DOCSLIB.ORG

A Introduction to Tensor Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Introduction to Tensor Analysis A Introduction to tensor analysis In this Appendix, assuming that the reader is acquainted with vector analy- sis, we present a short introduction to tensor analysis. However, since tensor analysis is a fundamental tool for understanding continuum mechanics, we strongly recommend a deeper study of this subject. Some of the books that can be used for that purpose are: (Synge & Schild 1949, McConnell 1957, Santaló 1961, Aris 1962, Sokolniko 1964, Fung 1965, Green & Zerna 1968, Flügge 1972, Chapelle & Bathe 2003). A.1 Coordinates transformation Let us assume that in a three-dimensional space ( 3) we can define a system of Cartesian coordinates: we call this space the Euclidean< space. In this Appendix we will restrict our presentation to the case of the Eu- clidean space. In the 3 spacewedefine a system of Cartesian coordinates } >= 1> 2> 3 , and< an arbitrary system of curvilinear coordinates l >l{=1> 2> 3 . The following} relations hold: { } l = l(} >=1> 2> 3) >l=1> 2> 3 = (A.1) The above functions are single-valued, continuous and with continuous first derivatives. We call M the Jacobian of the coordinates transformation defined by Eq. (A.1). Hence Cl = (A.2) M = "C} # An admissible transformation is one in which det M =0,thatistosay, a transformation where a region of nonzero volume in one6 system does not collapse into a point in the other system and vice versa. 214 Nonlinear continua A proper transformation is an admissible transformation in which det MA 0. A.1.1 Contravariant transformation rule From Eq. (A.1) we obtain Cl dl = d} = (A.3) C} When the coordinates system is changed, the mathematical entities dl at a certain point of 3 that transform following the same rule as does the coordinate dierentials< (Eq. (A.3)) are said to transform according to a con- travariant transformation rule. We indicate these mathematical entities using upper indices. l Now we consider two systems of curvilinear coordinates l and ˆ , related by the following equations: { } © ª l l ˆ = ˆ (m >m=1> 2> 3) >l=1> 2> 3 (A.4a) and o n = n(ˆ >o=1> 2> 3) >n=1> 2> 3 = (A.4b) We can write the coordinate dierentials as: l l C ˆm d = m d (A.4c) Cˆ l l Cˆ dˆ = dm = (A.4d) Cm In the same way, a contravariant mathematical entity can be defined in either of the two systems dl = dl(m >m=1> 2> 3) >l=1> 2> 3 (A.4e) m dˆl =ˆdl(ˆ >m=1> 2> 3) >l=1> 2> 3 (A.4f) and we transform it from one curvilinear system to the other following a trans- formation rule similar to the transformation rule followed by the coordinate dierentials: l l C m d = m dˆ >l=1> 2> 3 (A.4g) Cˆ l Cˆ dˆl = dm >l=1> 2> 3 = (A.4h) Cm A.2 Vectors 215 Although the contravariant transformation rule applies to dl and not to l, using a notation abuse, we follow the convention of using upper indices for the coordinates. A.1.2 Covariant transformation rule Given an arbitrary continuous and dierentiable function i(1>2>3) and using the chain rule, we write Ci Ci Cm l = m l >l=1> 2> 3 = (A.5a) Cˆ C Cˆ We define Ci dm = >m=1> 2> 3 = (A.5b) Cm l In the ˆ coordinate system we define { } Ci dˆm = m >m=1> 2> 3 (A.5c) Cˆ Cm dˆl = dm l >l=1> 2> 3 (A.5d) Cˆ m Cˆ dl =ˆdm >l=1> 2> 3 = (A.5e) Cl When the coordinates system is changed, the mathematical entities dl at a certain point of 3that transform following the same rule as does the derivatives of a scalar< function (Eqs. (A.5d) and (A.5e)) are said to transform accordingtoacovariant transformation rule. We indicate those mathematical entities using lower indices. A.2 Vectors There are some physical properties like mass, temperature, concentration of a given substance, etc., whose values do not change when the coordinate system used to describe the problem is changed. These variables are referred to as scalars. On the other hand, there are other physical variables like velocity, accel- eration, force, etc. that do not change their intensity and direction when the coordinate system used to describe the problem is changed. They are called vectors. In what follows, we will make use of the above intuitive definition of scalars and vectors. However, in Sect. A.4 we will see that they represent two partic- ular kinds of tensors (order 0 and 1, respectively). 