DOCSLIB.ORG

An Introduction to Differential Geometry Through Computation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Introduction to Differential Geometry Through Computation Utah State University DigitalCommons@USU Textbooks Open Texts Winter 1-26-2016 An Introduction to Differential Geometry through Computation Mark E. Fels Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/oer_textbooks Part of the Mathematics Commons Recommended Citation Fels, Mark E., "An Introduction to Differential Geometry through Computation" (2016). Textbooks. 2. https://digitalcommons.usu.edu/oer_textbooks/2 This Book is brought to you for free and open access by the Open Texts at DigitalCommons@USU. It has been accepted for inclusion in Textbooks by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. An Introduction to Differential Geometry through Computation Mark E. Fels c Draft date March 9, 2017 Contents Preface v 1 Preliminaries 1 1.1 Open sets . .1 1.2 Smooth functions . .2 1.3 Smooth Curves . .5 1.4 Composition and the Chain-rule . .6 1.5 Vector Spaces, Basis, and Subspaces . 10 1.6 Algebras . 17 1.7 Exercises . 21 2 Linear Transformations 25 2.1 Matrix Representation . 25 2.2 Kernel, Rank, and the Rank-Nullity Theorem . 30 2.3 Composition, Inverse and Isomorphism . 34 2.4 Exercises . 39 3 Tangent Vectors 43 3.1 Tangent Vectors and Curves . 43 3.2 Derivations . 45 3.3 Vector fields . 51 3.4 Exercises . 54 4 The push-forward and the Jacobian 57 4.1 The push-forward using curves . 57 4.2 The push-forward using derivations . 59 4.3 The Chain rule, Immersions, Submersions, and Diffeomorphisms 63 4.4 Change of Variables . 66 i ii CONTENTS 4.5 Exercises . 72 5 Differential One-forms and Metric Tensors 75 5.1 Differential One-Forms . 75 5.2 Bilinear forms and Inner Products . 81 5.3 Tensor product . 86 5.4 Metric Tensors . 89 5.4.1 Arc-length . 93 5.4.2 Orthonormal Frames . 95 5.5 Raising and Lowering Indices and the Gradient . 97 5.6 A tale of two duals . 105 5.7 Exercises . 107 6 The Pullback and Isometries 111 6.1 The Pullback of a Differential One-form . 111 6.2 The Pullback of a Metric Tensor . 117 6.3 Isometries . 122 6.4 Exercises . 131 7 Hypersurfaces 133 7.1 Regular Level Hyper-Surfaces . 133 7.2 Patches and Covers . 135 7.3 Maps between surfaces . 138 7.4 More General Surfaces . 139 7.5 Metric Tensors on Surfaces . 139 7.6 Exercises . 146 8 Flows, Invariants and the Straightening Lemma 149 8.1 Flows . 149 8.2 Invariants . 155 8.3 Invariants I . 158 8.4 Invariants II . 163 8.5 Exercises . 168 9 The Lie Bracket and Killing Vectors 171 9.1 Lie Bracket . 171 9.2 Killing vectors . 179 9.3 Exercises . 185 CONTENTS iii 10 Group actions and Multi-parameter Groups 187 10.1 Group Actions . 187 10.2 Infinitesimal Generators . 192 10.3 Right and Left Invariant Vector Fields . 194 10.4 Invariant Metric Tensors on Multi-parameter groups . 196 10.5 Exercises . 199 11 Computations in Maple 203 12 Algebraic Notions 207 12.1 Groups . 207 12.2 Rings . 209 Bibliography 213 iv CONTENTS Preface This book was conceived after numerous discussions with my colleague Ian Anderson about what to teach in an introductory one semester course in differential geometry. We found that after covering the classical differential geometry of curves and surfaces that it was difficult to make the transition to more advanced texts in differential geometry such as [4], or to texts which use differential geometry such as in differential equations [9] or general relativity [11], [13]. This book aims to make this transition more rapid, and to prepare upper level undergraduates and beginning level graduate students to be able to do some basic computational research on such topics as the isometries of metrics in general relativity or the symmetries of differential equations. This is not a book on classical differential geometry or tensor analysis, but rather a modern treatment of vector fields, push-forward by mappings, one-forms, metric tensor fields, isometries, and the infinitesimal generators of group actions, and some Lie group theory using only open sets in IR n. The definitions, notation and approach are taken from the corresponding concept on manifolds and developed in IR n. For example, tangent vectors are defined as derivations (on functions in IR n) and metric tensors are a field of positive definite symmetric bilinear functions on the tangent vectors. This approach introduces the student to these concepts in a familiar setting so that in the more abstract setting