DOCSLIB.ORG

Beyond Scalar, Vector and Tensor Harmonics in Maximally Symmetric Three-Dimensional Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Beyond Scalar, Vector and Tensor Harmonics in Maximally Symmetric Three-Dimensional Spaces Beyond scalar, vector and tensor harmonics in maximally symmetric three-dimensional spaces Cyril Pitrou1, ∗ and Thiago S. Pereira2, y 1Institut d'Astrophysique de Paris, CNRS UMR 7095, 98 bis Bd Arago, 75014 Paris, France. 2Departamento de F´ısica, Universidade Estadual de Londrina, Rod. Celso Garcia Cid, Km 380, 86057-970, Londrina, Paran´a,Brazil. (Dated: October 1, 2019) We present a comprehensive construction of scalar, vector and tensor harmonics on max- imally symmetric three-dimensional spaces. Our formalism relies on the introduction of spin-weighted spherical harmonics and a generalized helicity basis which, together, are ideal tools to decompose harmonics into their radial and angular dependencies. We provide a thorough and self-contained set of expressions and relations for these harmonics. Being gen- eral, our formalism also allows to build harmonics of higher tensor type by recursion among radial functions, and we collect the complete set of recursive relations which can be used. While the formalism is readily adapted to computation of CMB transfer functions, we also collect explicit forms of the radial harmonics which are needed for other cosmological ob- servables. Finally, we show that in curved spaces, normal modes cannot be factorized into a local angular dependence and a unit norm function encoding the orbital dependence of the harmonics, contrary to previous statements in the literature. 1. INTRODUCTION ables. Indeed, on cosmological scales, where lin- ear perturbation theory successfully accounts for the formation of structures, perturbation modes, Tensor harmonics are ubiquitous tools in that is the components in an expansion on tensor gravitational theories. Their applicability reach harmonics, evolve independently from one an- a wide spectrum of topics including black-hole other. This fact enormously simplifies the con- physics, gravitational waves, quantum-field the- struction of observables and the assessment of ory in curved spacetimes, and cosmology. In the their statistics. In particular, a decomposition particular context of cosmology, one is usually based on tensor harmonics is essential for the interested in the description of tensor harmonics computation of cosmic microwave background over maximally symmetric manifolds, since these (CMB) fluctuations around a maximally sym- are the spaces in better agreement with observa- metric (but possibly curved) space [5]. The nor- tions. In this work we revisit the construction mal modes which have been introduced in [6,7] of scalar, vector and tensor harmonics in sym- correspond to specific components of those of [5], metric three-dimensional spaces with particular and are consequently an equivalent presentation interest in { but not limited to { cosmological arXiv:1909.13687v1 [gr-qc] 30 Sep 2019 of them. An equivalent covariant formulation of applications. these normal modes is also presented in [8,9]. Scalar harmonics in symmetric spaces are In this article, we review the general con- well known among cosmologists [1], and they are struction of harmonics in maximally symmetric defined as a complete set of eigenfunctions of the three-dimensional spaces, along with the associ- Laplace-Beltrami operator. Vector- and tensor- ated normal modes, and show how they can be valued harmonics can be similarly defined, and systematically built by recursions. In doing so, their explicit forms were gathered in [2{4]. These we collect all explicit expressions of the normal objects are found nearly everywhere in cosmo- modes for scalar, vector and tensor harmonics. logical applications, specially in those related Throughout, we choose to use a modern formu- to large-scale structure and its related observ- lation based on spin-weighted spherical harmon- ics from which even and odd parts and also the ∗[email protected] general structure is more tractable. Hence, this [email protected] differs from the formulations given in [10, 11]. 