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Multivector Differentiation and Linear Algebra 0.5Cm 17Th Santaló
Multivector differentiation and Linear Algebra 17th Santalo´ Summer School 2016, Santander Joan Lasenby Signal Processing Group, Engineering Department, Cambridge, UK and Trinity College Cambridge [email protected], www-sigproc.eng.cam.ac.uk/ s jl 23 August 2016 1 / 78 Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. 2 / 78 Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. 3 / 78 Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. 4 / 78 Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... 5 / 78 Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary 6 / 78 We now want to generalise this idea to enable us to find the derivative of F(X), in the A ‘direction’ – where X is a general mixed grade multivector (so F(X) is a general multivector valued function of X). Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. Then, provided A has same grades as X, it makes sense to define: F(X + tA) − F(X) A ∗ ¶XF(X) = lim t!0 t The Multivector Derivative Recall our definition of the directional derivative in the a direction F(x + ea) − F(x) a·r F(x) = lim e!0 e 7 / 78 Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. -
1.2 Topological Tensor Calculus
PH211 Physical Mathematics Fall 2019 1.2 Topological tensor calculus 1.2.1 Tensor fields Finite displacements in Euclidean space can be represented by arrows and have a natural vector space structure, but finite displacements in more general curved spaces, such as on the surface of a sphere, do not. However, an infinitesimal neighborhood of a point in a smooth curved space1 looks like an infinitesimal neighborhood of Euclidean space, and infinitesimal displacements dx~ retain the vector space structure of displacements in Euclidean space. An infinitesimal neighborhood of a point can be infinitely rescaled to generate a finite vector space, called the tangent space, at the point. A vector lives in the tangent space of a point. Note that vectors do not stretch from one point to vector tangent space at p p space Figure 1.2.1: A vector in the tangent space of a point. another, and vectors at different points live in different tangent spaces and so cannot be added. For example, rescaling the infinitesimal displacement dx~ by dividing it by the in- finitesimal scalar dt gives the velocity dx~ ~v = (1.2.1) dt which is a vector. Similarly, we can picture the covector rφ as the infinitesimal contours of φ in a neighborhood of a point, infinitely rescaled to generate a finite covector in the point's cotangent space. More generally, infinitely rescaling the neighborhood of a point generates the tensor space and its algebra at the point. The tensor space contains the tangent and cotangent spaces as a vector subspaces. A tensor field is something that takes tensor values at every point in a space. -
Appendix A: Matrices and Tensors
Appendix A: Matrices and Tensors A.1 Introduction and Rationale The purpose of this appendix is to present the notation and most of the mathematical techniques that will be used in the body of the text. The audience is assumed to have been through several years of college level mathematics that included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in undergraduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles and the vector cross and triple scalar products. The solutions to ordinary differential equations are reviewed in the last two sections. The notation, as far as possible, will be a matrix notation that is easily entered into existing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad etc. The desire to represent the components of three-dimensional fourth order tensors that appear in anisotropic elasticity as the components of six-dimensional second order tensors and thus represent these components in matrices of tensor components in six dimensions leads to the nontraditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in Sect. A.11, along with the rationale for this approach. A.2 Definition of Square, Column, and Row Matrices An r by c matrix M is a rectangular array of numbers consisting of r rows and c columns, S.C. -
Matrices and Tensors
