Solutions of Laplace s equation in 3d
|
|
- Gwendolyn Robinson
- 7 years ago
- Views:
Transcription
1 Solutions of Laplace s equation in 3 Motivation The general form of Laplace s equation is: Ψ = 0 ; it contains the laplacian, an nothing else. This section will examine the form of the solutions of Laplaces equation in cartesian coorinates an in cylinrical an spherical polar coorinates. Of course it is nice to know how to solve Laplace s equation in these coorinate systems, particularly recalling that the choice of coorinate system is generally etermine by the symmetry of the bounary conitions. But there is a much more important reason why these solutions are of interest. Similar parts of ifferent partial ifferential equations separate off in the same way. Thus, for example, let s take the (time inepenent) Schröinger equation with a spherically symmetric potential. Now the angular part of the equation is all in the angular part of the laplacian. So the angular part of this Schröinger equation is equivalent to the angular part of Laplace s equation in spherical polar coorinates. Then the angular part of the solution to the Schröinger equation with a spherically symmetric potential will be exactly the same as the angular part of the solution to Laplace s equation in spherical polar coorinates: the spherical harmonics we shall iscover below. Of course the raial part of the solution will be ifferent because here the potential will have an effect. So although we are here examining solutions to Laplace s equation, the solutions we shall fin will have relevance to other equations which involve the laplacian. Rectangular Cartesian Coorinates In rectangular cartesian coorinates Laplace s equation takes the form + + Ψ Ψ Ψ = 0. x y z The solution by the separation of variables metho is accomplishe in a number of steps. Step 1: Write the fiel variable as a proct of functions of the inepenent variables. Ψ xyz,, = XxYyZz. Step : Substitute the proct solution into the partial ifferential equation. The erivatives are now total erivatives. X1x6 Y y Zz Y 0 y Z z X x Zz XxYy + + =. x y z Step 3: Divie through by the proct expression for the solution. PH130 Mathematical Methos 1
2 1 X x 1 Y y 1 Zz + + = 0. X x x Y y y Z z z Now the first term is a function of the inepenent variable x only, the secon term a function of the inepenent variable y only an the last term a function only of the inepenent variable z. Since x, y, an z are inepenent of each other, each of the three terms in the equation must be constant an their sum equal to zero. We will set the first term equal to a, the secon term equal to b, an the thir term equal to 1a+ b6. There may well be restrictions on the allowe values of the separation constants from the bounary conitions on the system. We have the three orinary ifferential equations X x ax x = 0 x Y y by 0 y = y Zz + a b Z z =. z These are three Simple Harmonic Oscillator equations an their solutions are ± ax X x = const e Y y Zz = const e ay ± i a+ bz = const e. The solution, corresponing to particular values of a an b is then ax ay i a bz Ψ ab xyz,, const e e e. = ± ± ± + But since the original equation is linear, any linear combination of possible solutions is also a solution. So we may write the general form of the solution as ± ax ± ay ± i a+ bz Ψ1xyz,, 6= Cab e e e. ab The constants C ab remain to be etermine from the bounary conitions that the particular solution must satisfy. In two imensions (let us say there is no epenence on the z coorinate), we may put a+ b=0 an we obtain solutions of the form ax i ay Ψ1xy, 6= const e ± e ±. ± PH130 Mathematical Methos
3 Cylinrical Polar Coorinates In cylinrical polar coorinates Laplace s equation takes the form Ψ Ψ 1 1 Ψ Ψ = 0. z We procee by the three stanar steps for solution by the separation of variables metho. Step 1: Write the fiel variable as a proct of functions of the inepenent variables. Ψ,,z = R Φ Z z. Step : Substitute the proct solution into the partial ifferential equation. The erivatives are now total erivatives. R16 1 R 1 Zz Zz 0 Φ Zz R Zz R Φ + Φ + + Φ =. z Step 3: Divie through by the proct expression for the solution. 1 R R Φ16 1 Zz = 0. R R Φ Zz z Now the last term is a function of z only, an so it must be a constant. Let s set this constant equal to a. Then we have an orinary ifferential equation for Z: Zz az z 0 =, z a familiar SHO equation. We are left with an equation in an : 1 R 1 1 R 1 1 Φ a = 0. R R Φ This may be separate by multiplying through by, giving 1 R16 1 R16 1 Φ a + = 0. R R Φ The last term may now be separate. For convenience we shall use the separation constant n ; we shall justify this choice later. Separating off the last term gives the orinary ifferential equation for Φas Φ + n Φ = 0, a straightforwar SHM equation, together with the ODE for the raial function, which we write in stanar form as: PH130 Mathematical Methos 3