2 Section 2.1 is dedicated to definitions and no- K 6= 0 we can further use units for which `c = 1, tation. In particular, we define the harmonics, that is, all lengths are expressed in units of the the helicity basis, the normal modes and the ra- curvature radius and, in the closed case, this im- dial functions of which many properties are col- plies 0 ≤ χ ≤ π. The general case `c 6= 1 can lected in the appendices. Section3 is dedicated be trivially restored from dimensional analysis if to the general construction of these radial func- needed. To emphasize our special choice of units, tions which characterize fully the harmonics and let us introduce a reduced curvature parameter most relations are collected in AppendixD. The 2 reader interested only in the actual expression K ≡ K`c = K=jKj (2.3) of the harmonics can jump directly to Section4 where the explicit expressions of the radial func- which assumes the value +1 (−1) in the closed tions are collected, or to AppendixF if inter- (open) case. ested in the flat case only. The normalization of The Riemann tensor of maximally symmetric harmonics is discussed in Section5, while plane spaces can be written directly in terms of the waves are built in Section6. The formalism is il- space metric and the constant K: lustrated in Section7 for the standard multipole R = K(g g − g g ) : (2.4) expansion of the CMB radiative transfer func- ijkl ik jl il jk tions. Finally the comparison of our results with This greatly simplifies identities involving com- previous references is detailed in AppendixG. mutators of covariant derivatives. One identity The tables of AppendixH gather the most im- that we shall need is portant ancillary notation used throughout. [∆; rk1 · · · rkn ]Ti1:::ij = 2K [n(n + 1)=2 + nj] 2. DEFINITIONS × rk1 · · · rkn Ti1:::ij (2.5) 2.1. Maximally symmetric spaces where ri is the covariant derivative associated We start be recalling some basic properties of with the metric (2.1) (i.e., rkgij = 0) and ∆ = j maximally symmetric spaces. A nice and physi- r rj is the Laplace-Beltrami operator. In what cist targeted introduction can be found in [12]. follows we shall adopt the unifying notation sin, Maximally symmetric spaces (as opposed to tan and cot for trigonometric functions, defined spacetime) are uniquely fixed by a real parame- as the usual functions when K > 0, and as their ter K, known as constant of curvature. In three hyperbolic counterparts when K < 0. dimensions, and using standard spherical coor- dinates (χ, θ; φ), the metrics of these spaces read 2.2. Helicity basis g dxidxj = dχ2 + r2(χ)[dθ2 + sin2 θdφ2] : (2.1) ij The notion of helicity (or spin) basis is more The radial coordinate χ is implicitly defined by conveniently introduced in terms of an orthonor- the function r(χ), which assumes different values mal triad of basis vectors according to the sign of the parameter K: nχ = @χ ; (2.6a) 8 ` sinh(χ/` ) ; (K < 0) ; n = r−1(χ)@ ; (2.6b) <> c c θ θ −1 r(χ) = `c sin(χ/`c) ; (K > 0) ; (2.2) nφ = r (χ) csc(θ)@φ ; (2.6c) :>χ ; (K = 0) : together with its dual basis Here, ` ≡ 1=pjKj is the curvature radius, c nχ = dχ ; (2.7a) which is related to the Ricci scalar by R = 6K. θ Clearly, K distinguishes between