Exam to General Relativity (Winter term 2011/2012) Prof. V. F. Mukhanov, LMU München Solution! Problem: 1a 1b 1c 1d 1e
|
|
- Πραξιτέλης Βικελίδης
- 6 χρόνια πριν
- Προβολές:
Transcript
1 Exam to General Relativity Winter term 211/212 Prof. V. F. Mukhanov, LMU München Solution! Problem: 1a 1b 1c 1d 1e Points:
2 1 Short Questions In the following questions, mark the correct answer with a cross. There is always only one correct answer. a In general relativity, which of the following statements is true in vacuum? The Riemann tensor is zero. The energy-momentum tensor is undetermined. The Ricci tensor is zero. There are no gravitational waves. 1 Point The Ricci tensor is zero according to the vacuum Einstein equations. b Which of the following is the correct transformation law for the energymomentum tensor T ν µ under a general coordinate change x x? T µ ν x γ = Tα β x γ xα x ν x µ x β T µ ν x γ = Tα β x γ xµ x ν x α x β T µ ν x γ = Tα β x γ xα x β x µ x ν T µ ν x γ = Tα β x γ xα x ν x µ x β 1 Point The correct answer is the fourth. c Let W β ɛ α γδ be an arbitrary tensor. Which of the following quantities does transform like a tensor under coordinate transformations? γ W γ δ W β δ W β γ α γδ ɛ α δδ δ α γδ 1 Point d Which of the following is the correct expression for the proper time in a Minkowski metric with signature η αβ = diag+1, 1, 1, 1? dτ 2 dτ 2 = η αβ dx α dx β = η αβ dx α dx β e Which of the following statements is true in General Relativity? 1 Point At an arbitrary point, the gravitational field can be removed by a coordinate transformation. In some finite neighborhood around a point the gravitational field can be removed by a coordinate transformation. 1 Point 2
3 2 Exotic spacetimes 1 Points Consider the following spacetime, which is described in the local coordinates x α = t, x, y, z by the metric ds 2 = dt + αxdx 2 dx 2 dy 2 dz 2. Here, x is a periodic coordinate, i.e., we identify x x+l, and αx is a periodic function of x only, satisfying αx + L = αx. a Read off and determine the matrices g αβ and g αβ. Compute the nonvanishing Christoffel symbols. b Show that this spacetime is flat by computing the Riemann tensor. 3 Points 2 Points c Make this flatness manifest by showing the existence of a coordinate system in which all metric components are constant. To this end, consider the coordinate transformation t t = t fx. Which conditions need f to satisfy in order to set αx to a constant A? Give an expression for A as an integral over α. d Consider a particle trajectory x α τ = τ1, k,,. 3 Points Assume that αx equals a constant A > 1. Determine a constant k such that this describes a light ray. 2 Points 3
4 a g αβ = 1 α α α 2 1 1, gαβ = 1 1 α 2 α α For the Christoffel symbols Γ γ αβ = 1 2 gγδ α g βδ + β g αδ δ g αβ many are manifestly zero. The non-zero candidates are Γ 11 = g 1 g g1 1 g 11 = 1 α 2 α αα2 1 = 1 α 2 α +α 2 α = α and Γ 1 11 = g 1 1 g g11 1 g 11 = αα 1 2 α2 1 =, where = x. The only non-vanishing Christoffel symbols are Γ 11 = α. b From the definition of the Riemann tensor R δ γαβ = α Γ δ βγ β Γ δ αγ + Γ δ αζ Γ ζ βγ Γδ βζ Γ ζ αγ we see that due to the antisymmetrization in α, β there are no nonvanishing components with only Γ 11 non-zero. Thus, this spacetime is flat. c The coordinate transformation implies dt = dt + f xdx dt + αxdx = dt + f x + αxdx Thus, in order to set αx to a constant A by this coordinate transformation, fx needs to satisfy Integration of this equation gives L f x + αx = A f xdx + L αxdx = LA For f to be a valid coordinate transformation it needs to respect the periodicity, i.e., we have f = fl. The first integral therefore vanishes and we conclude A = 1 L αxdx L 4