4 R 1 R n + + a R = 0. This is Bessel s equation of orer n. We have previously encountere Bessel s equation of orer zero when we stuie the (circularly symmetric) vibrations of a circular rum. Now we see how the equation arises in the general case. Allowe values for n The solution to the SHM equation for Φis, to within an arbitrary constant Φ= e in. Now there is an important property of angular variables: as the angle increases the path trace out returns again an again to the same point. But the fiel variable, which we are trying to solve for, must have a given value at a given point; it must be single-value. The physical requirement of single-valueness translates into the mathematical requirement that Φ = Φ1 + π6. So if Φ= e in is a solution to the equation, then so is in π Φ= e e in. An for this to be the case, we must have e inπ = 1, which is only true when n is a positive or negative integer. Previously we have foun that restrictions on the allowe values of the separation constants are impose by bounary conitions. In this case restrictions on the allowe values of the separation constant are impose by the requirements of single-valueness of functions of angular variables. This is a secon cause of quantisation. It follows from this iscussion that the raial solutions of cylinrical problems will involve Bessel functions of integer orer n. Spherical Polar Coorinates In spherical polar coorinates Laplace s equation takes the form Ψ Ψ r r 1 1 Ψ sinϑ r sinϑ ϑ ϑ sin ϑ + + = 0. We procee by the three stanar steps for solution by the separation of variables metho. Step 1: Write the fiel variable as a proct of functions of the inepenent variables. Ψ r, ϑ, = R r Θ ϑφ. Step : Substitute the proct solution into the partial ifferential equation. The erivatives are now total erivatives. PH130 Mathematical Methos 4
5 r r Rr r Rr Θ ϑ RrΘ ϑ Φ Θ ϑ Φ + sinϑ Φ + = sinϑ ϑ ϑ sin ϑ 0 Step 3: Divie through by the proct expression for the solution Rr r r Rr Θ ϑ + r Φ sinϑ + = Θ ϑ sinϑ ϑ ϑ Φ sin ϑ Now the first term is epenent only on r, thus it must be constant an we choose ll+1 as the separation constant. This choice must be justifie later. This gives us the orinary ifferential equation for the Rr function: 1 Rr r r Rr = ll r or r r Rr ll 1 + Rr = 0. r This equation has solutions of the form l l R r = Ar + Br 1 +. Because the solutions of this equation are quite simple, particularly since the solutions can be expresse in terms of the elementary functions, this equation has no special name. However for completeness, we shall refer to it as the spherical R equation. Now we have separate off the spherical R equation, the remainer of the Laplace equation is sinϑ Θ ll+ + 1 ϑ Φ sin ϑ sinϑ + = 0, Θ1ϑ6 ϑ ϑ Φ16 where we have tiie things up so that the last term is a function only of. The last term is thus now separable, an similar to the cylinrical case, we shall use m. The justification for this is the same as in the cylinrical case: the single-valueness of the solutions requires that m be a positive or negative integer. The equation for Φis then simply an SHM equation Φ + m Φ 0 =. : When the Φpart is separate off we are left with the equation for Θ ϑ ll 1 + m + 7 sin = 0 ϑ ϑ sin ϑ sin ϑ Θ ϑ Θ. ϑ ϑ This equation is conventionally transforme into one of the stanar equations through change of the inepenent variable ϑ to u by:. PH130 Mathematical Methos 5
6 u = cosϑ. Now = sinϑ ϑ so that = = sin ϑ. ϑ ϑ Substituting this into the equation gives ll + m + Θ 1 sin ϑ 7 Θ sin ϑ sin ϑ u u = 0 an eliminating ϑ in favour of u: Θ u 3ll u 7 m8 u + 1 u7 1 u = 0 u 7 Θ where now Θ is regare as a function of u. Conventionally this is written (not quite in stanar form) as u u m 1 u 7 Θ Θ u + ll+ 1 u = 0 u u Θ 1 u. This is the Associate Legenre equation. In the particular case that m = 0 we get the simpler equation u u 1 u 7 Θ Θ u + ll+ 1 u = 0 Θ known as Legenre s equation. You shoul recognise that the m = 0 case is when we are consiering cylinrically symmetric solutions. Summary of special equations We are now in a position to list the various orinary ifferential equations we have iscovere in looking at solutions of Laplace s equation in ifferent coorinate systems. 1 The SHM equation. Zz + nzz 0 = z Here the inepenent variable z can be a linear or an angular variable. In the linear case bounary conitions will usually restrict the allowe values of the separation constant n. In the angular case the requirement that the solution be single-value restricts the allowe values of n to integers. The solutions of the SHM equation are sines an cosines (equivalently complex exponentials). PH130 Mathematical Methos 6
