previous lecture we came up with several ways to analyze functions. We saw previously that the slope of a linear
More informationModule FP2. Further Pure 2. Cambridge University Press Further Pure 2 and 3 Hugh Neill and Douglas Quadling Excerpt More information
5548993  Further Pure an 3 Moule FP Further Pure 5548993  Further Pure an 3 Differentiating inverse trigonometric functions Throughout the course you have graually been increasing the number of functions
More informationMA Midterm Exam 1 Spring 2012
MA Miterm Eam Spring Hoffman. (7 points) Differentiate g() = sin( ) ln(). Solution: We use the quotient rule: g () = ln() (sin( )) sin( ) (ln()) (ln()) = ln()(cos( ) ( )) sin( )( ()) (ln()) = ln() cos(
More informationIMPLICIT DIFFERENTIATION
Mathematics Revision Guies Implicit Differentiation Page 1 of Author: Mark Kulowski MK HOME TUITION Mathematics Revision Guies Level: AS / A Level AQA : C4 Eecel: C4 OCR: C4 OCR MEI: C3 IMPLICIT DIFFERENTIATION
More informationComputing Derivatives
Chapter 2 Computing Derivatives 2.1 Elementary erivative rules Motivating Questions In this section, we strive to unerstan the ieas generate by the following important questions: What are alternate notations
More informationSection 2.1 The Derivative and the Tangent Line Problem
Chapter 2 Differentiation Course Number Section 2.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan
More informationElectric Charge and Electrostatic Force
PHY 049 Lecture Notes Chapter : Page 1 of 8 Electric Charge an Electrostatic Force Contemporary vision: all forces of nature can be viewe as interaction between "charges", specific funamental properties
More informationChapter 2. Exponential and Log functions. Contents
Chapter. Exponential an Log functions This material is in Chapter 6 of Anton Calculus. The basic iea here is mainly to a to the list of functions we know about (for calculus) an the ones we will stu all
More informationf(x) f(a) Limit definition of the at a point in slope notation.
Lesson 9: Orinary Derivatives Review Hanout Reference: Brigg s Calculus: Early Transcenentals, Secon Eition Topics: Chapter 3: Derivatives, p. 126235 Definition. Limit Definition of Derivatives at a point
More informationCHAPTER 3 DERIVATIVES (continued)
CHAPTER 3 DERIVATIVES (continue) 3.3. RULES FOR DIFFERENTIATION A. The erivative of a constant is zero: [c] = 0 B. The Power Rule: [n ] = n (n1) C. The Constant Multiple Rule: [c *f()] = c * f () D. The
More information12.5. Differentiation of vectors. Introduction. Prerequisites. Learning Outcomes
Differentiation of vectors 12.5 Introuction The area known as vector calculus is use to moel mathematically a vast range of engineering phenomena incluing electrostatics, electromagnetic fiels, air flow
More informationSome functions and their derivatives
Chapter Some functions an their erivatives. Derivative of x n for integer n Recall, from eqn (.6), for y = f (x), Also recall that, for integer n, Hence, if y = x n then y x = lim δx 0 (a + b) n = a n
More information1 The Derivative of ln(x)
Monay, December 3, 2007 The Derivative of ln() 1 The Derivative of ln() The first term or semester of most calculus courses will inclue the it efinition of the erivative an will work out, long han, a number
More informationThe derivative of a constant function is 0. That is,
NOTES 3: DIFFERENTIATION RULES Name: Date: Perio: LESSON 3. DERIVATIVE OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS Eample : Prove f ( ) 6 is not ifferentiable at 4. Practice Problems: Fin f '( ) using the
More informationRules of Differentiation. Lecture 12. Product and Quotient Rules.
Rules of Differentiation. Lecture 12. Prouct an Quotient Rules. We warne earlier that we can not calculate the erivative of a prouct as the prouct of the erivatives. It is easy to see that this is so.