of manifolds the role of the manifold can be emphasized. The book emphasizes liner algebra. The approach that I have taken is to provide a detailed review of a linear algebra concept and then translate the concept over to the field theoretic version in differential geometry. The level of preparation in linear algebra effects how many chapters can be covered in one semester. For example, there is quite a bit of detail on linear transformations and dual spaces which can be quickly reviewed for students with advanced training in linear algebra. v CONTENTS 1 The outline of the book is as follows. Chapter 1 reviews some basic facts about smooth functions from IR n to IR m, as well as the basic facts about vector spaces, basis, and algebras. Chapter 2 introduces tangent vec- tors and vector fields in IR n using the standard two approaches with curves and derivations. Chapter 3 reviews linear transformations and their matrix representation so that in Chapter 4 the push-forward as an abstract linear transformation can be defined and its matrix representation as the Jacobian can be derived. As an application, the change of variable formula for vector fields is derived in Chapter 4. Chapter 5 develops the linear algebra of the dual space and the space of bi-linear functions and demonstrates how these concepts are used in defining differential one-forms and metric tensor fields. Chapter 6 introduces the pullback map on one-forms and metric tensors from which the important concept of isometries is then defined. Chapter 7 inves- tigates hyper-surfaces in IR n, using patches and defines the induced metric tensor from Euclidean space. The change of coordinate formula on overlaps is then derived. Chapter 8 returns to IR n to define a flow and investigates the relationship between a flow and its infinitesimal generator. The theory of flow invariants is then investigated both infinitesimally and from the flow point of view with the goal of proving the rectification theorem for vector fields. Chapter 9 investigates the Lie bracket of vector-fields and Killing vec- tors for a metric. Chapter 10 generalizes chapter 8 and introduces the general notion of a group action with the goal of providing examples of metric tensors with a large number of Killing vectors. It also introduces a special family of Lie groups which I've called multi-parameter groups. These are Lie groups whose domain is an open set in IR n. The infinitesimal generators for these groups are used to construct the left and right invariant vector-fields on the group, as well as the Killing vectors for some special invariant metric tensors on the groups. 2 CONTENTS Chapter 1 Preliminaries 1.1 Open sets The components (or Cartesian coordinates ) of a point x 2 IR n will be denoted by x = (x1; x2; : : : ; xn): Note that the labels are in the up position. That is x2 is not the square of x unless we are working in IR 1; IR 2; IR 3 where we will use the standard notation of x; y; z. The position of indices is important, and make many formulas easier to remember or derive. The Euclidean distance between the points x = (x1; : : : ; xn) and y = (y1; : : : ; yn) is d(x; y) = p(x1 − y1)2 + ::: + (xn − yn)2: + n The open ball of radius r 2 IR at the point p 2 IR is the set Br(p) ⊂ IR n, defined by n Br(p) = f x 2 IR j d(x; p) < rg: A subset U ⊂ IR n is an open set if given any point p 2 U there exists + an r 2 IR (which depends on p) such that the open ball Br(p) satisfies Br(p) ⊂ U. The empty set is also taken to be open. Example 1.1.1. The set IR n is an open set. n + Example 1.1.2. Let p 2 IR and r 2 IR . Any open ball Br(p) is an open set. 1 2 CHAPTER 1. PRELIMINARIES Example 1.1.3. The upper half plane is the set U = f (x; y) 2 IR 2 j y > 0 g and is open. Example 1.1.4. The set V = f (x; y) 2 IR 2 j y ≥ 0 g is not open. Any point (x; 0) 2 V can not satisfy the open ball condition. Example 1.1.5. The unit n-sphere Sn ⊂ IR n+1 is the subset Sn = f x 2 IR n+1 j d(x; 0) = 1 g and Sn is not open. No point x 2 Sn satisfies the open ball condition. The n n+1 set S is the boundary of the open ball B1(0) ⊂ IR . Roughly speaking, open sets contain no boundary point. This can be made precise using some elementary topology. 1.2 Smooth functions In this section we recall some facts from multi-variable calculus. A real-valued function f : IR n ! IR has the form, f(x) = f(x1; x2; : : : ; xn): We will only be interested in functions whose domain Dom(f), is either all of IR n or an open subset U ⊂ IR n.