7 Bessel s equation R 1 R n + + a R 0 = Recall that n is restricte to integer values, following from the SHM equation for Φ The solutions to Bessel s equation are the Bessel functions Jn r an Yn. r These are known to Mathematica as BesselJ[n, r] an BesselY[n, r]. 3 The spherical R equation r r Rr ll 1 + Rr = 0. r This equation has simple solutions: l l R1r6 = Ar + Br 1 +; we on t nee any special functions. Although we have not shown this, the separation constant l is restricte to integer variables by the requirement that solutions to the Θ equation are a single-value function of its angular argument ϑ. 4 Legenre s equation u u 1 u 7 Θ Θ u + ll+ 1 u = 0 Θ. Expresse as functions of u = cosϑ, the solutions of Legenre s equation are polynomials, known as Legenre polynomials, Pu l. Since u = cosϑ, it follows that the variable u ranges between 1< u < 1an this explains the interval for Legenre polynomials. Mathematica knows the Legenre polynomials as LegenreP[l, u]. 5 The associate Legenre equation u u m 1 u 7 Θ Θ u + ll+ 1 u = 0 u u Θ 1 u. When expresse as functions of u = cosϑ, the solutions of the associate Legenre equation are polynomials, known as the associate Legenre polynomials, P m l 1u6. Mathematica knows the associate Legenre polynomials as LegenreP[l, m, u]. Spherical Harmonics The angular part of the solutions of Laplace s equation (an any other equation involving which has spherical symmetry) is containe in the proct of the azimuthal function Φ an the polar function Θ. ϑ The azimuthal function Φwill comprise a complex exponential e im an the polar function will be a solution of the associate. PH130 Mathematical Methos 7
8 Legenre equation P m l 1cosϑ6. Thus a solution corresponing to given values of the separation constants l an m will be m m Yl 1ϑ, 6 Pl 1cosϑ6e to within an arbitrary factor. Since these parts always go together in this way for spherical problems the Y m l ϑ, given their own name; they are calle spherical harmonics. By convention the assocoate Legenre polynomials an the spherical harmonic have their own (ifferent) normalisation conventions. Mathematica knows the spherical harmonics as SphericalHarmonicY[l, m, ϑ, ] im are PH130 Mathematical Methos 8
Lagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying
More informationElliptic Functions sn, cn, dn, as Trigonometry W. Schwalm, Physics, Univ. N. Dakota
Elliptic Functions sn, cn, n, as Trigonometry W. Schwalm, Physics, Univ. N. Dakota Backgroun: Jacobi iscovere that rather than stuying elliptic integrals themselves, it is simpler to think of them as inverses
More informationLecture L25-3D Rigid Body Kinematics
J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L25-3D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general three-imensional
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms
More informationMath 230.01, Fall 2012: HW 1 Solutions
Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The
More informationMathematics. Circles. hsn.uk.net. Higher. Contents. Circles 119 HSN22400
hsn.uk.net Higher Mathematics UNIT OUTCOME 4 Circles Contents Circles 119 1 Representing a Circle 119 Testing a Point 10 3 The General Equation of a Circle 10 4 Intersection of a Line an a Circle 1 5 Tangents
More informationf(x) = a x, h(5) = ( 1) 5 1 = 2 2 1
Exponential Functions an their Derivatives Exponential functions are functions of the form f(x) = a x, where a is a positive constant referre to as the base. The functions f(x) = x, g(x) = e x, an h(x)
More information2 Session Two - Complex Numbers and Vectors
PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coorinates Part : The Area Di erential in Polar Coorinates We can also aly the change of variable formula to the olar coorinate transformation x = r cos () ; y = r sin () However,
More informationy or f (x) to determine their nature.