open (K < 0), n = r(χ)dθ ; (2.7b) closed (K > 0) and flat (K = 0) spaces. When nφ = r(χ) sin(θ)dφ : (2.7c) 3 From this we can form the standard helicity vec- with s = 0. Up to normalisation differences, it tor (spin 1) basis as corresponds to the Legendre tensors introduced in [14] for the cases s = 0; 2. The explicit ex- 1 n± ≡ p (nθ ∓ inφ) ; pressions for j ≤ 3, and a collection of properties 2 (2.8) (which extends those already found in Appendix 1 n± ≡ p nθ ∓ inφ : A of Ref. [15] for the case s = 0), are given in 2 AppendixC. The set of ^ns with jsj ≤ j form a Ij basis for STF tensors with j free indices at each Given a unit vector n at the origin (χ = 0), the point. Their normalization, used for extraction pair (χ, n) denotes a point reached following a of components along that basis, is given by geodesic of length χ whose tangential direction at the origin is n. It is also obvious from the ±s Ij n^ n^ 0 = δss0 djs (2.12) spherical symmetry that the tangential vector of Ij ∓s the geodesic at that point is nχ. Hence it is cus- where tomary to use the symbol n for both nχ and its χ j! 1 dual n . The helicity basis vectors n± also de- djs ≡ 2 ; (2j − 1)!! (bjs) pend on the point (χ, n) considered, but they are s (2.13) parallel transported along a radial curve, that is p jsj (j!)2 b ≡ 2 : js (j + s)!(j − s)! k ± n rkni = 0 : (2.9) Thus, since they depend essentially only on the 2.3. Decomposition of tensor fields direction n, it is customary not to write this dependence explicitly. Any STF tensor field in a maximally sym- We now use the vector basis (2.8) to build metric (three-dimensional) space can be decom- a suitable tensor basis (spin s) for symmetric posed onto the generalized helicity basis using trace-free (STF) tensors. For 0 ≤ jsj ≤ j, we spin-weighted spherical harmonics. This decom- define position can be understood in two steps. At each ±s ± ± point the helicity basis is a basis for STF tensors n^ ≡ n : : : n ni : : : ni i ; (2.10) i1:::ij hi1 is s+1 j at that point, hence we can decompose the STF with a similar definition when free indices are tensor field as up. The angle brackets mean that we must form j X s the symmetric trace-free part on the enclosed TIj (χ, n) = sT (χ, n)^nIj : (2.14) indices, and this is performed in practice with s=−j (C.1).
Load more

Recommended publications
  • Introduction to Vector and Tensor Analysis
    Introduction to vector and tensor analysis Jesper Ferkinghoff-Borg September 6, 2007 Contents 1 Physical space 3 1.1 Coordinate systems . 3 1.2 Distances . 3 1.3 Symmetries . 4 2 Scalars and vectors 5 2.1 Definitions . 5 2.2 Basic vector algebra . 5 2.2.1 Scalar product . 6 2.2.2 Cross product . 7 2.3 Coordinate systems and bases . 7 2.3.1 Components and notation . 9 2.3.2 Triplet algebra . 10 2.4 Orthonormal bases . 11 2.4.1 Scalar product . 11 2.4.2 Cross product . 12 2.5 Ordinary derivatives and integrals of vectors . 13 2.5.1 Derivatives . 14 2.5.2 Integrals . 14 2.6 Fields . 15 2.6.1 Definition . 15 2.6.2 Partial derivatives . 15 2.6.3 Differentials . 16 2.6.4 Gradient, divergence and curl . 17 2.6.5 Line, surface and volume integrals . 20 2.6.6 Integral theorems . 24 2.7 Curvilinear coordinates . 26 2.7.1 Cylindrical coordinates . 27 2.7.2 Spherical coordinates . 28 3 Tensors 30 3.1 Definition . 30 3.2 Outer product . 31 3.3 Basic tensor algebra . 31 1 3.3.1 Transposed tensors . 32 3.3.2 Contraction . 33 3.3.3 Special tensors . 33 3.4 Tensor components in orthonormal bases . 34 3.4.1 Matrix algebra . 35 3.4.2 Two-point components . 38 3.5 Tensor fields . 38 3.5.1 Gradient, divergence and curl . 39 3.5.2 Integral theorems . 40 4 Tensor calculus 42 4.1 Tensor notation and Einsteins summation rule .