APPENDIX MATRICES AND TENSORS A.1. INTRODUCTION AND RATIONALE The purpose of this appendix is to present the notation and most of the mathematical tech- niques that are used in the body of the text. The audience is assumed to have been through sev- eral years of college-level mathematics, which included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly, and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in under- graduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles, and the vector cross and triple scalar prod- ucts. The notation, as far as possible, will be a matrix notation that is easily entered into exist- ing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad. The desire to represent the components of three-dimensional fourth-order tensors that appear in anisotropic elasticity as the components of six-dimensional second-order tensors and thus rep- resent these components in matrices of tensor components in six dimensions leads to the non- traditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in §A.11, along with the rationale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES An r-by-c matrix, M, is a rectangular array of numbers consisting of r rows and c columns: ¯MM.. -
General Relativity Fall 2019 Lecture 13: Geodesic Deviation; Einstein field Equations
General Relativity Fall 2019 Lecture 13: Geodesic deviation; Einstein field equations Yacine Ali-Ha¨ımoud October 11th, 2019 GEODESIC DEVIATION The principle of equivalence states that one cannot distinguish a uniform gravitational field from being in an accelerated frame. However, tidal fields, i.e. gradients of gravitational fields, are indeed measurable. Here we will show that the Riemann tensor encodes tidal fields. Consider a fiducial free-falling observer, thus moving along a geodesic G. We set up Fermi normal coordinates in µ the vicinity of this geodesic, i.e. coordinates in which gµν = ηµν jG and ΓνσjG = 0. Events along the geodesic have coordinates (x0; xi) = (t; 0), where we denote by t the proper time of the fiducial observer. Now consider another free-falling observer, close enough from the fiducial observer that we can describe its position with the Fermi normal coordinates. We denote by τ the proper time of that second observer. In the Fermi normal coordinates, the spatial components of the geodesic equation for the second observer can be written as d2xi d dxi d2xi dxi d2t dxi dxµ dxν = (dt/dτ)−1 (dt/dτ)−1 = (dt/dτ)−2 − (dt/dτ)−3 = − Γi − Γ0 : (1) dt2 dτ dτ dτ 2 dτ dτ 2 µν µν dt dt dt The Christoffel symbols have to be evaluated along the geodesic of the second observer. If the second observer is close µ µ λ λ µ enough to the fiducial geodesic, we may Taylor-expand Γνσ around G, where they vanish: Γνσ(x ) ≈ x @λΓνσjG + 2 µ 0 µ O(x ). -
Tensor Calculus and Differential Geometry
Course Notes Tensor Calculus and Differential Geometry 2WAH0 Luc Florack March 10, 2021 Cover illustration: papyrus fragment from Euclid’s Elements of Geometry, Book II [8]. Contents Preface iii Notation 1 1 Prerequisites from Linear Algebra 3 2 Tensor Calculus 7 2.1 Vector Spaces and Bases . .7 2.2 Dual Vector Spaces and Dual Bases . .8 2.3 The Kronecker Tensor . 10 2.4 Inner Products . 11 2.5 Reciprocal Bases . 14 2.6 Bases, Dual Bases, Reciprocal Bases: Mutual Relations . 16 2.7 Examples of Vectors and Covectors . 17 2.8 Tensors . 18 2.8.1 Tensors in all Generality . 18 2.8.2 Tensors Subject to Symmetries . 22 2.8.3 Symmetry and Antisymmetry Preserving Product Operators . 24 2.8.4 Vector Spaces with an Oriented Volume . 31 2.8.5 Tensors on an Inner Product Space . 34 2.8.6 Tensor Transformations . 36 2.8.6.1 “Absolute Tensors” . 37 CONTENTS i 2.8.6.2 “Relative Tensors” . 38 2.8.6.3 “Pseudo Tensors” . 41 2.8.7 Contractions . 43 2.9 The Hodge Star Operator . 43 3 Differential Geometry 47 3.1 Euclidean Space: Cartesian and Curvilinear Coordinates . 47 3.2 Differentiable Manifolds . 48 3.3 Tangent Vectors . 49 3.4 Tangent and Cotangent Bundle . 50 3.5 Exterior Derivative . 51 3.6 Affine Connection . 52 3.7 Lie Derivative . 55 3.8 Torsion . 55 3.9 Levi-Civita Connection . 56 3.10 Geodesics . 57 3.11 Curvature . 58 3.12 Push-Forward and Pull-Back . 59 3.13 Examples . 60 3.13.1 Polar Coordinates in the Euclidean Plane . -
The Language of Differential Forms