More information18.01 Single Variable Calculus Fall 2006
MIT OpenCourseWare http://ocw.mit.eu 8.0 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.eu/terms. Lecture 8.0 Fall 2006 Unit
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More information3.6. Implicit Differentiation. Implicitly Defined Functions
3.6 Implicit Differentiation 205 3.6 Implicit Differentiation 5 2 25 2 25 2 0 5 (3, ) Slope 3 FIGURE 3.36 The circle combines the graphs of two functions. The graph of 2 is the lower semicircle an passes
More informationThe derivative of a constant function is 0. That is,
NOTES : DIFFERENTIATION RULES Name: LESSON. DERIVATIVE OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS Date: Perio: Mrs. Nguyen s Initial: Eample : Prove f ( ) 4 is not ifferentiable at. Practice Problems: Fin
More informationCalculus Class Notes for the Combined Calculus and Physics Course Semester I
Calculus Class Notes for the Combine Calculus an Physics Course Semester I Kelly Black December 14, 2001 Support provie by the National Science Founation  NSFDUE9752485 1 Section 0 2 Contents 1 Average
More information12.11 Laplace s Equation in Cylindrical and
SEC. 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential 593 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential One of the most important PDEs in physics an engineering
More informationMATH , 06 Differential Equations Section 03: MWF 1:00pm1:50pm McLaury 306 Section 06: MWF 3:00pm3:50pm EEP 208
MATH 32103, 06 Differential Equations Section 03: MWF 1:00pm1:50pm McLaury 306 Section 06: MWF 3:00pm3:50pm EEP 208 Instructor: Brent Deschamp Email: brent.eschamp@ssmt.eu Office: McLaury 316B Phone:
More informationAdditional Derivative Topics
BARNMC04_0132328186.QXD 2/21/07 1:27 PM Page 216 Aitional Derivative Topics CHAPTER 4 41 The Constant e an Continuous Compoun Interest 42 Derivatives of Eponential an Logarithmic Functions 43 Derivatives
More informationTaylor Expansions in 2d
Taylor Expansions in 2 In your irst year Calculus course you evelope a amily o ormulae or approximating a unction F(t) or t near any ixe point t 0. The cruest approximation was just a constant. F(t 0 +
More information1 Heisenberg Representation
1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationLogarithmic, Exponential and Other Transcendental Functions
Logarithmic, Eponential an Other Transcenental Fnctions 5: The Natral Logarithmic Fnction: Differentiation The Definition First, yo mst know the real efinition of the natral logarithm: ln= t (where > 0)
More informationIntroduction to Markov Processes
Introuction to Markov Processes Connexions moule m44014 Zzis law Gustav) Meglicki, Jr Office of the VP for Information Technology Iniana University RCS: Section2.tex,v 1.24 2012/12/21 18:03:08 gustav
More informationChapter 9 Method of Weighted Residuals
Chapter 9 Metho of Weighte Resiuals 9 Introuction Metho of Weighte Resiuals (MWR) is an approimate technique for solving bounary value problems. It utilizes a trial functions satisfying the prescribe
More informationChapter 2 The Derivative Business Calculus 155
Chapter The Derivative Business Calculus 155 Section 11: Implicit Differentiation an Relate Rates In our work up until now, the functions we neee to ifferentiate were either given explicitly, x such as
More informationApplications of the Wronskian to ordinary linear differential equations
Physics 116C Fall 2011 Applications of the Wronskian to orinary linear ifferential equations Consier a of n continuous functions y i (x) [i = 1,2,3,...,n], each of which is ifferentiable at least n times.