Load more

Recommended publications
  • Topology and Physics 2019 - Lecture 2
    Topology and Physics 2019 - lecture 2 Marcel Vonk February 12, 2019 2.1 Maxwell theory in differential form notation Maxwell's theory of electrodynamics is a great example of the usefulness of differential forms. A nice reference on this topic, though somewhat outdated when it comes to notation, is [1]. For notational simplicity, we will work in units where the speed of light, the vacuum permittivity and the vacuum permeability are all equal to 1: c = 0 = µ0 = 1. 2.1.1 The dual field strength In three dimensional space, Maxwell's electrodynamics describes the physics of the electric and magnetic fields E~ and B~ . These are three-dimensional vector fields, but the beauty of the theory becomes much more obvious if we (a) use a four-dimensional relativistic formulation, and (b) write it in terms of differential forms. For example, let us look at Maxwells two source-free, homogeneous equations: r · B = 0;@tB + r × E = 0: (2.1) That these equations have a relativistic flavor becomes clear if we write them out in com- ponents and organize the terms somewhat suggestively: x y z 0 + @xB + @yB + @zB = 0 x z y −@tB + 0 − @yE + @zE = 0 (2.2) y z x −@tB + @xE + 0 − @zE = 0 z y x −@tB − @xE + @yE + 0 = 0 Note that we also multiplied the last three equations by −1 to clarify the structure. All in all, we see that we have four equations (one for each space-time coordinate) which each contain terms in which the four coordinate derivatives act. Therefore, we may be tempted to write our set of equations in more \relativistic" notation as ^µν @µF = 0 (2.3) 1 with F^µν the coordinates of an antisymmetric two-tensor (i.
    [Show full text]
  • Tensor Manipulation in GPL Maxima
    Tensor Manipulation in GPL Maxima Viktor Toth http://www.vttoth.com/ February 1, 2008 Abstract GPL Maxima is an open-source computer algebra system based on DOE-MACSYMA. GPL Maxima included two tensor manipulation packages from DOE-MACSYMA, but these were in various states of disrepair. One of the two packages, CTENSOR, implemented component-based tensor manipulation; the other, ITENSOR, treated tensor symbols as opaque, manipulating them based on their index properties. The present paper describes the state in which these packages were found, the steps that were needed to make the packages fully functional again, and the new functionality that was implemented to make them more versatile. A third package, ATENSOR, was also implemented; fully compatible with the identically named package in the commercial version of MACSYMA, ATENSOR implements abstract tensor algebras. 1 Introduction GPL Maxima (GPL stands for the GNU Public License, the most widely used open source license construct) is the descendant of one of the world’s first comprehensive computer algebra systems (CAS), DOE-MACSYMA, developed by the United States Department of Energy in the 1960s and the 1970s. It is currently maintained by 18 volunteer developers, and can be obtained in source or object code form from http://maxima.sourceforge.net/. Like other computer algebra systems, Maxima has tensor manipulation capability. This capability was developed in the late 1970s. Documentation is scarce regarding these packages’ origins, but a select collection of e-mail messages by various authors survives, dating back to 1979-1982, when these packages were actively maintained at M.I.T. When this author first came across GPL Maxima, the tensor packages were effectively non-functional.
