MATHEMATICAL PHYSICS UNIT – 7 Tensor Algebra
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Tensors and Differential Forms on Vector Spaces
APPENDIX A TENSORS AND DIFFERENTIAL FORMS ON VECTOR SPACES Since only so much of the vast and growing field of differential forms and differentiable manifolds will be actually used in this survey, we shall attempt to briefly review how the calculus of exterior differential forms on vector spaces can serve as a replacement for the more conventional vector calculus and then introduce only the most elementary notions regarding more topologically general differentiable manifolds, which will mostly be used as the basis for the discussion of Lie groups, in the following appendix. Since exterior differential forms are special kinds of tensor fields – namely, completely-antisymmetric covariant ones – and tensors are important to physics, in their own right, we shall first review the basic notions concerning tensors and multilinear algebra. Presumably, the reader is familiar with linear algebra as it is usually taught to physicists, but for the “basis-free” approach to linear and multilinear algebra (which we shall not always adhere to fanatically), it would also help to have some familiarity with the more “abstract-algebraic” approach to linear algebra, such as one might learn from Hoffman and Kunze [ 1], for instance. 1. Tensor algebra. – A tensor algebra is a type of algebra in which multiplication takes the form of the tensor product. a. Tensor product. – Although the tensor product of vector spaces can be given a rigorous definition in a more abstract-algebraic context (See Greub [ 2], for instance), for the purposes of actual calculations with tensors and tensor fields, it is usually sufficient to say that if V and W are vector spaces of dimensions n and m, respectively, then the tensor product V ⊗ W will be a vector space of dimension nm whose elements are finite linear combinations of elements of the form v ⊗ w, where v is a vector in V and w is a vector in W. -
Tensors Notation • Contravariant Denoted by Superscript Ai Took a Vector and Gave for Vector Calculus Us a Vector
Tensors Notation • contravariant denoted by superscript Ai took a vector and gave For vector calculus us a vector • covariant denoted by subscript Ai took a scaler and gave us a vector Review To avoid confusion in cartesian coordinates both types are the same so • Vectors we just opt for the subscript. Thus a vector x would be x1,x2,x3 R3 • Summation representation of an n by n array As it turns out in an cartesian space and other rectilinear coordi- nate systems there is no difference between contravariant and covariant • Gradient, Divergence and Curl vectors. This will not be the case for other coordinate systems such a • Spherical Harmonics (maybe) curvilinear coordinate systems or in 4 dimensions. These definitions are closely related to the Jacobian. Motivation If you tape a book shut and try to spin it in the air on each indepen- Definitions for Tensors of Rank 2 dent axis you will notice that it spins fine on two axes but not on the Rank 2 tensors can be written as a square array. They have con- third. That’s the inertia tensor in your hands. Similar are the polar- travariant, mixed, and covariant forms. As we might expect in cartesian izations tensor, index of refraction tensor and stress tensor. But tensors coordinates these are the same. also show up in all sorts of places that don’t connect to an anisotropic material property, in fact even spherical harmonics are tensors. What are the similarities and differences between such a plethora of tensors? Vector Calculus and Identifers The mathematics of tensors is particularly useful for de- Tensor analysis extends deep into coordinate transformations of all scribing properties of substances which vary in direction– kinds of spaces and coordinate systems. -
A Some Basic Rules of Tensor Calculus
A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a b + ... +c d. The scalars, vectors and tensors are handled as invariant (independent⊗ from the choice⊗ of the coordinate system) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write a = aig , A = aibj + ... + cidj g g i i ⊗ j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k k ik ik a gk ∑ a gk, A bk ∑ A bk ≡ k=1 ≡ k=1 In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e. -
Multilinear Algebra