Level C5 of challenge: D C5 Fining stationar points of cubic functions functions Mathematical goals Starting points Materials require Time neee To enable learners to: fin the stationar points of a cubic
More informationInverse Trig Functions
Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that
More informationRules for Finding Derivatives
3 Rules for Fining Derivatives It is teious to compute a limit every time we nee to know the erivative of a function. Fortunately, we can evelop a small collection of examples an rules that allow us to
More informationis identically equal to x 2 +3x +2
Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3
More informationSection 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of
More informationCalculating Viscous Flow: Velocity Profiles in Rivers and Pipes
previous inex next Calculating Viscous Flow: Velocity Profiles in Rivers an Pipes Michael Fowler, UVa 9/8/1 Introuction In this lecture, we ll erive the velocity istribution for two examples of laminar
More informationDIFFRACTION AND INTERFERENCE
DIFFRACTION AND INTERFERENCE In this experiment you will emonstrate the wave nature of light by investigating how it bens aroun eges an how it interferes constructively an estructively. You will observe
More informationFactoring Dickson polynomials over finite fields
Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ 08544 manjul@math.princeton.eu Michael Zieve Department of Mathematics, University
More information10.2 Systems of Linear Equations: Matrices
SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix
More informationLagrange s equations of motion for oscillating central-force field
Theoretical Mathematics & Applications, vol.3, no., 013, 99-115 ISSN: 179-9687 (print), 179-9709 (online) Scienpress Lt, 013 Lagrange s equations of motion for oscillating central-force fiel A.E. Eison
More informationAn Introduction to Partial Differential Equations in the Undergraduate Curriculum
An Introduction to Partial Differential Equations in the Undergraduate Curriculum J. Tolosa & M. Vajiac LECTURE 11 Laplace s Equation in a Disk 11.1. Outline of Lecture The Laplacian in Polar Coordinates
More informationMSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUM-LIKELIHOOD ESTIMATION
MAXIMUM-LIKELIHOOD ESTIMATION The General Theory of M-L Estimation In orer to erive an M-L estimator, we are boun to make an assumption about the functional form of the istribution which generates the
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationarxiv:1309.1857v3 [gr-qc] 7 Mar 2014
Generalize holographic equipartition for Friemann-Robertson-Walker universes Wen-Yuan Ai, Hua Chen, Xian-Ru Hu, an Jian-Bo Deng Institute of Theoretical Physics, LanZhou University, Lanzhou 730000, P.
More informationTrigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
More informationExponential Functions: Differentiation and Integration. The Natural Exponential Function
46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 5.4 f () = e f() = ln The inverse function of the natural logarithmic function is the natural eponential
More informationf(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.
Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,
More informationIntroduction to Integration Part 1: Anti-Differentiation
Mathematics Learning Centre Introuction to Integration Part : Anti-Differentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction
More information15.2. First-Order Linear Differential Equations. First-Order Linear Differential Equations Bernoulli Equations Applications
00 CHAPTER 5 Differential Equations SECTION 5. First-Orer Linear Differential Equations First-Orer Linear Differential Equations Bernoulli Equations Applications First-Orer Linear Differential Equations
More information3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes
Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same
More informationDifferentiability of Exponential Functions
Differentiability of Exponential Functions Philip M. Anselone an John W. Lee Philip Anselone (panselone@actionnet.net) receive his Ph.D. from Oregon State in 1957. After a few years at Johns Hopkins an
More informationAnswers to the Practice Problems for Test 2
Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationSecond Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.
Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard
More informationGiven three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B);
1.1.4. Prouct of three vectors. Given three vectors A, B, anc. We list three proucts with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); a 1 a 2 a 3 (A B) C = b 1 b 2 b 3 c 1 c 2 c 3 where the
More informationThe Quick Calculus Tutorial
The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,
More informationJON HOLTAN. if P&C Insurance Ltd., Oslo, Norway ABSTRACT
OPTIMAL INSURANCE COVERAGE UNDER BONUS-MALUS CONTRACTS BY JON HOLTAN if P&C Insurance Lt., Oslo, Norway ABSTRACT The paper analyses the questions: Shoul or shoul not an iniviual buy insurance? An if so,
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationQUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE
MODULE - 1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write
More informationHere the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and
Lecture 9 : Derivatives of Trigonometric Functions (Please review Trigonometry uner Algebra/Precalculus Review on the class webpage.) In this section we will look at the erivatives of the trigonometric
More informationMathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)
( : Focus on free Education) Miscellaneous Exercise on chapter 5 Question 1: Evaluate: Answer 1: 1 ( : Focus on free Education) Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) =
More informationPythagorean Triples Over Gaussian Integers
International Journal of Algebra, Vol. 6, 01, no., 55-64 Pythagorean Triples Over Gaussian Integers Cheranoot Somboonkulavui 1 Department of Mathematics, Faculty of Science Chulalongkorn University Bangkok
More informationElectrostatics I. Potential due to Prescribed Charge Distribution, Dielectric Properties, Electric Energy and Force
Chapter Electrostatics I. Potential ue to Prescribe Charge Distribution, Dielectric Properties, Electric Energy an Force. Introuction In electrostatics, charges are assume to be stationary. Electric charges
More information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More informationTo differentiate logarithmic functions with bases other than e, use
To ifferentiate logarithmic functions with bases other than e, use 1 1 To ifferentiate logarithmic functions with bases other than e, use log b m = ln m ln b 1 To ifferentiate logarithmic functions with
More informationis identically equal to x 2 +3x +2
Partial fractions.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as + for any
More informationExample Optimization Problems selected from Section 4.7
Example Optimization Problems selecte from Section 4.7 19) We are aske to fin the points ( X, Y ) on the ellipse 4x 2 + y 2 = 4 that are farthest away from the point ( 1, 0 ) ; as it happens, this point
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More informationStudent Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain
More informationMeasures of distance between samples: Euclidean
4- Chapter 4 Measures of istance between samples: Eucliean We will be talking a lot about istances in this book. The concept of istance between two samples or between two variables is funamental in multivariate
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationChapter 2 Kinematics of Fluid Flow
Chapter 2 Kinematics of Flui Flow The stuy of kinematics has flourishe as a subject where one may consier isplacements an motions without imposing any restrictions on them; that is, there is no nee to
More information20. Product rule, Quotient rule
20. Prouct rule, 20.1. Prouct rule Prouct rule, Prouct rule We have seen that the erivative of a sum is the sum of the erivatives: [f(x) + g(x)] = x x [f(x)] + x [(g(x)]. One might expect from this that
More informationLecture Notes on Polynomials
Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex
More information+ 4θ 4. We want to minimize this function, and we know that local minima occur when the derivative equals zero. Then consider
Math Xb Applications of Trig Derivatives 1. A woman at point A on the shore of a circular lake with radius 2 miles wants to arrive at the point C diametrically opposite A on the other side of the lake
More information7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( )
34 7. Beats 7.1. What beats are. Musicians tune their instruments using beats. Beats occur when two very nearby pitches are sounded simultaneously. We ll make a mathematical study of this effect, using
More informationHow to Avoid the Inverse Secant (and Even the Secant Itself)
How to Avoi the Inverse Secant (an Even the Secant Itself) S A Fulling Stephen A Fulling (fulling@mathtamue) is Professor of Mathematics an of Physics at Teas A&M University (College Station, TX 7783)
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More informationSIGNAL PROCESSING & SIMULATION NEWSLETTER
1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty
More informationVieta s Formulas and the Identity Theorem
Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion
More informationIntegrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
More informationHow To Find Out How To Calculate Volume Of A Sphere
Contents High-Dimensional Space. Properties of High-Dimensional Space..................... 4. The High-Dimensional Sphere......................... 5.. The Sphere an the Cube in Higher Dimensions...........