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More informationCMSC 313 Preview Slides
CMSC 33 Preview Slies These are raft slies. The actual slies presente in lecture may be ifferent ue to last minute changes, scheule slippage,... UMBC, CMSC33, Richar Chang CMSC 33 Lecture
More informationComputing Derivatives Solutions
Stuent Stuy Session Solutions We have intentionally inclue more material than can be covere in most Stuent Stuy Sessions to account for groups that are able to answer the questions at a faster rate. Use
More informationUsing the definition of the derivative of a function is quite tedious. f (x + h) f (x)
Derivative Rules Using te efinition of te erivative of a function is quite teious. Let s prove some sortcuts tat we can use. Recall tat te efinition of erivative is: Given any number x for wic te limit
More informationDifferentiability, Computing Derivatives, Trig Review
Unit #3 : Differentiability, Computing Derivatives, Trig Review Goals: Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an original function Compute
More informationFlash Card Construction Instructions
Flash Car Construction Instructions *** THESE CARDS ARE FOR CALCULUS HONORS, AP CALCULUS AB AND AP CALCULUS BC. AP CALCULUS BC WILL HAVE ADDITIONAL CARDS FOR THE COURSE (IN A SEPARATE FILE). The left column
More information11.4. Differentiating ProductsandQuotients. Introduction. Prerequisites. Learning Outcomes
Differentiating ProductsandQuotients 11.4 Introduction We have seen, in the first three Sections, how standard functions like n, e a, sin a, cos a, ln a may be differentiated. In this Section we see how
More informationELEC3114 Control Systems 1
ELEC34 Control Systems Linear Systems  Moelling  Some Issues Session 2, 2007 Introuction Linear systems may be represente in a number of ifferent ways. Figure shows the relationship between various representations.
More informationSYNCHRONOUS SEQUENTIAL CIRCUITS
CHAPTER SYNCHRONOUS SEUENTIAL CIRCUITS Registers an counters, two very common synchronous sequential circuits, are introuce in this chapter. Register is a igital circuit for storing information. Contents
More informationTrigonometric Functions
4 Trigonometric Functions So far we have use only algebraic functions as examples when fining erivatives, that is, functions that can be built up by the usual algebraic operations of aition, subtraction,
More informationProduct and Quotient Rules and HigherOrder Derivatives. The Product Rule
330_003.q 11/3/0 :3 PM Page 119 SECTION.3 Prouct an Quotient Rules an HigherOrer Derivatives 119 Section.3 Prouct an Quotient Rules an HigherOrer Derivatives Fin the erivative o a unction using the Prouct
More informationThe Kepler Problem. 1 Features of the Ellipse: Geometry and Analysis
The Kepler Problem For the Newtonian 1/r force law, a miracle occurs all of the solutions are perioic instea of just quasiperioic. To put it another way, the twoimensional tori are further ecompose into
More informationAccelerate Implementation of Forwaring Control Laws using Composition Methos Yves Moreau an Roolphe Sepulchre June 1997 Abstract We use a metho of int
Katholieke Universiteit Leuven Departement Elektrotechniek ESATSISTA/TR 199711 Accelerate Implementation of Forwaring Control Laws using Composition Methos 1 Yves Moreau, Roolphe Sepulchre, Joos Vanewalle
More information4. Important theorems in quantum mechanics
TFY4215 Kjemisk fysikk og kvantemekanikk  Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking threeimensional potentials in the next chapter, we shall in chapter 4 of this
More informationBasic IIR Digital Filter Structures
Basic IIR Digital Filter Structures The causal IIR igital filters we are concerne with in this course are characterie by a real rational transfer function of or, equivalently by a constant coefficient
More informationDefine each term or concept.