216 Nonlinear continua A.2.1 Base vectors Asetofn linearly independent vectors is a basis of the space q and any othervectorin q can be constructed as a linear combination of< those base vectors. < Let us consider the three linearly independent vectors g (l =1> 2> 3) in l 3. Any vector v inthesamespacecanbewrittenas: < v = yl g = (A.6) l The mathematical entities yl (l =1> 2> 3) are the components of v in the basis g (l =1> 2> 3). l Example A.1. JJJJJ In a Cartesian system } >=1> 2> 3 thebasevectorsare { } e1 =(1> 0> 0) > e2 =(0> 1> 0) > e3 =(0> 0> 1) > where we have indicated the projection of the base vectors on the Cartesian axes. The position vector r of a point S in 3 is < r = } e = Hence, dr = d} e > but also, Cr dr = d} = C} Therefore, we get Cr e = >=1> 2> 3 = C} JJJJJ A.2.2 Covariant base vectors In the arbitrary curvilinear system l >l=1> 2> 3 we can write, at any point S of the space, { } Cr dr = dl = (A.7a) Cl Since A.2 Vectors 217 Fig. A.1. Covariant base vectors at a point P dr = dlg (A.7b) l we obtain Cr g = >=1> 2> 3 = (A.7c) l Cl The vectors g ,defined with the above equation, are the covariant base l vectors of the curvilinear coordinate system l at the point S . From its definition, the vector g is tangent{ to} the line, = ( ) and 1 2 2 S 3 = 3 (S ) . Similar conclusions can be reached for the covariant base vectors g and 2 g . 3 In a Cartesian system, we can write Eq. (A.7c) as: C} g = e >l=1> 2> 3 = (A.8) l Cl l In a second curvilinear system ˆ >l=1> 2> 3 , { } 218 Nonlinear continua m l Cˆ dr = dˆ gˆ = dl gˆ = (A.9a) l l C m Hence, we have m Cˆ g = gˆ >l=1> 2> 3 = (A.9b) l Cl m Due to the similarity between Eqs. (A.9b) and Eq. (A.5e) the base vectors g are called covariant base vectors. l A.2.3 Contravariant base vectors In an arbitrary curvilinear coordinate system l >l=1> 2> 3 we define the contravariant base vectors (dual basis) (gl >l{=1> 2> 3) with the} equation gl g = l > (A.10) · m m where the dot indicates a scalar product (“dot product”) between two vectors l l l and m is the Kronecker delta (m =1for l = m and m =0for l = m ). l 6 Defining in 3 two curvilinear systems l and ˆ and using Eq. (A.9b), we obtain < { } { } Cp gˆl gˆ = gˆl g = l = (A.11a) m m p m · · Cˆ Hence, using Eq. (A.10), we obtain Cp gl g = gˆl g = (A.11b) m m p · · Cˆ If we define l Cˆ gˆl = go >l=1> 2> 3 (A.11c) Co from Eq. (A.11b), we obtain l Cˆ Cp gl g = go g (A.11d) m o m p · C Cˆ · and l l Cˆ Cp Cˆ gl g = o = = l > (A.11e) m o m p m m · C Cˆ Cˆ where we can see that the relation (A.11a) is satisfied. Therefore, Eq.(A.11c) can be considered the transformation rule for the contravariant base vectors. Due to the similarity between Eqs. (A.11c) and (A.4h) the base vectors gl are called contravariant base vectors. A.3 Metric of a coordinates system 219 Fig. A.2. Covariant and contravariant base vectors In Fig.A.2 we represent in a space 2,atapointS ,thecovariantand contravariant base vectors, in order to provide< the reader with a useful geo- metrical insight. It is important to note that in a Cartesian system the covariant and contravariant base vectors are coincident. A.3 Metric of a coordinates system If a position vector r defines a point S in 3 and a position vector (r +dr) < defines a neighboring point S 0, the distance between these points is given by dv = dr dr = (A.12) · p A.3.1 Cartesian coordinates In a Cartesian system in 3,wehave < dv2 =d} d} (e e ) = (A.13a) · 220 Nonlinear continua If we call e e = > then · 2 dv =d} d} = (A.13b) We call the nine number the Cartesian components of the metric ten- sor at S (this notation will be clarified in Sect. A.4.3). From its definition, it is obvious that =1if = and =0 if = . 6 A.3.2 Curvilinear coordinates. Covariant metric components In an arbitrary curvilinear system l >l=1> 2> 3 the distance between S { } and S 0 is given by dv2 =dr dr =dl dm (g g ) = (A.14a) · l · m We call j = j = g g (A.14b) lm ml l · m the covariant components of the metric tensor at S in the curvilinear system l (this notation will be clarified in Section A.4.3.).