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More informationDefinition of the spin current: The angular spin current and its physical consequences
Definition of the spin current: The angular spin current an its physical consequences Qing-feng Sun 1, * an X. C. Xie 2,3 1 Beijing National Lab for Conense Matter Physics an Institute of Physics, Chinese
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty
More informationPartial Fractions. p(x) q(x)
Partial Fractions Introduction to Partial Fractions Given a rational function of the form p(x) q(x) where the degree of p(x) is less than the degree of q(x), the method of partial fractions seeks to break
More informationDigital barrier option contract with exponential random time
IMA Journal of Applie Mathematics Avance Access publishe June 9, IMA Journal of Applie Mathematics ) Page of 9 oi:.93/imamat/hxs3 Digital barrier option contract with exponential ranom time Doobae Jun
More informationCalculus Refresher, version 2008.4. c 1997-2008, Paul Garrett, garrett@math.umn.edu http://www.math.umn.edu/ garrett/
Calculus Refresher, version 2008.4 c 997-2008, Paul Garrett, garrett@math.umn.eu http://www.math.umn.eu/ garrett/ Contents () Introuction (2) Inequalities (3) Domain of functions (4) Lines (an other items
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationScalar : Vector : Equal vectors : Negative vectors : Proper vector : Null Vector (Zero Vector): Parallel vectors : Antiparallel vectors :
ELEMENTS OF VECTOS 1 Scalar : physical quantity having only magnitue but not associate with any irection is calle a scalar eg: time, mass, istance, spee, work, energy, power, pressure, temperature, electric
More informationIn mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.
MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target
More informationA Brief Review of Elementary Ordinary Differential Equations
1 A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationDIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents
DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationOptimal Control Policy of a Production and Inventory System for multi-product in Segmented Market
RATIO MATHEMATICA 25 (2013), 29 46 ISSN:1592-7415 Optimal Control Policy of a Prouction an Inventory System for multi-prouct in Segmente Market Kuleep Chauhary, Yogener Singh, P. C. Jha Department of Operational
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationSecond Order Linear Partial Differential Equations. Part I
Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables; - point boundary value problems; Eigenvalues and Eigenfunctions Introduction
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationFluid Pressure and Fluid Force
0_0707.q //0 : PM Page 07 SECTION 7.7 Section 7.7 Flui Pressure an Flui Force 07 Flui Pressure an Flui Force Fin flui pressure an flui force. Flui Pressure an Flui Force Swimmers know that the eeper an
More informationAPPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS
Application of Calculus in Commerce an Economics 41 APPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS æ We have learnt in calculus that when 'y' is a function of '', the erivative of y w.r.to i.e. y ö
More information2.6 Exponents and Order of Operations
2.6 Exponents and Order of Operations We begin this section with exponents applied to negative numbers. The idea of applying an exponent to a negative number is identical to that of a positive number (repeated
More informationPolynomial Invariants
Polynomial Invariants Dylan Wilson October 9, 2014 (1) Today we will be interested in the following Question 1.1. What are all the possible polynomials in two variables f(x, y) such that f(x, y) = f(y,
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationFirewall Design: Consistency, Completeness, and Compactness
C IS COS YS TE MS Firewall Design: Consistency, Completeness, an Compactness Mohame G. Goua an Xiang-Yang Alex Liu Department of Computer Sciences The University of Texas at Austin Austin, Texas 78712-1188,
More informationSections 3.1/3.2: Introducing the Derivative/Rules of Differentiation
Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here
More informationSo far, we have looked at homogeneous equations
Chapter 3.6: equations Non-homogeneous So far, we have looked at homogeneous equations L[y] = y + p(t)y + q(t)y = 0. Homogeneous means that the right side is zero. Linear homogeneous equations satisfy
More informationCURVE FITTING LEAST SQUARES APPROXIMATION
CURVE FITTING LEAST SQUARES APPROXIMATION Data analysis and curve fitting: Imagine that we are studying a physical system involving two quantities: x and y Also suppose that we expect a linear relationship
More informationDRAFT. Further mathematics. GCE AS and A level subject content
Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed
More informationASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1
19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A Two-DOF Mass-Spring-Dashpot Dynamic System Consider the lumped-parameter, mass-spring-dashpot dynamic system shown in the Figure. It has two point
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationMathematics Review for Economists
Mathematics Review for Economists by John E. Floy University of Toronto May 9, 2013 This ocument presents a review of very basic mathematics for use by stuents who plan to stuy economics in grauate school
More informationExamples of Functions
Examples of Functions In this document is provided examples of a variety of functions. The purpose is to convince the beginning student that functions are something quite different than polynomial equations.
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More informationCONTENTS 1. Peter Kahn. Spring 2007
CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More informationTHE COMPLEX EXPONENTIAL FUNCTION
Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula
More information