    [Show full text]
  • SPINORS and SPACE–TIME ANISOTROPY
    Sergiu Vacaru and Panayiotis Stavrinos SPINORS and SPACE{TIME ANISOTROPY University of Athens ————————————————— c Sergiu Vacaru and Panyiotis Stavrinos ii - i ABOUT THE BOOK This is the first monograph on the geometry of anisotropic spinor spaces and its applications in modern physics. The main subjects are the theory of grav- ity and matter fields in spaces provided with off–diagonal metrics and asso- ciated anholonomic frames and nonlinear connection structures, the algebra and geometry of distinguished anisotropic Clifford and spinor spaces, their extension to spaces of higher order anisotropy and the geometry of gravity and gauge theories with anisotropic spinor variables. The book summarizes the authors’ results and can be also considered as a pedagogical survey on the mentioned subjects. ii - iii ABOUT THE AUTHORS Sergiu Ion Vacaru was born in 1958 in the Republic of Moldova. He was educated at the Universities of the former URSS (in Tomsk, Moscow, Dubna and Kiev) and reveived his PhD in theoretical physics in 1994 at ”Al. I. Cuza” University, Ia¸si, Romania. He was employed as principal senior researcher, as- sociate and full professor and obtained a number of NATO/UNESCO grants and fellowships at various academic institutions in R. Moldova, Romania, Germany, United Kingdom, Italy, Portugal and USA. He has published in English two scientific monographs, a university text–book and more than hundred scientific works (in English, Russian and Romanian) on (super) gravity and string theories, extra–dimension and brane gravity, black hole physics and cosmolgy, exact solutions of Einstein equations, spinors and twistors, anistoropic stochastic and kinetic processes and thermodynamics in curved spaces, generalized Finsler (super) geometry and gauge gravity, quantum field and geometric methods in condensed matter physics.
    [Show full text]
  • General Relativity Fall 2019 Lecture 11: the Riemann Tensor
    General Relativity Fall 2019 Lecture 11: The Riemann tensor Yacine Ali-Ha¨ımoud October 8th 2019 The Riemann tensor quantifies the curvature of spacetime, as we will see in this lecture and the next. RIEMANN TENSOR: BASIC PROPERTIES α γ Definition { Given any vector field V , r[αrβ]V is a tensor field. Let us compute its components in some coordinate system: σ σ λ σ σ λ r[µrν]V = @[µ(rν]V ) − Γ[µν]rλV + Γλ[µrν]V σ σ λ σ λ λ ρ = @[µ(@ν]V + Γν]λV ) + Γλ[µ @ν]V + Γν]ρV 1 = @ Γσ + Γσ Γρ V λ ≡ Rσ V λ; (1) [µ ν]λ ρ[µ ν]λ 2 λµν where all partial derivatives of V µ cancel out after antisymmetrization. σ Since the left-hand side is a tensor field and V is a vector field, we conclude that R λµν is a tensor field as well { this is the tensor division theorem, which I encourage you to think about on your own. You can also check that explicitly from the transformation law of Christoffel symbols. This is the Riemann tensor, which measures the non-commutation of second derivatives of vector fields { remember that second derivatives of scalar fields do commute, by assumption. It is completely determined by the metric, and is linear in its second derivatives. Expression in LICS { In a LICS the Christoffel symbols vanish but not their derivatives. Let us compute the latter: 1 1 @ Γσ = @ gσδ (@ g + @ g − @ g ) = ησδ (@ @ g + @ @ g − @ @ g ) ; (2) µ νλ 2 µ ν λδ λ νδ δ νλ 2 µ ν λδ µ λ νδ µ δ νλ since the first derivatives of the metric components (thus of its inverse as well) vanish in a LICS.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • Low-Level Image Processing with the Structure Multivector
    Low-Level Image Processing with the Structure Multivector Michael Felsberg Bericht Nr. 0202 Institut f¨ur Informatik und Praktische Mathematik der Christian-Albrechts-Universitat¨ zu Kiel Olshausenstr. 40 D – 24098 Kiel e-mail: [email protected] 12. Marz¨ 2002 Dieser Bericht enthalt¨ die Dissertation des Verfassers 1. Gutachter Prof. G. Sommer (Kiel) 2. Gutachter Prof. U. Heute (Kiel) 3. Gutachter Prof. J. J. Koenderink (Utrecht) Datum der mundlichen¨ Prufung:¨ 12.2.2002 To Regina ABSTRACT The present thesis deals with two-dimensional signal processing for computer vi- sion. The main topic is the development of a sophisticated generalization of the one-dimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance – equivariance constraint, and by its relation to complex analysis and potential theory, a two-dimensional approach is derived. This method is called the monogenic signal and it is based on the Riesz transform instead of the Hilbert transform. By means of this linear approach it is possible to estimate the local orientation and the local phase of signals which are projections of one-dimensional functions to two dimensions. For general two-dimensional signals, however, the monogenic signal has to be further extended, yielding the structure multivector. The latter approach combines the ideas of the structure tensor and the quaternionic analytic signal. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitudes, the local anisotropy, the local orientation, and two local phases. Both, the monogenic signal and the struc- ture multivector are combined with an appropriate scale-space approach, resulting in generalized quadrature filters.