    [Show full text]
  • The Grassmannian Sigma Model in SU(2) Yang-Mills Theory
    The Grassmannian σ-model in SU(2) Yang-Mills Theory David Marsh Undergraduate Thesis Supervisor: Antti Niemi Department of Theoretical Physics Uppsala University Contents 1 Introduction 4 1.1 Introduction ............................ 4 2 Some Quantum Field Theory 7 2.1 Gauge Theories .......................... 7 2.1.1 Basic Yang-Mills Theory ................. 8 2.1.2 Faddeev, Popov and Ghosts ............... 10 2.1.3 Geometric Digression ................... 14 2.2 The nonlinear sigma models .................. 15 2.2.1 The Linear Sigma Model ................. 15 2.2.2 The Historical Example ................. 16 2.2.3 Geometric Treatment ................... 17 3 Basic properties of the Grassmannian 19 3.1 Standard Local Coordinates ................... 19 3.2 Plücker coordinates ........................ 21 3.3 The Grassmannian as a homogeneous space .......... 23 3.4 The complexied four-vector ................... 25 4 The Grassmannian σ Model in SU(2) Yang-Mills Theory 30 4.1 The SU(2) Yang-Mills Theory in spin-charge separated variables 30 4.1.1 Variables .......................... 30 4.1.2 The Physics of Spin-Charge Separation ......... 34 4.2 Appearance of the ostensible model ............... 36 4.3 The Grassmannian Sigma Model ................ 38 4.3.1 Gauge Formalism ..................... 38 4.3.2 Projector Formalism ................... 39 4.3.3 Induced Metric Formalism ................ 40 4.4 Embedding Equations ...................... 43 4.5 Hodge Star Decomposition .................... 48 2 4.6 Some Complex Dierential Geometry .............. 50 4.7 Quaternionic Decomposition ................... 54 4.7.1 Verication ........................ 56 5 The Interaction 58 5.1 The Interaction Terms ...................... 58 5.1.1 Limits ........................... 59 5.2 Holomorphic forms and Dolbeault Cohomology ........ 60 5.3 The Form of the interaction ..................
    [Show full text]
  • Mach's Principle: Exact Frame-Dragging Via
    Mach’s principle: Exact frame-dragging via gravitomagnetism in perturbed Friedmann-Robertson-Walker universes with K =( 1, 0) ± Christoph Schmid∗ ETH Zurich, Institute for Theoretical Physics, 8093 Zurich, Switzerland (Dated: October 30, 2018) We show that there is exact dragging of the axis directions of local inertial frames by a weighted average of the cosmological energy currents via gravitomagnetism for all linear perturbations of all Friedmann-Robertson-Walker (FRW) universes and of Einstein’s static closed universe, and for all energy-momentum-stress tensors and in the presence of a cosmolgical constant. This includes FRW universes arbitrarily close to the Milne Universe and the de Sitter universe. Hence the postulate formulated by Ernst Mach about the physical cause for the time-evolution of inertial axes is shown to hold in General Relativity for linear perturbations of FRW universes. — The time-evolution of local inertial axes (relative to given local fiducial axes) is given experimentally by the precession angular velocity Ω~ gyro of local gyroscopes, which in turn gives the operational definition of the gravitomagnetic field: B~ g ≡−2 Ω~ gyro. The gravitomagnetic field is caused by energy currents J~ε via 0ˆ − 2 ~ − ~ ~ ~ the momentum constraint, Einstein’s G ˆi equation, ( ∆+ µ )Ag = 16πGNJε with Bg = curl Ag. This equation is analogous to Amp`ere’s law, but it holds for all time-dependent situations. ∆ is the de Rham-Hodge Laplacian, and ∆ = −curl curl for the vorticity sector in Riemannian 3- space. — In the solution for an open universe the 1/r2-force of Amp`ere is replaced by a Yukawa force Yµ(r)=(−d/dr)[(1/R) exp(−µr)], form-identical for FRW backgrounds with K = (−1, 0).