Load more

Recommended publications
  • Geometrical Tools for Embedding Fields, Submanifolds, and Foliations Arxiv
    Geometrical tools for embedding fields, submanifolds, and foliations Antony J. Speranza∗ Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, ON N2L 2Y5, Canada Abstract Embedding fields provide a way of coupling a background structure to a theory while preserving diffeomorphism-invariance. Examples of such background structures include embedded submani- folds, such as branes; boundaries of local subregions, such as the Ryu-Takayanagi surface in holog- raphy; and foliations, which appear in fluid dynamics and force-free electrodynamics. This work presents a systematic framework for computing geometric properties of these background structures in the embedding field description. An overview of the local geometric quantities associated with a foliation is given, including a review of the Gauss, Codazzi, and Ricci-Voss equations, which relate the geometry of the foliation to the ambient curvature. Generalizations of these equations for cur- vature in the nonintegrable normal directions are derived. Particular care is given to the question of which objects are well-defined for single submanifolds, and which depend on the structure of the foliation away from a submanifold. Variational formulas are provided for the geometric quantities, which involve contributions both from the variation of the embedding map as well as variations of the ambient metric. As an application of these variational formulas, a derivation is given of the Jacobi equation, describing perturbations of extremal area surfaces of arbitrary codimension. The embedding field formalism is also applied to the problem of classifying boundary conditions for general relativity in a finite subregion that lead to integrable Hamiltonians. The framework developed in this paper will provide a useful set of tools for future analyses of brane dynamics, fluid arXiv:1904.08012v2 [gr-qc] 26 Apr 2019 mechanics, and edge modes for finite subregions of diffeomorphism-invariant theories.
    [Show full text]
  • 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.
    [Show full text]
  • C:\Book\Booktex\C1s2.DVI
    35 1.2 TENSOR CONCEPTS AND TRANSFORMATIONS § ~ For e1, e2, e3 independent orthogonal unit vectors (base vectors), we may write any vector A as ~ b b b A = A1 e1 + A2 e2 + A3 e3 ~ where (A1,A2,A3) are the coordinates of A relativeb to theb baseb vectors chosen. These components are the projection of A~ onto the base vectors and A~ =(A~ e ) e +(A~ e ) e +(A~ e ) e . · 1 1 · 2 2 · 3 3 ~ ~ ~ Select any three independent orthogonal vectors,b b (E1, Eb 2, bE3), not necessarilyb b of unit length, we can then write ~ ~ ~ E1 E2 E3 e1 = , e2 = , e3 = , E~ E~ E~ | 1| | 2| | 3| and consequently, the vector A~ canb be expressedb as b A~ E~ A~ E~ A~ E~ A~ = · 1 E~ + · 2 E~ + · 3 E~ . ~ ~ 1 ~ ~ 2 ~ ~ 3 E1 E1 ! E2 E2 ! E3 E3 ! · · · Here we say that A~ E~ · (i) ,i=1, 2, 3 E~ E~ (i) · (i) ~ ~ ~ ~ are the components of A relative to the chosen base vectors E1, E2, E3. Recall that the parenthesis about the subscript i denotes that there is no summation on this subscript. It is then treated as a free subscript which can have any of the values 1, 2or3. Reciprocal Basis ~ ~ ~ Consider a set of any three independent vectors (E1, E2, E3) which are not necessarily orthogonal, nor of unit length. In order to represent the vector A~ in terms of these vectors we must find components (A1,A2,A3) such that ~ 1 ~ 2 ~ 3 ~ A = A E1 + A E2 + A E3. This can be done by taking appropriate projections and obtaining three equations and three unknowns from which the components are determined.