5 d The 4-velocity is given by u α = dxα dτ = 1, k,, and the trajectory describes a light ray if the 4-velocity is null, = g αβ u α u β = g + 2kg 1 + k 2 g 11 This gives a quadratic equation for k, which is solved by 1 + 2Ak + A 2 1k 2 = k = A or k = 1 1 A 5
6 3 Differential Bianchi identities 1 Points a Prove that for any torsion-free connection the commutator of covariant derivatives gives the Riemann tensor, [ Dα, D β Vγ := D α D β D β D α V γ = R δ γαβ V δ. 1 b Prove that the differential Bianchi identity 4 Points D α R βγδζ + D β R γαδζ + D γ R αβδζ = 2 follows as a consequence of the Jacobi identity [[ Dα, D β, Dγ + [[ Dβ, D γ, Dα + [[ Dγ, D α, Dβ =. 3 4 Points Hint: Act with the left-hand side of 3 on a vector V δ and use 1. You may use the algebraic Bianchi identity R δ γαβ + R δ αβγ + R δ βγα =. c Derive the contracted differential Bianchi identities D α R αβγδ D γ R δβ + D δ R γβ =, D α R αβ 1 2 Rg αβ = where R αβ is the Ricci tensor and R the Ricci scalar. 2 Points 6
7 a D α D β V γ α β = α D β V γ Γ δ αβd δ V γ Γ δ αγd β V δ α β = α β V γ Γ δ βγv δ Γ δ αγ β V δ Γ ζ βδ V ζ α β = α Γ δ βγ β Γ δ αγ + Γ δ αζ Γ ζ βγ Γδ βζ Γ ζ αγ Vδ = R δ γαβv δ b = [Dα, D β, Dγ + cycl. V δ = [ D α, D β Dγ V δ D γ [ Dα, D β Vδ + cycl. = R ζ γαβd ζ V δ R ζ δαβd γ V ζ + D γ R ζ δαβ V ζ + cycl. = R ζ γαβd ζ V δ + D γ R ζ δαβ Vζ + cycl. The cyclic sum of the first term vanishes due to the algebraic Bianchi identity. Thus the cyclic sum of the second term has to vanish, implying the differential Bianchi identity c Contracting in 4 α, ζ we obtain D α R ζ δβγ + D β R ζ δγα + D γ R ζ δαβ = 4 Contracting further with g δγ we get D α R αδβγ D β R δγ + D γ R δβ = = D α R αβ D β R + D γ R γβ = 2D α R αβ 1 2 g αβr 7
8 4 Metric Perturbations 1 Points a Prove that the transformation law for metric perturbations δg αβ in an arbitrary curved background spacetime g αβ under an infinitesimal coordinate transformation x µ x µ = x µ + ξ µ is given by δ g αβ = δg αβ g αβ,γ ξ γ g γβ ξ γ,α g αγ ξ γ,β 4 Points Hint: Split the metric into background and perturbation parts as g αβ = g αβ + δg αβ and use the usual transformation law for tensors, i.e. Keep only the terms linear in δg and ξ. g αβ x = xµ x α x ν x β g µνx. 5 b Show that the transformation law derived in a can be written in a compact form as δ g αβ = δg αβ g µβ ξ µ ;α g αµ ξ µ ;β 6 where the semicolon denotes the covariant derivative with respect to the background metric g αβ. 3 Points Hint: Write out the covariant derivatives in 6 explicitly and compare with your result from a. Use the Christoffel symbols Γ µ αβ = 1 2 g µν g αν,β + g βν,α g αβ,ν. c Consider the Friedmann metric g αβ dx α dx β = a 2 η dη 2 δ ij dx i dx j 7 with small perturbations, g αβ = g αβ + δg αβ. Using the transformation law derived in a find the explicit form of the transformed metric perturbations δ g, δ g i, δ g ij in the new coordinate system { x µ }. For this split the infinitesimal vector as ξ µ ξ, ξ i + ζ,i with ξ,i i =. Also use the explicit form of the background metric given in 7. 3 Points Hint: For comparison we give the correct answer for δ g = δg 2a aξ where x = η. 8