Appendix A Multilinear Algebra This chapter presents concepts from multilinear algebra based on the basic properties of finite dimensional vector spaces and linear maps. The primary aim of the chapter is to give a concise introduction to alternating tensors which are necessary to define differential forms on manifolds. Many of the stated definitions and propositions can be found in Lee [1], Chaps. 11, 12 and 14. Some definitions and propositions are complemented by short and simple examples. First, in Sect. A.1 dual and bidual vector spaces are discussed. Subsequently, in Sects. A.2–A.4, tensors and alternating tensors together with operations such as the tensor and wedge product are introduced. Lastly, in Sect. A.5, the concepts which are necessary to introduce the wedge product are summarized in eight steps. A.1 The Dual Space Let V be a real vector space of finite dimension dim V = n.Let(e1,...,en) be a basis of V . Then every v ∈ V can be uniquely represented as a linear combination i v = v ei , (A.1) where summation convention over repeated indices is applied. The coefficients vi ∈ R arereferredtoascomponents of the vector v. Throughout the whole chapter, only finite dimensional real vector spaces, typically denoted by V , are treated. When not stated differently, summation convention is applied. Definition A.1 (Dual Space)Thedual space of V is the set of real-valued linear functionals ∗ V := {ω : V → R : ω linear} . (A.2) The elements of the dual space V ∗ are called linear forms on V . © Springer International Publishing Switzerland 2015 123 S.R. -
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 . -
Tensor Algebra
TENSOR ALGEBRA Continuum Mechanics Course (MMC) - ETSECCPB - UPC Introduction to Tensors Tensor Algebra 2 Introduction SCALAR , , ... v VECTOR vf, , ... MATRIX σε,,... ? C,... 3 Concept of Tensor A TENSOR is an algebraic entity with various components which generalizes the concepts of scalar, vector and matrix. Many physical quantities are mathematically represented as tensors. Tensors are independent of any reference system but, by need, are commonly represented in one by means of their “component matrices”. The components of a tensor will depend on the reference system chosen and will vary with it. 4 Order of a Tensor The order of a tensor is given by the number of indexes needed to specify without ambiguity a component of a tensor. a Scalar: zero dimension 3.14 1.2 v 0.3 a , a Vector: 1 dimension i 0.8 0.1 0 1.3 2nd order: 2 dimensions A, A E 02.40.5 ij rd A , A 3 order: 3 dimensions 1.3 0.5 5.8 A , A 4th order … 5 Cartesian Coordinate System Given an orthonormal basis formed by three mutually perpendicular unit vectors: eeˆˆ12,, ee ˆˆ 23 ee ˆˆ 31 Where: eeeˆˆˆ1231, 1, 1 Note that 1 if ij eeˆˆi j ij 0 if ij 6 Cylindrical Coordinate System x3 xr1 cos x(,rz , ) xr2 sin xz3 eeeˆˆˆr cosθθ 12 sin eeeˆˆˆsinθθ cos x2 12 eeˆˆz 3 x1 7 Spherical Coordinate System x3 xr1 sin cos xrxr, , 2 sin sin xr3 cos ˆˆˆˆ x2 eeeer sinθφ sin 123sin θ cos φ cos θ eeeˆˆˆ cosφφ 12sin x1 eeeeˆˆˆˆφ cosθφ sin 123cos θ cos φ sin θ 8 Indicial or (Index) Notation Tensor Algebra 9 Tensor Bases – VECTOR A vector v can be written as a unique linear combination of the three vector basis eˆ for i 1, 2, 3 . -
Geometric Algebra Techniques for General Relativity
Geometric Algebra Techniques for General Relativity Matthew R. Francis∗ and Arthur Kosowsky† Dept. of Physics and Astronomy, Rutgers University 136 Frelinghuysen Road, Piscataway, NJ 08854 (Dated: February 4, 2008) Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the straightforwardness of coordinate methods. We focus our attention on orthonormal frames and the associated connection bivector, using them to find the Schwarzschild and Kerr solutions, along with a detailed exposition of the Petrov types for the Weyl tensor. PACS numbers: 02.40.