    [Show full text]
  • Chapter 9 Symmetries and Transformation Groups
    Chapter 9 Symmetries and transformation groups When contemporaries of Galilei argued against the heliocentric world view by pointing out that we do not feel like rotating with high velocity around the sun he argued that a uniform motion cannot be recognized because the laws of nature that govern our environment are invariant under what we now call a Galilei transformation between inertial systems. Invariance arguments have since played an increasing role in physics both for conceptual and practical reasons. In the early 20th century the mathematician Emmy Noether discovered that energy con- servation, which played a central role in 19th century physics, is just a special case of a more general relation between symmetries and conservation laws. In particular, energy and momen- tum conservation are equivalent to invariance under translations of time and space, respectively. At about the same time Einstein discovered that gravity curves space-time so that space-time is in general not translation invariant. As a consequence, energy is not conserved in cosmology. In the present chapter we discuss the symmetries of non-relativistic and relativistic kine- matics, which derive from the geometrical symmetries of Euclidean space and Minkowski space, respectively. We decompose transformation groups into discrete and continuous parts and study the infinitesimal form of the latter. We then discuss the transition from classical to quantum mechanics and use rotations to prove the Wigner-Eckhart theorem for matrix elements of tensor operators. After discussing the discrete symmetries parity, time reversal and charge conjugation we conclude with the implications of gauge invariance in the Aharonov–Bohm effect.
    [Show full text]
  • Spatiospectral Concentration of Vector Fields on a Sphere
    Spatiospectral concentration of vector fields on a sphere Alain Plattner a;∗ and Frederik J. Simons a aDepartment of Geosciences, Princeton University, Princeton NJ 08544, U.S.A. Received 1 November 2018; revised 1 November 2018; accepted 1 November 2018 Abstract We construct spherical vector bases that are bandlimited and spatially concentrated, or alternatively, spacelimited and spectrally concentrated, suitable for the analysis and representation of real-valued vector fields on the surface of the unit sphere, as arises in the natural and biomedical sciences, and engineering. Building on the original approach of Slepian, Landau, and Pollak we concentrate the energy of our function basis into arbitrarily shaped regions of interest on the sphere and within a certain bandlimit in the vector spherical-harmonic domain. As with the concentration prob- lem for scalar functions on the sphere, which has been treated in detail elsewhere, the vector basis can be constructed by solving a finite-dimensional algebraic eigenvalue problem. The eigenvalue problem decouples into separate prob- lems for the radial, and tangential components. For regions with advanced symmetry such as latitudinal polar caps, the spectral concentration kernel matrix is very easily calculated and block-diagonal, which lends itself to efficient diago- nalization. The number of spatiospectrally well-concentrated vector fields is well estimated by a Shannon number that only depends on the area of the target region and the maximal spherical harmonic degree or bandwidth. The Slepian spherical vector basis is doubly orthogonal, both over the entire sphere and over the geographic target regions. Like its scalar counterparts it should be a powerful tool in the inversion, approximation and extension of bandlimited fields on the sphere: vector fields such as gravity and magnetism in the earth and planetary sciences, or electromagnetic fields in optics, antenna theory and medical imaging.
    [Show full text]
  • Pseudodifferential Equations on Spheres with Spherical
    SCIENTIA MANU E T MENTE Pseudodifferential equations on spheres with spherical radial basis functions and spherical splines 31st March, 2011 A thesis presented to The School of Mathematics and Statistics The University of New South Wales in fulfilment of the thesis requirement for the degree of Doctor of Philosophy by Duong Thanh Pham 2 THE UNIVERSITY OF NEW SOUTH WALES Thesis/Dissertation Sheet Surname or Family name: PHAM Firstname: Duong Thanh Other names: Abbreviation for degree as given in the University calendar: PhD School: Mathematics and Statistics Faculty: Science Title: Pseudodifferential equations on spheres with spherical radial basis functions and spherical splines Abstract 350 words maximum n Pseudodifferential equations on the unit sphere in R , n ≥ 3, are considered. The class of pseudodiffrential operators have long been used as a modern and powerful tool to tackle linear boundary-value problems. These equations arise in geophysics, where the sphere of interest is the earth. Efficient solutions to these equations on the sphere become more demanding when given data are collected by satellites. In this dissertation, firstly we solve these equations by using spherical radial basis functions. The use of these functions results in meshless methods, which have recently become more and more popular. In this dissertation, the collocation and Galerkin methods are used to solve pseudodifferential equations. From the point of view of application, the collocation method is easier to implement, in particular when the given data are scattered. However, it is well-known that the collocation methods in general elicit a complicated error analysis. A salient feature of our work is that error estimates for collocation methods are obtained as a by-product of the analysis for the Galerkin method.