Appendix A The Language of Differential Forms This appendix—with the only exception of Sect.A.4.2—does not contain any new physical notions with respect to the previous chapters, but has the purpose of deriving and rewriting some of the previous results using a different language: the language of the so-called differential (or exterior) forms. Thanks to this language we can rewrite all equations in a more compact form, where all tensor indices referred to the diffeomorphisms of the curved space–time are “hidden” inside the variables, with great formal simplifications and benefits (especially in the context of the variational computations). The matter of this appendix is not intended to provide a complete nor a rigorous introduction to this formalism: it should be regarded only as a first, intuitive and oper- ational approach to the calculus of differential forms (also called exterior calculus, or “Cartan calculus”). The main purpose is to quickly put the reader in the position of understanding, and also independently performing, various computations typical of a geometric model of gravity. The readers interested in a more rigorous discussion of differential forms are referred, for instance, to the book [22] of the bibliography. Let us finally notice that in this appendix we will follow the conventions introduced in Chap. 12, Sect. 12.1: latin letters a, b, c,...will denote Lorentz indices in the flat tangent space, Greek letters μ, ν, α,... tensor indices in the curved manifold. For the matter fields we will always use natural units = c = 1. Also, unless otherwise stated, in the first three Sects. -
Appendix a Multilinear Algebra and Index Notation
Appendix A Multilinear algebra and index notation Contents A.1 Vector spaces . 164 A.2 Bases, indices and the summation convention 166 A.3 Dual spaces . 169 A.4 Inner products . 170 A.5 Direct sums . 174 A.6 Tensors and multilinear maps . 175 A.7 The tensor product . 179 A.8 Symmetric and exterior algebras . 182 A.9 Duality and the Hodge star . 188 A.10 Tensors on manifolds . 190 If linear algebra is the study of vector spaces and linear maps, then multilinear algebra is the study of tensor products and the natural gener- alizations of linear maps that arise from this construction. Such concepts are extremely useful in differential geometry but are essentially algebraic rather than geometric; we shall thus introduce them in this appendix us- ing only algebraic notions. We'll see finally in A.10 how to apply them to tangent spaces on manifolds and thus recoverx the usual formalism of tensor fields and differential forms. Along the way, we will explain the conventions of \upper" and \lower" index notation and the Einstein sum- mation convention, which are standard among physicists but less familiar in general to mathematicians. 163 164 APPENDIX A. MULTILINEAR ALGEBRA A.1 Vector spaces and linear maps We assume the reader is somewhat familiar with linear algebra, so at least most of this section should be review|its main purpose is to establish notation that is used in the rest of the notes, as well as to clarify the relationship between real and complex vector spaces. Throughout this appendix, let F denote either of the fields R or C; we will refer to elements of this field as scalars. -
3. Tensor Fields
3.1 The tangent bundle. So far we have considered vectors and tensors at a point. We now wish to consider fields of vectors and tensors. The union of all tangent spaces is called the tangent bundle and denoted TM: TM = ∪x∈M TxM . (3.1.1) The tangent bundle can be given a natural manifold structure derived from the manifold structure of M. Let π : TM → M be the natural projection that associates a vector v ∈ TxM to the point x that it is attached to. Let (UA, ΦA) be an atlas on M. We construct an atlas on TM as follows. The domain of a −1 chart is π (UA) = ∪x∈UA TxM, i.e. it consists of all vectors attached to points that belong to UA. The local coordinates of a vector v are (x1,...,xn, v1,...,vn) where (x1,...,xn) are the coordinates of x and v1,...,vn are the components of the vector with respect to the coordinate basis (as in (2.2.7)). One can easily check that a smooth coordinate trasformation on M induces a smooth coordinate transformation on TM (the transformation of the vector components is given by (2.3.10), so if M is of class Cr, TM is of class Cr−1). In a similar way one defines the cotangent bundle ∗ ∗ T M = ∪x∈M Tx M , (3.1.2) the tensor bundles s s TMr = ∪x∈M TxMr (3.1.3) and the bundle of p-forms p p Λ M = ∪x∈M Λx . (3.1.4) 3.2 Vector and tensor fields. -
Vectors and Matrices Notes