-k; 04.20.Cv Keywords: General relativity; Clifford algebras; solution techniques I. INTRODUCTION Geometric (or Clifford) algebra provides a simple and natural language for describing geometric concepts, a point which has been argued persuasively by Hestenes [1] and Lounesto [2] among many others. Geometric algebra (GA) unifies many other mathematical formalisms describing specific aspects of geometry, including complex variables, matrix algebra, projective geometry, and differential geometry. Gravitation, which is usually viewed as a geometric theory, is a natural candidate for translation into the language of geometric algebra. This has been done for some aspects of gravitational theory; notably, Hestenes and Sobczyk have shown how geometric algebra greatly simplifies certain calculations involving the curvature tensor and provides techniques for classifying the Weyl tensor [3, 4]. Lasenby, Doran, and Gull [5] have also discussed gravitation using geometric algebra via a reformulation in terms of a gauge principle. In this paper, we formulate standard general relativity in terms of geometric algebra. A comprehensive overview like the one presented here has not previously appeared in the literature, although unpublished works of Hestenes and of Doran take significant steps in this direction. -
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362 INDEX A Cauchy stress law 216 Absolute differentiation 120 Cauchy-Riemann equations 293,321 Absolute scalar field 43 Charge density 323 Absolute tensor 45,46,47,48 Christoffel symbols 108,110,111 Acceleration 121, 190, 192 Circulation 293 Action integral 198 Codazzi equations 139 Addition of systems 6, 51 Coefficient of viscosity 285 Addition of tensors 6, 51 Cofactors 25, 26, 32 Adherence boundary condition 294 Compatibility equations 259, 260, 262 Aelotropic material 245 Completely skew symmetric system 31 Affine transformation 86, 107 Compound pendulum 195,209 Airy stress function 264 Compressible material 231 Almansi strain tensor 229 Conic sections 151 Alternating tensor 6,7 Conical coordinates 74 Ampere’s law 176,301,337,341 Conjugate dyad 49 Angle between vectors 80, 82 Conjugate metric tensor 36, 77 Angular momentum 218, 287 Conservation of angular momentum 218, 295 Angular velocity 86,87,201,203 Conservation of energy 295 Arc length 60, 67, 133 Conservation of linear momentum 217, 295 Associated tensors 79 Conservation of mass 233, 295 Auxiliary Magnetic field 338 Conservative system 191, 298 Axis of symmetry 247 Conservative electric field 323 B Constitutive equations 242, 251,281, 287 Continuity equation 106,234, 287, 335 Basic equations elasticity 236, 253, 270 Contraction 6, 52 Basic equations for a continuum 236 Contravariant components 36, 44 Basic equations of fluids 281, 287 Contravariant tensor 45 Basis vectors 1,2,37,48 Coordinate curves 37, 67 Beltrami 262 Coordinate surfaces 37, 67 Bernoulli’s Theorem 292 Coordinate transformations -
A Mathematica Package for Doing Tensor Calculations in Differential Geometry User's Manual
Ricci A Mathematica package for doing tensor calculations in differential geometry User’s Manual Version 1.32 By John M. Lee assisted by Dale Lear, John Roth, Jay Coskey, and Lee Nave 2 Ricci A Mathematica package for doing tensor calculations in differential geometry User’s Manual Version 1.32 By John M. Lee assisted by Dale Lear, John Roth, Jay Coskey, and Lee Nave Copyright c 1992–1998 John M. Lee All rights reserved Development of this software was supported in part by NSF grants DMS-9101832, DMS-9404107 Mathematica is a registered trademark of Wolfram Research, Inc. This software package and its accompanying documentation are provided as is, without guarantee of support or maintenance. The copyright holder makes no express or implied warranty of any kind with respect to this software, including implied warranties of merchantability or fitness for a particular purpose, and is not liable for any damages resulting in any way from its use. Everyone is granted permission to copy, modify and redistribute this software package and its accompanying documentation, provided that: 1. All copies contain this notice in the main program file and in the supporting documentation. 2. All modified copies carry a prominent notice stating who made the last modifi- cation and the date of such modification. 3. No charge is made for this software or works derived from it, with the exception of a distribution fee to cover the cost of materials and/or transmission. John M. Lee Department of Mathematics Box 354350 University of Washington Seattle, WA 98195-4350 E-mail: [email protected] Web: http://www.math.washington.edu/~lee/ CONTENTS 3 Contents 1 Introduction 6 1.1Overview.............................. -
Appendix a Relations Between Covariant and Contravariant Bases