    [Show full text]
  • Tensors 7.1 Introduction in Two Dimensions
    Physics 411 Lecture 7 Tensors Lecture 7 Physics 411 Classical Mechanics II September 12th 2007 In Electrodynamics, the implicit law governing the motion of particles is F α = m x¨α. This is also true, of course, for most of classical physics and the details of the physical principle one is discussing are hidden in F α, and potentially, its potential. That is what defines the interaction. In general relativity, the motion of particles will be described by µ µ α β x¨ + Γ αβ x_ x_ = 0 (7.1) and this will occur in a four-dimensional space-time { but that doesn't con- cern us for now. The point of the above is that it lacks a potential, and can be connected in a natural way to the metric. 7.1 Introduction in Two Dimensions We will start with some basic examples in two-dimensions for concreteness. Here we will always begin in a Cartesian-parametrized plane, with basis vectors x^ and y^. 7.1.1 Rotation To begin, consider a simple rotation of the usual coordinate axes through an angle θ (counterclockwise) as shown in Figure 7.1. From the figure, we can easily relate the coordinates w.r.t. the new axes (¯x andy ¯) to coordinates w.r.t. the \usual" axes (x and y) { define ` = 1 of 12 7.1. INTRODUCTION IN TWO DIMENSIONS Lecture 7 yˆ y¯ y x¯ ! x¯ y¯ ψ θ xˆ x Figure 7.1: Two sets of axes, rotated through an angle θ with respect to each other. px2 + y2, then the invariance of length allows us to write x¯ = ` cos( − θ) = ` cos cos θ + ` sin sin θ = x cos θ + y sin θ (7.2) y¯ = ` sin( − θ) = ` sin cos θ − ` cos sin θ = y cos θ − x sin θ; which can be written in matrix form as: x¯ cos θ sin θ x = : (7.3) y¯ − sin θ cos θ y | {z α } ≡R _=R β If we think of infinitesimal displacements centered at the origin, then we α α β would write dx¯ = R β dx .
    [Show full text]
  • On Wigner's D-Matrix and Angular Momentum
    On Wigner's D-matrix and Angular Momentum Mehdi Hage-Hassan Lebanese University, Faculty of Sciences - Section 1 Hadath - Beirut Abstract Using the Analytic Oscillator space (Fock Bargmann space) and the gener- ating function of Wigner's D-matrix we will determine the generating functions of spherical harmonics and the 3-j symbols of SU(2). We will find also a relation between the Gegenbauer Polynomials and Wigner's d-matrix. We drive the 3-j symbols and we find simply the expression of 3-j symbols with magnetic momen- tum nulls. We will find the generating function the 3 − lm symbols of SO(3) and a new relation between the 3-j symbols. We will derive also the generating function of 6-j symbols. We will apply our method for the determination of the generating function and the spherical basis of four dimensional R4 Harmonic oscillators. 1 Introduction The Harmonic oscillator and the Angular Momentum are parts of the course of quan- tum mechanics and has been studied since time ago by many scientists using different methods [1,7]. Racah has made an important contribution to the study of coefficients of 3-j and 6-j symbols. Furthermore Wigner fined the representation matrix of rota- tion. Then Schwinger studied the rotation groups starting from oscillator basis and found the generating functions of 3-j and 6-j. By analogy with the Schwinger work, Bargmann used the Fock space to study the angular momentum. We observe that many spaces of the oscillators that have wave functions are function of complex numbers and it is conjugated.