    [Show full text]
  • Arxiv:1510.06157V2 [Math.DG] 24 Aug 2017 2
    DETERMINATION OF A RIEMANNIAN MANIFOLD FROM THE DISTANCE DIFFERENCE FUNCTIONS MATTI LASSAS AND TEEMU SAKSALA Abstract. Let (N; g) be a Riemannian manifold with the dis- tance function d(x; y) and an open subset M ⊂ N. For x 2 M we denote by Dx the distance difference function Dx : F × F ! R, given by Dx(z1; z2) = d(x; z1) − d(x; z2), z1; z2 2 F = N n M. We consider the inverse problem of determining the topological and the differentiable structure of the manifold M and the metric gjM on it when we are given the distance difference data, that is, the set F , the metric gjF , and the collection D(M) = fDx; x 2 Mg. Moreover, we consider the embedded image D(M) of the manifold M, in the vector space C(F × F ), as a representation of manifold M. The inverse problem of determining (M; g) from D(M) arises e.g. in the study of the wave equation on R×N when we observe in F the waves produced by spontaneous point sources at unknown points (t; x) 2 R × M. Then Dx(z1; z2) is the difference of the times when one observes at points z1 and z2 the wave produced by a point source at x that goes off at an unknown time. The prob- lem has applications in hybrid inverse problems and in geophysical imaging. Keywords: Inverse problems, distance functions, embeddings of man- ifolds, wave equation. Contents 1. Introduction 2 1.1. Motivation of the problem 2 1.2. Definitions and the main result 2 1.3.
    [Show full text]
  • Weyl's Spin Connection
    THE SPIN CONNECTION IN WEYL SPACE c William O. Straub, PhD Pasadena, California “The use of general connections means asking for trouble.” —Abraham Pais In addition to his seminal 1929 exposition on quantum mechanical gauge invariance1, Hermann Weyl demonstrated how the concept of a spinor (essentially a flat-space two-component quantity with non-tensor- like transformation properties) could be carried over to the curved space of general relativity. Prior to Weyl’s paper, spinors were recognized primarily as mathematical objects that transformed in the space of SU (2), but in 1928 Dirac showed that spinors were fundamental to the quantum mechanical description of spin—1/2 particles (electrons). However, the spacetime stage that Dirac’s spinors operated in was still Lorentzian. Because spinors are neither scalars nor vectors, at that time it was unclear how spinors behaved in curved spaces. Weyl’s paper provided a means for this description using tetrads (vierbeins) as the necessary link between Lorentzian space and curved Riemannian space. Weyl’selucidation of spinor behavior in curved space and his development of the so-called spin connection a ab ! band the associated spin vector ! = !ab was noteworthy, but his primary purpose was to demonstrate the profound connection between quantum mechanical gauge invariance and the electromagnetic field. Weyl’s 1929 paper served to complete his earlier (1918) theory2 in which Weyl attempted to derive electrodynamics from the geometrical structure of a generalized Riemannian manifold via a scale-invariant transformation of the metric tensor. This attempt failed, but the manifold he discovered (known as Weyl space), is still a subject of interest in theoretical physics.