Chapter Differentiation Course Number Section.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan
More informationBy writing (1) as y (x 5 1). (x 5 1), we can find the derivative using the Product Rule: y (x 5 1) 2. we know this from (2)
3.5 Chain Rule 149 3.5 Chain Rule Introuction As iscusse in Section 3.2, the Power Rule is vali for all real number exponents n. In this section we see that a similar rule hols for the erivative of a power
More informationNotes on Lie Groups, Lie algebras, and the Exponentiation Map Mitchell Faulk
Notes on Lie Groups, Lie algebras, an the Exponentiation Map Mitchell Faulk 1. Preliminaries. In these notes, we concern ourselves with special objects calle matrix Lie groups an their corresponing Lie
More information7 Wilson Coefficients and Hard Dynamics
7 Wilson Coefficients an Har Dynamics 7 WILSON COEFFICIENTS AND HARD DYNAMICS We now turn to the ynamics of SCET at one loop. An interesting aspect of loops in the effective theory is that often a full
More informationTransformations of Random Variables
Transformations of Ranom Variables September, 2009 We begin with a ranom variable an we want to start looking at the ranom variable Y = g() = g where the function g : R R. The inverse image of a set A,
More informationMath Review for Physical Chemistry
Chemistry 362 Spring 27 Dr. Jean M. Stanar January 25, 27 Math Review for Physical Chemistry I. Algebra an Trigonometry A. Logarithms an Exponentials General rules for logarithms These rules, except where
More informationThe PressSchechter mass function
The PressSchechter mass function To state the obvious: It is important to relate our theories to what we can observe. We have looke at linear perturbation theory, an we have consiere a simple moel for
More informationComputing Derivatives J. Douglas Child, Ph.D. Rollins College Winter Park, FL
Computing Derivatives by J. Douglas Chil, Ph.D. Rollins College Winter Park, FL ii Computing Inefinite Integrals Important notice regaring book materials Texas Instruments makes no warranty, either express
More information27. Fitting a Model to Data I. A Model of Direct Variation. Lesson. Mental Math
Lesson 27 Fitting a Moel to Data I BIG IDEA If you etermine from a particular set of ata that y varies irectly or inversely as, you can graph the ata to see what relationship is reasonable. Using that
More informationG j dq i + G j. q i. = a jt. and
Lagrange Multipliers Wenesay, 8 September 011 Sometimes it is convenient to use reunant coorinates, an to effect the variation of the action consistent with the constraints via the metho of Lagrange unetermine
More informationPure Further Mathematics 1. Revision Notes
Pure Further Mathematics Revision Notes June 20 2 FP JUNE 20 SDB Further Pure Complex Numbers... 3 Definitions an arithmetical operations... 3 Complex conjugate... 3 Properties... 3 Complex number plane,
More informationAntiderivatives Introduction
Antierivatives 40. Introuction So far much of the term has been spent fining erivatives or rates of change. But in some circumstances we alreay know the rate of change an we wish to etermine the original
More informationPhysics 115C Homework 4
Physics 115C Homework 4 Problem 1 a In the Heisenberg picture, the ynamical equation is the Heisenberg equation of motion: for any operator Q H, we have Q H = 1 t i [Q H,H]+ Q H t where the partial erivative
More informationTAYLOR S POLYNOMIAL APPROXIMATION FOR FUNCTIONS
MISN04 TAYLOR S POLYNOMIAL APPROXIMATION FOR FUNCTIONS f(x ± ) = f(x) ± f ' (x) + f '' (x) 2 ±... 1! 2! = 1.000 ± 0.100 + 0.005 ±... TAYLOR S POLYNOMIAL APPROXIMATION FOR FUNCTIONS by Peter Signell 1.
More informationMATHEMATICS BONUS FILES for faculty and students
MATHMATI BONU FIL for faculty an stuents http://www.onu.eu/~mcaragiu1/bonus_files.html RIVD: May 15, 9 PUBLIHD: May 5, 9 toffel 1 Maxwell s quations through the Major Vector Theorems Joshua toffel Department
More informationAntiderivatives and Indefinite Integration
60_00.q //0 : PM Page 8 8 CHAPTER Integration Section. EXPLORATION Fining Antierivatives For each erivative, escribe the original function F. a. F b. F c. F. F e. F f. F cos What strateg i ou use to fin
More informationShape functions in 1D
MAE 44 & CIV 44 Introuction to Finite Elements Reaing assignment: ecture notes, ogan.,. Summary: Prof. Suvranu De Shape functions in D inear shape functions in D Quaratic an higher orer shape functions
More informationq = F If we integrate this equation over all the mass in a star, we have q dm = F (M) F (0)
Astronomy 112: The Physics of Stars Class 4 Notes: Energy an Chemical Balance in Stars In the last class we introuce the iea of hyrostatic balance in stars, an showe that we coul use this concept to erive
More informationand from it produce the action integral whose variation we set to zero:
Lagrange Multipliers Monay, 6 September 01 Sometimes it is convenient to use reunant coorinates, an to effect the variation of the action consistent with the constraints via the metho of Lagrange unetermine
More informationChapter 6: EnergyMomentum Tensors
49 Chapter 6: EnergyMomentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energymomentum tensors, then applies these ieas to the case of Bohm's moel.