    [Show full text]
  • Approximation of Parabolic Pdes on Spheres Using Spherical Basis Functions ∗
    Advances in Computational Mathematics (2005) 22: 377–397 Springer 2005 Approximation of parabolic PDEs on spheres using spherical basis functions ∗ Q.T. Le Gia Department of Mathematics, Texas A&M University, TX 77843-3368, USA E-mail: [email protected] Received 17 June 2003; accepted 29 September 2003 Communicated by C.A. Micchelli In this paper we investigate the approximation of a class of parabolic partial differential + equations on the unit spheres Sn ⊂ Rn 1 using spherical basis functions. Error estimates in the Sobolev norm are derived. Keywords: heat equation, radial basis functions, collocation method, spheres AMS subject classification: 35K05, 65M70, 46E22 1. Introduction Approximation of partial differential equations on spheres has many applications in physical geodesy, potential theory, oceanography, and meteorology [2,17,18]. Evolu- tion equations on spherical geometry such as shallow water equations have been studied in weather forecasting services [3,23]. The geometry of the sphere is a major obsta- cle in constructing the approximation space for the solution of the PDEs. One way to overcome the obstacle is to construct basis functions which depend only on the geodesic distance between two points on the sphere, which are called spherical basis functions in literature [2,5,13]. Error estimates of pseudo-differential operator (which are time- independent) were studied in [2,9] but error estimates for the evolution equations remain unexplored. In this paper we consider the following parabolic partial differential equation de- fined on the unit sphere Sn ⊂ Rn+1: ∂ u(x, t) − u(x, t) = F(x,t) ∂t (1) u(x, 0) = f(x), f ∈ H 2σ Sn , ∗ The results presented in this paper are taken from the author’s Ph.D.
    [Show full text]
  • Math 321 Vector and Complex Calculus for the Physical Sciences
    Math 321 Vector and Complex Calculus for the Physical Sciences c Fabian Waleffe Department of Mathematics & Department of Engineering Physics University of Wisconsin, Madison 2016/1/18 ii \Any fool can know. The point is to understand." | Albert Einstein Contents 1 Vector Geometry and Algebra 1 1 Get your bearings . .1 1.1 Magnitude and direction . .1 1.2 Representations of 2D vectors . .2 1.3 Representations of 3D vectors . .3 2 Addition and scaling of vectors . .6 3 General Vector Spaces . .7 3.1 Axiomatic definition of a vector space . .7 3.2 The vector space Rn .....................7 4 Bases and Components . .8 5 Points and Coordinates . 11 5.1 Position vector . 11 5.2 Lines and Planes . 13 6 Dot Product . 14 7 Orthonormal bases . 18 8 Dot product and norm in Rn (Optional) . 20 9 Cross Product . 22 10 Index notation . 29 10.1 Levi-Civita symbol . 29 10.2 Einstein summation convention . 30 11 Mixed product and Determinant . 34 12 Points, Lines, Planes, etc. 39 13 Orthogonal Transformations and Matrices . 42 13.1 Change of cartesian basis . 42 13.2 Matrices . 46 13.3 Spherical coordinates with a twist: Euler Angles . 48 13.4 Determinant of a matrix (Optional) . 52 13.5 Three views of Ax = b (Optional) . 53 13.6 Eigenvalues and Eigenvectors (Math 320 not 321) . 55 2 Vector Calculus 57 1 Vector function of a scalar variable . 57 2 Motion of a particle . 59 3 Motion of a system of particles (optional) . 62 iii iv 4 Motion of a rigid body (optional) . 63 5 Curves .