    [Show full text]
  • Differential Forms on Riemannian (Lorentzian) and Riemann-Cartan
    Differential Forms on Riemannian (Lorentzian) and Riemann-Cartan Structures and Some Applications to Physics∗ Waldyr Alves Rodrigues Jr. Institute of Mathematics Statistics and Scientific Computation IMECC-UNICAMP CP 6065 13083760 Campinas SP Brazil e-mail: [email protected] or [email protected] May 28, 2018 Abstract In this paper after recalling some essential tools concerning the theory of differential forms in the Cartan, Hodge and Clifford bundles over a Riemannian or Riemann-Cartan space or a Lorentzian or Riemann-Cartan spacetime we solve with details several exercises involving different grades of difficult. One of the problems is to show that a recent formula given in [10] for the exterior covariant derivative of the Hodge dual of the torsion 2-forms is simply wrong. We believe that the paper will be useful for students (and eventually for some experts) on applications of differential geometry to some physical problems. A detailed account of the issues discussed in the paper appears in the table of contents. Contents 1 Introduction 3 arXiv:0712.3067v6 [math-ph] 4 Dec 2008 2 Classification of Metric Compatible Structures (M; g;D) 4 2.1 Levi-Civita and Riemann-Cartan Connections . 5 2.2 Spacetime Structures . 5 3 Absolute Differential and Covariant Derivatives 5 ∗published in: Ann. Fond. L. de Broglie 32 (special issue dedicate to torsion), 424-478 (2008). 1 ^ 4 Calculus on the Hodge Bundle ( T ∗M; ·; τg) 8 4.1 Exterior Product . 8 4.2 Scalar Product and Contractions . 9 4.3 Hodge Star Operator ? ........................ 9 4.4 Exterior derivative d and Hodge coderivative δ .
    [Show full text]
  • Special Relativity: Tensor Calculus and Four-Vectors
    Special Relativity: Tensor Calculus and Four-Vectors Looking ahead to general relativity, where such things are more important, we will now introduce the mathematics of tensors and four-vectors. The Mathematics of Spacetime Let's start by de¯ning some geometric objects. Bear with me for the ¯rst couple, which seem obvious but lay the groundwork for the less obvious sequels. Scalar.|A scalar is a pure number, meaning that all observers will agree on its value. For example, the number \3" is a scalar. Okay, that one's trivial, but there are others that aren't so obvious, and we'll get to those in a bit. Event.|Next we have an event. An event is e®ectively a \point" in spacetime. More generally, if you have an N-dimensional space, you need N numbers to label it uniquely. For example, in two dimensions you need two numbers; e.g., x and y for a plane, or and Á for the surface of a sphere. For spacetime, you need four numbers: e.g., t, x, y, and z. Naturally, the essence of the event isn't changed if you relabel the coordinates. I want to stress this, because something that is obvious for events but may not be obvious for some other geometric objects is that although when you ¯nally calculate something you may choose a coordinate system and break things into components, there is also an independent reality (well, within the math at least!) of the objects. Going with the coordinate-free representation has proved very helpful in proving theorems about spacetime, but when doing astrophysics it is usually best to investigate components in some given system.
    [Show full text]
  • Metric Tensor & Chriostofell Symbols
    MATHEMATICAL PHYSICS UNIT – 8 Metric Tensor & Chriostofell Symbols PRESENTED BY: DR. RAJESH MATHPAL ACADEMIC CONSULTANT SCHOOL OF SCIENCES U.O.U. TEENPANI, HALDWANI UTTRAKHAND MOB:9758417736,7983713112 STRUCTURE OF UNIT 8.1. INTRODUCTION 8.2. RIEMANNIAN SPACE: METRIC TENSOR jk 풋 8.3. FUNDAMENTAL TENSORS gjk, g AND 휹풌 8.4. CHRISTOFELL’S 3-INDEX SYMBOLS 8.5. GEODESICS 8.1. INTRODUCTION In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold. A metric tensor is called positive-definite if it assigns a positive value g(v, v) > 0 to every nonzero vector v. A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function d(p, q) whose value at a pair of points p and q is the distance from p to q.
    [Show full text]