9 a g αβ x = xµ ξ µ x α x ν ξ ν x β g µν x + δg µν = δ µ α α ξ µ δ ν β β ξ ν g µν x + δg µν = g αβ x + δg αβ g µβ α ξ µ g αµ β ξ µ + Oξ 2, δg ξ Then = g αβ x + δ g αβ = g αβ x + µ g αβ ξ µ + δ g αβ δ g αβ = δg αβ µ g αβ ξ µ g µβ α ξ µ g αµ β ξ µ. b δ g αβ = δg αβ g µβ ξ µ,α + Γ µ αλ ξλ g αµ ξ µ,β + Γ µ βλ ξλ = δg αβ g µβ ξ µ,α g µβ ξ λ 1 g µν g αν,λ + g λν,α g αλ,ν 2 g µα ξ µ,β 1 g µα g µν ξ λ g βν,λ + g λν,β g βλ,ν 2 = δg αβ g µβ ξ µ,α g µα ξ µ,β 1 2 ξλ g αβ,λ + g λβ,α g αλ,β + + g βα,λ + g αλ,β g βλ,α = δg αβ µ g αβ ξ µ g µβ α ξ µ g αµ β ξ µ. c δ g = δg ξ µ µ g g µ ξ µ g µ ξ µ = δg ξ g 2 g ξ = δg ξ 2aa 2a 2 ξ = δg 2a aξ δ g i = δg i ξ µ,i g µ ξ µ, g µi ξ λ g i,λ =δg i ξ,i g ξ j, g ij =δg i ξ,ia 2 ξ j + ζ,j δ ij a 2 [ =δg i + a 2 ξ i + ζ ξ,i where I have used the convention ξ i ξ i and ζ,i ζ,i. d δ g ij = δg ij g ik ξ k,j g kj ξ k,i ξ g ij, =δg ij + a 2 ξ i,j + a 2 ξ j,i + ξ 2aa δ ij [ =δg ij + a 2 2 a a δ ijξ + 2ζ,ij + ξ i,j + ξ j,i 9
10 5 Perturbations of the energy-momentum tensor 5 Points The background metric of a flat FLRW universe can be written as: g αβ dx α dx β = a 2 η dη 2 δ ij dx i dx j, 8 where η is the conformal time. For this metric, the Einstein tensor is: G = 3H a 2 G i =, where H = a /a and x = η. G i j = 1 a 2 2H + Hδ i j, 9 a Consider fluctuations of the energy-momentum tensor around a background, Tβ α = Tβ α + δt β α. As in exercise 4, the fluctuations transform under infinitesimal coordinate transformations x µ x µ = x µ + ξ µ as δ T α β = δt α β T α β,γ ξ γ T α γ ξ γ,β + T γ β ξα,γ. 1 Determine the transformation rules of δt, δti and δtj i the parameter ξ µ ξ, ξ i + ζ,i with ξ,i i =. with respect to Hint: Use the background Einstein equations G α β = 8πG Tβ α. 2 Points b The metric fluctuations transform as follows B B ζ + ξ E E ζ 11 Verify that the quantities δt = δt + T B E δt i = δti + T Tk k /3 B E,i 12 are gauge invariant. δ = δ + B E Solution: 3 Points a By using the background Einstein equations, we can immediately infer from eqs.9 that: T i =, δ i j 13 and that the spatial derivatives of T β α vanish. We can now calculate the components of the energy-momentum tensor fluctuation from eqs. 1: δ T = δt T,γ ξ γ T γ ξ γ, + T γ ξ,γ 1
11 = δt T ξ T ξ + T ξ = δt T ξ 14 δ T i = δt i T i,γ ξ γ T γ ξ γ,i + T γ i ξ,γ = δti T ξ,i + 1 Tk k δ j i 3 ξ,j = δti T Tk k /3 ξ,i 15 δ = δ,γ ξ γ T i γ ξ γ,j + T γ j ξi,γ = δ ξ T i k ξ k,j + T k j ξ i,k = δ ξ 16 b Applying the transformation rules derived above and eqs.11, we find: δt δt T ξ + B T ζ + ξ E + ζ = δt + T B E 17 δt i δti T Tk k /3 ξ,i + B T Tk /3 k ζ + ξ E + ζ,i = δti + T Tk k /3 B E,i 18 δ δ ξ + B ζ + ξ E + ζ = δ + B E 19 11
Tutorial problem set 6,
GENERAL RELATIVITY Tutorial problem set 6, 01.11.2013. SOLUTIONS PROBLEM 1 Killing vectors. a Show that the commutator of two Killing vectors is a Killing vector. Show that a linear combination with constant
Διαβάστε περισσότεραHigher Derivative Gravity Theories
Higher Derivative Gravity Theories Black Holes in AdS space-times James Mashiyane Supervisor: Prof Kevin Goldstein University of the Witwatersrand Second Mandelstam, 20 January 2018 James Mashiyane WITS)
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραMATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Monday 6 June, 2005 9 to 12 PAPER 60 GENERAL RELATIVITY Attempt THREE questions. There are FOUR questions in total. The questions carry equal weight. The signature is ( + ),
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραGeodesic Equations for the Wormhole Metric
Geodesic Equations for the Wormhole Metric Dr R Herman Physics & Physical Oceanography, UNCW February 14, 2018 The Wormhole Metric Morris and Thorne wormhole metric: [M S Morris, K S Thorne, Wormholes