    [Show full text]
  • Deformation and Buckling of Isolated and Interacting Thin Shells in an Elastic Medium
    DEFORMATION AND BUCKLING OF ISOLATED AND INTERACTING THIN SHELLS IN AN ELASTIC MEDIUM A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2016 By Maria Thorpe School of Mathematics Contents Abstract 17 Declaration 18 Copyright 19 Acknowledgements 20 1 Introduction 21 1.1 Introduction.................................. 21 2 Background 26 2.1 Tensor notation and coordinate systems . 26 2.1.1 Curvilinear coordinates: tensors and vectors . ...... 27 2.1.2 Vectors ................................ 28 2.1.3 Tensors ................................ 28 2.1.4 Cartesiancoordinates . 29 2.1.5 Spherical polar coordinates . 30 2.2 ElasticityTheory............................... 31 2.2.1 LinearElasticity ........................... 32 2.2.2 Equations of linear elasticity in spherical coordinates . 34 2.2.3 Constitutive equations in spherical coordinates . ....... 35 2.3 Nonlinear elasticity for shells . 36 2.3.1 The Boussinesq-Papkovich stress functions . ..... 43 2.3.2 Multipolesolutions. 45 2.4 Singlecavityproblems . 48 2.4.1 Steady state heat conduction in an infinite medium containing a sphericalcavity............................ 49 2.4.2 Spherical cavity in an elastic medium: A spherically symmetric problem................................ 51 2.4.3 Spherical cavity in an elastic medium: An axisymmetric problem 54 2 2.5 Singleshellproblems. .. .. .. .. .. .. .. .. 61 2.5.1 Thick spherical shell embedded in an elastic medium: Spherically symmetricproblem.......................... 61 2.5.2 Thin spherical shell embedded in an elastic medium: Spherically symmetricproblem-asecondapproach . 71 2.5.3 Spherical shell embedded in an elastic medium: Axisymmetric problem................................ 77 2.5.4 Shellunderuniaxialpressure . 88 3 Single shell buckling problems 100 3.1 Bucklingtheory................................ 100 3.1.1 Previouswork............................
    [Show full text]
  • EN221 Summary
    EN221 Summary 1 Tensor Stuff • Divergence : T ∇ · u = ∂iui ∇ · u = u nda, ZR Z∂R T ∇ · T = ∂iTije j ∇ · T = T nda. ZR Z∂R ∇ ⊗ u = Jacobian ∇ ⊗ u = u ⊗ nda, ZR Z∂R (matrix divergence: columns stay separate) • Box product : [ a , b, c] = a · ( b ∧ c) • Levi-Civita tensor: δi, 1 δj, 1 δk, 1 1 ( ijk) an even permut. of ( 123) , εijk = det δi, 2 δj, 2 δk, 2 = [ ei , e j , ek ] = − 1 ( ijk) an odd permut. of ( 123) , δ δ δ 0 if not. i, 3 j, 3 k, 3 e j ∧ ek = εijkei. det( abc) = εijkaibjck εijkεilm = δjl δkm − δjmδkl , εijkεijl = 2δkl , εijkεijk = 6 • Principal Invariants: IA = λ 1 + λ2 + λ 3 = ([ Aa, b, c] + [ a , Ab, c] + [ a , b, Ac])/[ a , b, c] = tr A, 1 II = λ λ + λ λ + λ λ = ([ Aa , Ab, c] + [ Aa , b, Ac] + [ a , Ab, Ac])/[ a , b, c] = [ tr2 A − tr A2 ] , A 1 2 2 3 1 3 2 IIIA = λ 1 λ 2 λ 3 = [ Aa , Ab, Ac]/[ a , b, c] = det A. T • Adjugate/Cofactor of a Tensor: A∗ ( a ∧ b) = ( Aa) ∧ ( Ab) ⇒ A∗ = det A( A− ) . 1 ∂t det A( t) = det A tr(( ∂tA) A− ) • Tensor Product : TO ⊗ FROM T ei ⊗ e j = eie j ( u ⊗ v) a = u( v · a) ( u ⊗ v)( w ⊗ x) = v · w( u ⊗ x) ( u ⊗ v) A = u ⊗ ( ATv) • Skewsymmetric matrices: Rotation around axis Ω given by orthogonal matrix Q( t) . T x˙ = Q˙ x ⇒ ∂t( Q Q) = 0, W = Q˙ Q T, W = − WT. Wx = Ω ∧ x.
    [Show full text]