Vectors and Matrices Notes. Jonathan Coulthard [email protected] 1 Index Notation Index notation may seem quite intimidating at first, but once you get used to it, it will allow us to prove some very tricky vector and matrix identities with very little effort. As with most things, it will only become clearer with practice, and so it is a good idea to work through the examples for yourself, and try out some of the exercises. Example: Scalar Product Let's start off with the simplest possible example: the dot product. For real column vectors a and b, 0 1 b1 Bb2C a · b = aT b = a a a ··· B C = a b + a b + a b + ··· (1) 1 2 3 Bb3C 1 1 2 2 3 3 @ . A . or, written in a more compact notation X a · b = aibi; (2) i where the σ means that we sum over all values of i. Example: Matrix-Column Vector Product Now let's take matrix-column vector multiplication, Ax = b. 2 3 2 3 2 3 A11 A12 A23 ··· x1 b1 6A21 A22 A23 ··· 7 6x27 6b27 6 7 6 7 = 6 7 (3) 6A31 A32 A33 ···7 6x37 6b37 4 . 5 4 . 5 4 . 5 . .. You are probably used to multiplying matrices by visualising multiplying the elements high- lighted in the red boxes. Written out explicitly, this is b2 = A21x1 + A22x2 + A23x3 + ··· (4) If we were to shift the A box and the b box down one place, we would instead get b3 = A31x1 + A32x2 + A33x3 + ··· (5) 1 It should be clear then, that in general, for the ith element of b, we can write bi = Ai1x1 + Ai2x2 + Ai3x3 + ··· (6) Or, in our more compact notation, X bi = Aijxj: (7) j Note that if the matrix A had only one column, then i would take only one value (i = 1). -
Appendix: an Index Notation for Multivariate Statistical Analysis
19 : APPENDIX An Index Notation for Multivariate Analysis 1. Tensor Products and Formal Orderings The algebra of multivariate statistical analysis is, predominantly, the al- gebra of tensor products, of which the Kronecker product of matrices is a particular instance. The Kronecker product of the t × n matrix B =[blj] and the s × m matrix A =[aki] is defined as an ts×nm matrix B ⊗A =[bljA] whose ljth partition is bljA. In many statistical texts, this definition provides the basis for subsequent alge- braic developments. The disadvantage of using the definition without support from the theory of multilinear algebra is that the results which are generated often seem purely technical and lacking in intuitive appeal. Another approach to the algebra of tensor products relies upon the algebra of abstract vector spaces. Thus The tensor product U⊗Vof two finite-dimensional vector spaces U and V may be defined as the dual of the vector space of all bilinear functionals on U and V. This definition facilitates the development of a rigorous abstract theory. In particular, U⊗V, defined in this way, already has all the features of a vector space. However, the definition also leads to acute technical difficulties when we seek to represent the resulting algebra in terms of coordinate vectors and matrices. The approach which we shall adopt here is to define tensor products in terms of formal products. According to this approach, a definition of U⊗V may ⊗ be obtained by considering the set of all objects of the form i j xij(ui vj), where ui ⊗vj, which is described as an elementary or decomposable tensor prod- uct, comprises an ordered pair of elements taken from the two vector spaces. -
Operator-Valued Tensors on Manifolds: a Framework for Field
Operator-Valued Tensors on Manifolds: A Framework for Field Quantization 1 2 H. Feizabadi , N. Boroojerdian ∗ 1,2Department of pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran. Abstract In this paper we try to prepare a framework for field quantization. To this end, we aim to replace the field of scalars R by self-adjoint elements of a commu- tative C⋆-algebra, and reach an appropriate generalization of geometrical concepts on manifolds. First, we put forward the concept of operator-valued tensors and extend semi-Riemannian metrics to operator valued metrics. Then, in this new geometry, some essential concepts of Riemannian geometry such as curvature ten- sor, Levi-Civita connection, Hodge star operator, exterior derivative, divergence,... will be considered. Keywords: Operator-valued tensors, Operator-Valued Semi-Riemannian Met- rics, Levi-Civita Connection, Curvature, Hodge star operator MSC(2010): Primary: 65F05; Secondary: 46L05, 11Y50. 0 Introduction arXiv:1501.05065v2 [math-ph] 27 Jan 2015 The aim of the present paper is to extend the theory of semi-Riemannian metrics to operator-valued semi-Riemannian metrics, which may be used as a framework for field quantization. Spaces of linear operators are directly related to C∗-algebras, so C∗- algebras are considered as a basic concept in this article. This paper has its roots in Hilbert C⋆-modules, which are frequently used in the theory of operator algebras, allowing one to obtain information about C⋆-algebras by studying Hilbert C⋆-modules over them. Hilbert C⋆-modules provide a natural generalization of Hilbert spaces arising when the field of scalars C is replaced by an arbitrary C⋆-algebra.