    [Show full text]
  • ROTATION: a Review of Useful Theorems Involving Proper Orthogonal Matrices Referenced to Three- Dimensional Physical Space
    Unlimited Release Printed May 9, 2002 ROTATION: A review of useful theorems involving proper orthogonal matrices referenced to three- dimensional physical space. Rebecca M. Brannon† and coauthors to be determined †Computational Physics and Mechanics T Sandia National Laboratories Albuquerque, NM 87185-0820 Abstract Useful and/or little-known theorems involving33× proper orthogonal matrices are reviewed. Orthogonal matrices appear in the transformation of tensor compo- nents from one orthogonal basis to another. The distinction between an orthogonal direction cosine matrix and a rotation operation is discussed. Among the theorems and techniques presented are (1) various ways to characterize a rotation including proper orthogonal tensors, dyadics, Euler angles, axis/angle representations, series expansions, and quaternions; (2) the Euler-Rodrigues formula for converting axis and angle to a rotation tensor; (3) the distinction between rotations and reflections, along with implications for “handedness” of coordinate systems; (4) non-commu- tivity of sequential rotations, (5) eigenvalues and eigenvectors of a rotation; (6) the polar decomposition theorem for expressing a general deformation as a se- quence of shape and volume changes in combination with pure rotations; (7) mix- ing rotations in Eulerian hydrocodes or interpolating rotations in discrete field approximations; (8) Rates of rotation and the difference between spin and vortici- ty, (9) Random rotations for simulating crystal distributions; (10) The principle of material frame indifference (PMFI); and (11) a tensor-analysis presentation of classical rigid body mechanics, including direct notation expressions for momen- tum and energy and the extremely compact direct notation formulation of Euler’s equations (i.e., Newton’s law for rigid bodies).
    [Show full text]
  • Spin in Relativistic Quantum Theory∗
    Spin in relativistic quantum theory∗ W. N. Polyzou, Department of Physics and Astronomy, The University of Iowa, Iowa City, IA 52242 W. Gl¨ockle, Institut f¨urtheoretische Physik II, Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany H. Wita la M. Smoluchowski Institute of Physics, Jagiellonian University, PL-30059 Krak´ow,Poland today Abstract We discuss the role of spin in Poincar´einvariant formulations of quan- tum mechanics. 1 Introduction In this paper we discuss the role of spin in relativistic few-body models. The new feature with spin in relativistic quantum mechanics is that sequences of rotationless Lorentz transformations that map rest frames to rest frames can generate rotations. To get a well-defined spin observable one needs to define it in one preferred frame and then use specific Lorentz transformations to relate the spin in the preferred frame to any other frame. Different choices of the preferred frame and the specific Lorentz transformations lead to an infinite number of possible observables that have all of the properties of a spin. This paper provides a general discussion of the relation between the spin of a system and the spin of its elementary constituents in relativistic few-body systems. In section 2 we discuss the Poincar´egroup, which is the group relating inertial frames in special relativity. Unitary representations of the Poincar´e group preserve quantum probabilities in all inertial frames, and define the rel- ativistic dynamics. In section 3 we construct a large set of abstract operators out of the Poincar´egenerators, and determine their commutation relations and ∗During the preparation of this paper Walter Gl¨ockle passed away.