More informationOutline. Calculus for the Life Sciences II. Introduction. Tides Introduction. Lecture Notes Differentiation of Trigonometric Functions
Calculus for the Life Sciences II c Functions Joseph M. Mahaffy, mahaffy@math.ssu.eu Department of Mathematics an Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State
More informationTorque OBJECTIVE INTRODUCTION APPARATUS THEORY
Torque OBJECTIVE To verify the rotational an translational conitions for equilibrium. To etermine the center of ravity of a rii boy (meter stick). To apply the torque concept to the etermination of an
More informationEVALUATING HIGHER DERIVATIVE TENSORS BY FORWARD PROPAGATION OF UNIVARIATE TAYLOR SERIES
MATHEMATICS OF COMPUTATION Volume 69, Number 231, Pages 1117 1130 S 00255718(00)011200 Article electronically publishe on February 17, 2000 EVALUATING HIGHER DERIVATIVE TENSORS BY FORWARD PROPAGATION
More informationFundamental Laws of Motion for Particles, Material Volumes, and Control Volumes
Funamental Laws of Motion for Particles, Material Volumes, an Control Volumes Ain A. Sonin Department of Mechanical Engineering Massachusetts Institute of Technology Cambrige, MA 02139, USA August 2001
More informationConnecting Algebra to Calculus Indefinite Integrals
Connecting Algebra to Calculus Inefinite Integrals Objective: Fin Antierivatives an use basic integral formulas to fin Inefinite Integrals an make connections to Algebra an Algebra. Stanars: Algebra.0,
More informationHarmonic Modelling of Thyristor Bridges using a Simplified Time Domain Method
1 Harmonic Moelling of Thyristor Briges using a Simplifie Time Domain Metho P. W. Lehn, Senior Member IEEE, an G. Ebner Abstract The paper presents time omain methos for harmonic analysis of a 6pulse
More information1 Introuction In the past few years there has been renewe interest in the nerson impurity moel. This moel was originally propose by nerson [2], for a
Theory of the nerson impurity moel: The Schrieer{Wol transformation re{examine Stefan K. Kehrein 1 an nreas Mielke 2 Institut fur Theoretische Physik, uprecht{karls{universitat, D{69120 Heielberg, F..
More informationPart 1. The Quantum Particle
CONTACT CONTACT Introuction to Nanoelectronics Part. The Quantum Particle This class is concerne with the propagation of electrons in conuctors. Here in Part, we will begin by introucing the tools from
More informationarxiv:physics/ v2 [physics.edph] 23 Sep 2003
Mass reistribution in variable mass systems Célia A. e Sousa an Vítor H. Rorigues Departamento e Física a Universiae e Coimbra, P3004516 Coimbra, Portugal arxiv:physics/0211075v2 [physics.eph] 23 Sep
More informationChapter 7. Integrals and Transcendental Functions
7. The Logarithm Define as an Integral Chapter 7. Integrals an Transcenental Functions 7.. The Logarithm Define as an Integral Note. In this section, we introuce the natural logarithm function using efinite
More informationThe Derivative and the Tangent Line Problem. The Tangent Line Problem
96 CHAPTER Differentiation Section ISAAC NEWTON (6 77) In aition to his work in calculus, Newton mae revolutionar contributions to phsics, incluing the Law of Universal Gravitation an his three laws of
More informationSHORTCUTS TO DIFFERENTIATION
Chapter Three SHORTCUTS TO DIFFERENTIATION In Chapter, we efine the erivative function f () = lim h 0 f( + h) f() h an saw how the erivative represents a slope an a rate of change. We learne how to approimate
More informationCalculus I Homework: Related Rates Page 1
Calculus I Homework: Relate Rates Page 1 Relate Rates in General Relate rates means relate rates of change, an since rates of changes are erivatives, relate rates really means relate erivatives. The only
More informationWe want to look at some special functions that can arise, especially in trying to solve certain types of rather simple equations.
Chapter 9 Special Functions We want to look at some special functions that can arise, especially in trying to solve certain types of rather simple equations. 9.1 Hyperbolic Trigonometric Functions The
More information