Appendix A Relations Between Covariant and Contravariant Bases The contravariant basis vector gk of the curvilinear coordinate of uk at the point P is perpendicular to the covariant bases gi and gj, as shown in Fig. A.1.This contravariant basis gk can be defined as or or a gk g  g ¼  ðA:1Þ i j oui ou j where a is the scalar factor; gk is the contravariant basis of the curvilinear coordinate of uk. Multiplying Eq. (A.1) by the covariant basis gk, the scalar factor a results in k k ðgi  gjÞ: gk ¼ aðg : gkÞ¼ad ¼ a ÂÃk ðA:2Þ ) a ¼ðgi  gjÞ : gk gi; gj; gk The scalar triple product of the covariant bases can be written as pffiffiffi a ¼ ½¼ðg1; g2; g3 g1  g2Þ : g3 ¼ g ¼ J ðA:3Þ where Jacobian J is the determinant of the covariant basis tensor G. The direction of the cross product vector in Eq. (A.1) is opposite if the dummy indices are interchanged with each other in Einstein summation convention. Therefore, the Levi-Civita permutation symbols (pseudo-tensor components) can be used in expression of the contravariant basis. ffiffiffi p k k g g ¼ J g ¼ðgi  gjÞ¼Àðgj  giÞ eijkðgi  gjÞ eijkðgi  gjÞ ðA:4Þ ) gk ¼ pffiffiffi ¼ g J where the Levi-Civita permutation symbols are defined by 8 <> þ1ifði; j; kÞ is an even permutation; eijk ¼ > À1ifði; j; kÞ is an odd permutation; : A:5 0ifi ¼ j; or i ¼ k; or j ¼ k ð Þ 1 , e ¼ ði À jÞÁðj À kÞÁðk À iÞ for i; j; k ¼ 1; 2; 3 ijk 2 H. -
Tensor, Exterior and Symmetric Algebras
Tensor, Exterior and Symmetric Algebras Daniel Murfet May 16, 2006 Throughout this note R is a commutative ring, all modules are left R-modules. If we say a ring is noncommutative, we mean it is not necessarily commutative. Unless otherwise specified, all rings are noncommutative (except for R). If A is a ring then the center of A is the set of all x ∈ A with xy = yx for all y ∈ A. Contents 1 Definitions 1 2 The Tensor Algebra 5 3 The Exterior Algebra 6 3.1 Dimension of the Exterior Powers ............................ 11 3.2 Bilinear Forms ...................................... 14 3.3 Other Properties ..................................... 18 3.3.1 The determinant formula ............................ 18 4 The Symmetric Algebra 19 1 Definitions Definition 1. A R-algebra is a ring morphism φ : R −→ A where A is a ring and the image of φ is contained in the center of A. This is equivalent to A being an R-module and a ring, with r · (ab) = (r · a)b = a(r · b), via the identification of r · 1 and φ(r). A morphism of R-algebras is a ring morphism making the appropriate diagram commute, or equivalently a ring morphism which is also an R-module morphism. In this section RnAlg will denote the category of these R-algebras. We use RAlg to denote the category of commutative R-algebras. A graded ring is a ring A together with a set of subgroups Ad, d ≥ 0 such that A = ⊕d≥0Ad as an abelian group, and st ∈ Ad+e for all s ∈ Ad, t ∈ Ae. -
Mechanics of Continuous Media in $(\Bar {L} N, G) $-Spaces. I
Mechanics of Continuous Media in (Ln, g)-spaces. I. Introduction and mathematical tools S. Manoff Bulgarian Academy of Sciences, Institute for Nuclear Research and Nuclear Energy, Department of Theoretical Physics, Blvd. Tzarigradsko Chaussee 72 1784 Sofia - Bulgaria e-mail address: [email protected] Abstract Basic notions and mathematical tools in continuum media mechanics are recalled. The notion of exponent of a covariant differential operator is intro- duced and on its basis the geometrical interpretation of the curvature and the torsion in (Ln,g)-spaces is considered. The Hodge (star) operator is general- ized for (Ln,g)-spaces. The kinematic characteristics of a flow are outline in brief. PACS numbers: 11.10.-z; 11.10.Ef; 7.10.+g; 47.75.+f; 47.90.+a; 83.10.Bb 1 Introduction 1.1 Differential geometry and space-time geometry arXiv:gr-qc/0203016v1 5 Mar 2002 In the last years, the evolution of the relations between differential geometry and space-time geometry has made some important steps toward applications of more comprehensive differential-geometric structures in the models of space-time than these used in (pseudo) Riemannian spaces without torsion (Vn-spaces) [1], [2]. 1. Recently, it has been proved that every differentiable manifold with one affine connection and metrics [(Ln,g)-space] [3], [4] could be used as a model for a space- time. In it, the equivalence principle (related to the vanishing of the components of an affine connection at a point or on a curve in a manifold) holds [5] ÷ [11], [12]. Even if the manifold has two different (not only by sign) connections for tangent and co-tangent vector fields [(Ln,g)-space] [13], [14] the principle of equivalence is fulfilled at least for one of the two types of vector fields [15].