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραPractice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 s Free graviton Hamiltonian Show that the free graviton action we discussed in class (before making it gauge- and Lorentzinvariant), S 0 = α d 4 x µ h ij µ h ij, () yields the correct free Hamiltonian
Διαβάστε περισσότεραExercises to Statistics of Material Fatigue No. 5
Prof. Dr. Christine Müller Dipl.-Math. Christoph Kustosz Eercises to Statistics of Material Fatigue No. 5 E. 9 (5 a Show, that a Fisher information matri for a two dimensional parameter θ (θ,θ 2 R 2, can
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότεραSpace-Time Symmetries
Chapter Space-Time Symmetries In classical fiel theory any continuous symmetry of the action generates a conserve current by Noether's proceure. If the Lagrangian is not invariant but only shifts by a
Διαβάστε περισσότεραProblem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.
Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραCosmological Space-Times
Cosmological Space-Times Lecture notes compiled by Geoff Bicknell based primarily on: Sean Carroll: An Introduction to General Relativity plus additional material 1 Metric of special relativity ds 2 =
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Διαβάστε περισσότερα= {{D α, D α }, D α }. = [D α, 4iσ µ α α D α µ ] = 4iσ µ α α [Dα, D α ] µ.
PHY 396 T: SUSY Solutions for problem set #1. Problem 2(a): First of all, [D α, D 2 D α D α ] = {D α, D α }D α D α {D α, D α } = {D α, D α }D α + D α {D α, D α } (S.1) = {{D α, D α }, D α }. Second, {D
Διαβάστε περισσότεραSixth lecture September 21, 2006
Sixth lecture September, 006 Web Page: http://www.colorado.edu/physics/phys7840 NOTE: Next lectures Tuesday, Sept. 6; noon Thursday, Sept. 8; noon Tuesday, Oct. 3; noon Thursday, Oct. 5; noon more????
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραParallel transport and geodesics
Parallel transport and geodesics February 4, 3 Parallel transport Before defining a general notion of curvature for an arbitrary space, we need to know how to compare vectors at different positions on
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότερα3+1 Splitting of the Generalized Harmonic Equations
3+1 Splitting of the Generalized Harmonic Equations David Brown North Carolina State University EGM June 2011 Numerical Relativity Interpret general relativity as an initial value problem: Split spacetime
Διαβάστε περισσότεραforms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with
Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We
Διαβάστε περισσότεραOrbital angular momentum and the spherical harmonics
Orbital angular momentum and the spherical harmonics March 8, 03 Orbital angular momentum We compare our result on representations of rotations with our previous experience of angular momentum, defined
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότερα9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr
9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values
Διαβάστε περισσότεραSolution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Διαβάστε περισσότεραCHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραRiemannian Curvature
Riemannian Curvature February 6, 013 We now generalize our computation of curvature to arbitrary spaces. 1 Parallel transport around a small closed loop We compute the change in a vector, w, which we parallel
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότερα( ) 2 and compare to M.
Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8
Διαβάστε περισσότεραPhys624 Quantization of Scalar Fields II Homework 3. Homework 3 Solutions. 3.1: U(1) symmetry for complex scalar
Homework 3 Solutions 3.1: U(1) symmetry for complex scalar 1 3.: Two complex scalars The Lagrangian for two complex scalar fields is given by, L µ φ 1 µ φ 1 m φ 1φ 1 + µ φ µ φ m φ φ (1) This can be written
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραD Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
Διαβάστε περισσότεραPhysics 554: HW#1 Solutions
Physics 554: HW#1 Solutions Katrin Schenk 7 Feb 2001 Problem 1.Properties of linearized Riemann tensor: Part a: We want to show that the expression for the linearized Riemann tensor, given by R αβγδ =
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραParametrized Surfaces
Parametrized Surfaces Recall from our unit on vector-valued functions at the beginning of the semester that an R 3 -valued function c(t) in one parameter is a mapping of the form c : I R 3 where I is some
Διαβάστε περισσότεραb. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Διαβάστε περισσότεραSymmetric Stress-Energy Tensor
Chapter 3 Symmetric Stress-Energy ensor We noticed that Noether s conserved currents are arbitrary up to the addition of a divergence-less field. Exploiting this freedom the canonical stress-energy tensor
Διαβάστε περισσότεραAppendix A. Curvilinear coordinates. A.1 Lamé coefficients. Consider set of equations. ξ i = ξ i (x 1,x 2,x 3 ), i = 1,2,3
Appendix A Curvilinear coordinates A. Lamé coefficients Consider set of equations ξ i = ξ i x,x 2,x 3, i =,2,3 where ξ,ξ 2,ξ 3 independent, single-valued and continuous x,x 2,x 3 : coordinates of point
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραVariational Wavefunction for the Helium Atom
Technische Universität Graz Institut für Festkörperphysik Student project Variational Wavefunction for the Helium Atom Molecular and Solid State Physics 53. submitted on: 3. November 9 by: Markus Krammer
Διαβάστε περισσότεραSymmetry. March 31, 2013
Symmetry March 3, 203 The Lie Derivative With or without the covariant derivative, which requires a connection on all of spacetime, there is another sort of derivation called the Lie derivative, which
Διαβάστε περισσότεραMath 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Διαβάστε περισσότεραORDINAL ARITHMETIC JULIAN J. SCHLÖDER
ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left- Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.
Διαβάστε περισσότεραΗΜΥ 220: ΣΗΜΑΤΑ ΚΑΙ ΣΥΣΤΗΜΑΤΑ Ι Ακαδημαϊκό έτος Εαρινό Εξάμηνο Κατ οίκον εργασία αρ. 2
ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΗΜΥ 220: ΣΗΜΑΤΑ ΚΑΙ ΣΥΣΤΗΜΑΤΑ Ι Ακαδημαϊκό έτος 2007-08 -- Εαρινό Εξάμηνο Κατ οίκον εργασία αρ. 2 Ημερομηνία Παραδόσεως: Παρασκευή
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραNotes on the Open Economy
Notes on the Open Econom Ben J. Heijdra Universit of Groningen April 24 Introduction In this note we stud the two-countr model of Table.4 in more detail. restated here for convenience. The model is Table.4.
Διαβάστε περισσότερα1 String with massive end-points
1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε
Διαβάστε περισσότεραSolutions Ph 236a Week 8
Solutions Ph 236a Week 8 Page 1 of 9 Solutions Ph 236a Week 8 Kevin Barkett, Jonas Lippuner, and Mark Scheel December 1, 2015 Contents Problem 1................................... 2...................................