    [Show full text]
  • Infeld–Van Der Waerden Wave Functions for Gravitons and Photons
    Vol. 38 (2007) ACTA PHYSICA POLONICA B No 8 INFELD–VAN DER WAERDEN WAVE FUNCTIONS FOR GRAVITONS AND PHOTONS J.G. Cardoso Department of Mathematics, Centre for Technological Sciences-UDESC Joinville 89223-100, Santa Catarina, Brazil [email protected] (Received March 20, 2007) A concise description of the curvature structures borne by the Infeld- van der Waerden γε-formalisms is provided. The derivation of the wave equations that control the propagation of gravitons and geometric photons in generally relativistic space-times is then carried out explicitly. PACS numbers: 04.20.Gr 1. Introduction The most striking physical feature of the Infeld-van der Waerden γε-formalisms [1] is related to the possible occurrence of geometric wave functions for photons in the curvature structures of generally relativistic space-times [2]. It appears that the presence of intrinsically geometric elec- tromagnetic fields is ultimately bound up with the imposition of a single condition upon the metric spinors for the γ-formalism, which bears invari- ance under the action of the generalized Weyl gauge group [1–5]. In the case of a spacetime which admits nowhere-vanishing Weyl spinor fields, back- ground photons eventually interact with underlying gravitons. Nevertheless, the corresponding coupling configurations turn out to be in both formalisms exclusively borne by the wave equations that control the electromagnetic propagation. Indeed, the same graviton-photon interaction prescriptions had been obtained before in connection with a presentation of the wave equations for the ε-formalism [6, 7]. As regards such earlier works, the main procedure seems to have been based upon the implementation of a set of differential operators whose definitions take up suitably contracted two- spinor covariant commutators.
    [Show full text]
  • APPLIED ELASTICITY, 2Nd Edition Matrix and Tensor Analysis of Elastic Continua
    APPLIED ELASTICITY, 2nd Edition Matrix and Tensor Analysis of Elastic Continua Talking of education, "People have now a-days" (said he) "got a strange opinion that every thing should be taught by lectures. Now, I cannot see that lectures can do so much good as reading the books from which the lectures are taken. I know nothing that can be best taught by lectures, expect where experiments are to be shewn. You may teach chymestry by lectures. — You might teach making of shoes by lectures.' " James Boswell: Lifeof Samuel Johnson, 1766 [1709-1784] ABOUT THE AUTHOR In 1947 John D Renton was admitted to a Reserved Place (entitling him to free tuition) at King Edward's School in Edgbaston, Birmingham which was then a Grammar School. After six years there, followed by two doing National Service in the RAF, he became an undergraduate in Civil Engineering at Birmingham University, and obtained First Class Honours in 1958. He then became a research student of Dr A H Chilver (now Lord Chilver) working on the stability of space frames at Fitzwilliam House, Cambridge. Part of the research involved writing the first computer program for analysing three-dimensional structures, which was used by the consultants Ove Arup in their design project for the roof of the Sydney Opera House. He won a Research Fellowship at St John's College Cambridge in 1961, from where he moved to Oxford University to take up a teaching post at the Department of Engineering Science in 1963. This was followed by a Tutorial Fellowship to St Catherine's College in 1966.