Διαβάστε περισσότεραThe kinetic and potential energies as T = 1 2. (m i η2 i k(η i+1 η i ) 2 ). (3) The Hooke s law F = Y ξ, (6) with a discrete analog
Lecture 12: Introduction to Analytical Mechanics of Continuous Systems Lagrangian Density for Continuous Systems The kinetic and potential energies as T = 1 2 i η2 i (1 and V = 1 2 i+1 η i 2, i (2 where
Διαβάστε περισσότεραAppendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee
Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset
Διαβάστε περισσότερα1 Lorentz transformation of the Maxwell equations
1 Lorentz transformation of the Maxwell equations 1.1 The transformations of the fields Now that we have written the Maxwell equations in covariant form, we know exactly how they transform under Lorentz
Διαβάστε περισσότεραTHE ENERGY-MOMENTUM TENSOR IN CLASSICAL FIELD THEORY
THE ENERGY-MOMENTUM TENSOR IN CLASSICAL FIELD THEORY Walter Wyss Department of Physics University of Colorado Boulder, CO 80309 (Received 14 July 2005) My friend, Asim Barut, was always interested in classical
Διαβάστε περισσότεραExercise 1.1. Verify that if we apply GS to the coordinate basis Gauss form ds 2 = E(u, v)du 2 + 2F (u, v)dudv + G(u, v)dv 2
Math 209 Riemannian Geometry Jeongmin Shon Problem. Let M 2 R 3 be embedded surface. Then the induced metric on M 2 is obtained by taking the standard inner product on R 3 and restricting it to the tangent
Διαβάστε περισσότεραRelativistic particle dynamics and deformed symmetry
Relativistic particle dynamics and deformed Poincare symmetry Department for Theoretical Physics, Ivan Franko Lviv National University XXXIII Max Born Symposium, Wroclaw Outline Lorentz-covariant deformed
Διαβάστε περισσότερα[Note] Geodesic equation for scalar, vector and tensor perturbations
[Note] Geodesic equation for scalar, vector and tensor perturbations Toshiya Namikawa 212 1 Curl mode induced by vector and tensor perturbation 1.1 Metric Perturbation and Affine Connection The line element
Διαβάστε περισσότεραCongruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Διαβάστε περισσότεραOn geodesic mappings of Riemannian spaces with cyclic Ricci tensor
Annales Mathematicae et Informaticae 43 (2014) pp. 13 17 http://ami.ektf.hu On geodesic mappings of Riemannian spaces with cyclic Ricci tensor Sándor Bácsó a, Robert Tornai a, Zoltán Horváth b a University
Διαβάστε περισσότεραLinearized Conformal gravity
Utah State University From the SelectedWors of James Thomas Wheeler Winter January 28, 206 Linearized Conformal gravity James Thomas Wheeler Available at: https://wors.bepress.com/james_wheeler/8/ Linearized
Διαβάστε περισσότεραA Short Introduction to Tensors
May 2, 2007 Tensors: Scalars and Vectors Any physical quantity, e.g. the velocity of a particle, is determined by a set of numerical values - its components - which depend on the coordinate system. Studying
Διαβάστε περισσότεραPaper Reference. Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced. Thursday 11 June 2009 Morning Time: 1 hour 30 minutes
Centre No. Candidate No. Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced Thursday 11 June 2009 Morning Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότεραGeometry of the 2-sphere
Geometry of the 2-sphere October 28, 2 The metric The easiest way to find the metric of the 2-sphere (or the sphere in any dimension is to picture it as embedded in one higher dimension of Euclidean space,
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραENGR 691/692 Section 66 (Fall 06): Machine Learning Assigned: August 30 Homework 1: Bayesian Decision Theory (solutions) Due: September 13
ENGR 69/69 Section 66 (Fall 06): Machine Learning Assigned: August 30 Homework : Bayesian Decision Theory (solutions) Due: Septemer 3 Prolem : ( pts) Let the conditional densities for a two-category one-dimensional
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότεραMA 342N Assignment 1 Due 24 February 2016
M 342N ssignment Due 24 February 206 Id: 342N-s206-.m4,v. 206/02/5 2:25:36 john Exp john. Suppose that q, in addition to satisfying the assumptions from lecture, is an even function. Prove that η(λ = 0,
Διαβάστε περισσότερα