    [Show full text]
  • Dual Electromagnetism: Helicity, Spin, Momentum, and Angular Momentum Konstantin Y
    Dual electromagnetism: Helicity, spin, momentum, and angular momentum Konstantin Y. Bliokh1,2, Aleksandr Y. Bekshaev3, and Franco Nori1,4 1Advanced Science Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan 2A. Usikov Institute of Radiophysics and Electronics, NASU, Kharkov 61085, Ukraine 3I. I. Mechnikov National University, Dvorianska 2, Odessa, 65082, Ukraine 4Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA The dual symmetry between electric and magnetic fields is an important intrinsic property of Maxwell equations in free space. This symmetry underlies the conservation of optical helicity, and, as we show here, is closely related to the separation of spin and orbital degrees of freedom of light (the helicity flux coincides with the spin angular momentum). However, in the standard field-theory formulation of electromagnetism, the field Lagrangian is not dual symmetric. This leads to problematic dual-asymmetric forms of the canonical energy-momentum, spin, and orbital angular momentum tensors. Moreover, we show that the components of these tensors conflict with the helicity and energy conservation laws. To resolve this discrepancy between the symmetries of the Lagrangian and Maxwell equations, we put forward a dual-symmetric Lagrangian formulation of classical electromagnetism. This dual electromagnetism preserves the form of Maxwell equations, yields meaningful canonical energy-momentum and angular momentum tensors, and ensures a self-consistent separation of the spin and orbital degrees of freedom. This provides rigorous derivation of results suggested in other recent approaches. We make the Noether analysis of the dual symmetry and all the Poincaré symmetries, examine both local and integral conserved quantities, and show that only the dual electromagnetism naturally produces a complete self-consistent set of conservation laws.
    [Show full text]
  • Arxiv:1903.03835V1 [Gr-Qc] 9 Mar 2019 [15–17] Thus Modifying Significantly the Orbit of 3
    Spinning particle orbits around a black hole in an expanding background I. Antoniou,1, ∗ D. Papadopoulos,2, y and L. Perivolaropoulos1, z 1Department of Physics, University of Ioannina, GR-45110, Ioannina, Greece 2Department of Physics, University of Thessaloniki, Thessaloniki,54124, Greece (Dated: March 12, 2019) We investigate analytically and numerically the orbits of spinning particles around black holes in the post Newtonian limit and in the presence of cosmic expansion. We show that orbits that are circular in the absence of spin, get deformed when the orbiting particle has spin. We show that the origin of this deformation is twofold: a. the background expansion rate which induces an attractive (repulsive) interaction due to the cosmic background fluid when the expansion is decelerating (accelerating) and b. a spin-orbit interaction which can be attractive or repulsive depending on the relative orientation between spin and orbital angular momentum and on the expansion rate. PACS numbers: 04.20.Cv, 97.60.Lf, 98.62.Dm I. INTRODUCTION bound systems in the presence of phantom dark energy with equation of state parameter w < −1 Even though most astrophysical bodies have [2{4]. In the absence of expansion but in the spins and evolve in an expanding cosmological presence of spin for the test particles it has been background, their motion is described well by ig- shown that spin-orbit and spin-spin interactions noring the cosmic expansion and under the non- in a Kerr spacetime can lead to deformations of spinning test particle approximation for large dis- circular orbits for large spin values [14].
    [Show full text]
  • Ch.2. Deformation and Strain
    CH.2. DEFORMATION AND STRAIN Multimedia Course on Continuum Mechanics Overview Introduction Lecture 1 Deformation Gradient Tensor Material Deformation Gradient Tensor Lecture 2 Lecture 3 Inverse (Spatial) Deformation Gradient Tensor Displacements Lecture 4 Displacement Gradient Tensors Strain Tensors Green-Lagrange or Material Strain Tensor Lecture 5 Euler-Almansi or Spatial Strain Tensor Variation of Distances Stretch Lecture 6 Unit elongation Variation of Angles Lecture 7 2 Overview (cont’d) Physical interpretation of the Strain Tensors Lecture 8 Material Strain Tensor, E Spatial Strain Tensor, e Lecture 9 Polar Decomposition Lecture 10 Volume Variation Lecture 11 Area Variation Lecture 12 Volumetric Strain Lecture 13 Infinitesimal Strain Infinitesimal Strain Theory Strain Tensors Stretch and Unit Elongation Lecture 14 Physical Interpretation of Infinitesimal Strains Engineering Strains Variation of Angles 3 Overview (cont’d) Infinitesimal Strain (cont’d) Polar Decomposition Lecture 15 Volumetric Strain Strain Rate Spatial Velocity Gradient Tensor Lecture 16 Strain Rate Tensor and Rotation Rate Tensor or Spin Tensor Physical Interpretation of the Tensors Material Derivatives Lecture 17 Other Coordinate Systems Cylindrical Coordinates Lecture 18 Spherical Coordinates 4 2.1 Introduction Ch.2. Deformation and Strain 5 Deformation Deformation: transformation of a body from a reference configuration to a current configuration. Focus on the relative movement of a given particle w.r.t. the particles in its neighbourhood (at differential level). It includes changes of size and shape. 6 2.2 Deformation Gradient Tensors Ch.2. Deformation and Strain 7 Continuous Medium in Movement Ω0: non-deformed (or reference) Ω or Ωt: deformed (or present) configuration, at reference time t0. configuration, at present time t.
    [Show full text]
  • A Vectors, Tensors, and Their Operations
    A Vectors, Tensors, and Their Operations A.1 Vectors and Tensors A spatial description of the fluid motion is a geometrical description, of which the essence is to ensure that relevant physical quantities are invariant under artificially introduced coordinate systems. This is realized by tensor analysis (cf. Aris 1962). Here we introduce the concept of tensors in an informal way, through some important examples in fluid mechanics. A.1.1 Scalars and Vectors Scalars and vectors are geometric entities independent of the choice of coor- dinate systems. This independence is a necessary condition for an entity to represent some physical quantity. A scalar, say the fluid pressure p or den- sity ρ, obviously has such independence. For a vector, say the fluid velocity u, although its three components (u1,u2,u3) depend on the chosen coordi- nates, say Cartesian coordinates with unit basis vectors (e1, e2, e3), as a single geometric entity the one-form of ei, u = u1e1 + u2e2 + u3e3 = uiei,i=1, 2, 3, (A.1) has to be independent of the basis vectors. Note that Einstein’s convention has been used in the last expression of (A.1): unless stated otherwise, a repeated index always implies summation over the dimension of the space. The inner (scalar) and cross (vector) products of two vectors are familiar. If θ is the angle between the directions of a and b, these operations give a · b = |a||b| cos θ, |a × b| = |a||b| sin θ. While the inner product is a projection operation, the cross-product produces a vector perpendicular to both a and b with magnitude equal to the area of the parallelogram spanned by a and b.Thusa×b determines a vectorial area 694 A Vectors, Tensors, and Their Operations with unit vector n normal to the (a, b) plane, whose direction follows from a to b by the right-hand rule.
    [Show full text]
  • Mathematical Techniques
    Chapter 8 Mathematical Techniques This chapter contains a review of certain basic mathematical concepts and tech- niques that are central to metric prediction generation. A mastery of these fun- damentals will enable the reader not only to understand the principles underlying the prediction algorithms, but to be able to extend them to meet future require- ments. 8.1. Scalars, Vectors, Matrices, and Tensors While the reader may have encountered the concepts of scalars, vectors, and ma- trices in their college courses and subsequent job experiences, the concept of a tensor is likely an unfamiliar one. Each of these concepts is, in a sense, an exten- sion of the preceding one. Each carries a notion of value and operations by which they are transformed and combined. Each is used for representing increas- ingly more complex structures that seem less complex in the higher-order form. Tensors were introduced in the Spacetime chapter to represent a complexity of perhaps an unfathomable incomprehensibility to the untrained mind. However, to one versed in the theory of general relativity, tensors are mathematically the simplest representation that has so far been found to describe its principles. 8.1.1. The Hierarchy of Algebras As a precursor to the coming material, it is perhaps useful to review some ele- mentary algebraic concepts. The material in this subsection describes the succes- 1 2 Explanatory Supplement to Metric Prediction Generation sion of algebraic structures from sets to fields that may be skipped if the reader is already aware of this hierarchy. The basic theory of algebra begins with set theory .
    [Show full text]