Computing Higher Order Derivatives of Matrix and Tensor Expressions
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Mathematics for Dynamicists - Lecture Notes Thomas Michelitsch
Mathematics for Dynamicists - Lecture Notes Thomas Michelitsch To cite this version: Thomas Michelitsch. Mathematics for Dynamicists - Lecture Notes. Doctoral. Mathematics for Dynamicists, France. 2006. cel-01128826 HAL Id: cel-01128826 https://hal.archives-ouvertes.fr/cel-01128826 Submitted on 10 Mar 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Mathematics for Dynamicists Lecture Notes c 2004-2006 Thomas Michael Michelitsch1 Department of Civil and Structural Engineering The University of Sheffield Sheffield, UK 1Present address : http://www.dalembert.upmc.fr/home/michelitsch/ . Dedicated to the memory of my parents Michael and Hildegard Michelitsch Handout Manuscript with Fractal Cover Picture by Michael Michelitsch https://www.flickr.com/photos/michelitsch/sets/72157601230117490 1 Contents Preface 1 Part I 1.1 Binomial Theorem 1.2 Derivatives, Integrals and Taylor Series 1.3 Elementary Functions 1.4 Complex Numbers: Cartesian and Polar Representation 1.5 Sequences, Series, Integrals and Power-Series 1.6 Polynomials and Partial Fractions -
Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E
BOOK REVIEW Einstein’s Italian Mathematicians: Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E. Rowe Einstein’s Italian modern Italy. Nor does the author shy away from topics Mathematicians: like how Ricci developed his absolute differential calculus Ricci, Levi-Civita, and the as a generalization of E. B. Christoffel’s (1829–1900) work Birth of General Relativity on quadratic differential forms or why it served as a key By Judith R. Goodstein tool for Einstein in his efforts to generalize the special theory of relativity in order to incorporate gravitation. In This delightful little book re- like manner, she describes how Levi-Civita was able to sulted from the author’s long- give a clear geometric interpretation of curvature effects standing enchantment with Tul- in Einstein’s theory by appealing to his concept of parallel lio Levi-Civita (1873–1941), his displacement of vectors (see below). For these and other mentor Gregorio Ricci Curbastro topics, Goodstein draws on and cites a great deal of the (1853–1925), and the special AMS, 2018, 211 pp. 211 AMS, 2018, vast secondary literature produced in recent decades by the world that these and other Ital- “Einstein industry,” in particular the ongoing project that ian mathematicians occupied and helped to shape. The has produced the first 15 volumes of The Collected Papers importance of their work for Einstein’s general theory of of Albert Einstein [CPAE 1–15, 1987–2018]. relativity is one of the more celebrated topics in the history Her account proceeds in three parts spread out over of modern mathematical physics; this is told, for example, twelve chapters, the first seven of which cover episodes in [Pais 1982], the standard biography of Einstein. -
Non-Metric Generalizations of Relativistic Gravitational Theory and Observational Data Interpretation
Non-metric Generalizations of Relativistic Gravitational Theory and Observational Data Interpretation А. N. Alexandrov, I. B. Vavilova, V. I. Zhdanov Astronomical Observatory,Taras Shevchenko Kyiv National University 3 Observatorna St., Kyiv 04053 Ukraine Abstract We discuss theoretical formalisms concerning with experimental verification of General Relativity (GR). Non-metric generalizations of GR are considered and a system of postulates is formulated for metric-affine and Finsler gravitational theories. We consider local observer reference frames to be a proper tool for comparing predictions of alternative theories with each other and with the observational data. Integral formula for geodesic deviation due to the deformation of connection is obtained. This formula can be applied for calculations of such effects as the bending of light and time- delay in presence of non-metrical effects. 1. Introduction General Relativity (GR) experimental verification is a principal problem of gravitational physics. Research of the GR application limits is of a great interest for the fundamental physics and cosmology, whose alternative lines of development are connected with the corresponding changes of space-time model. At present, various aspects of GR have been tested with rather a high accuracy (Will, 2001). General Relativity is well tested experimentally for weak gravity fields, and it is adopted as a basis for interpretation of the most precise measurements in astronomy and geodynamics (Soffel et al., 2003). Considerable efforts have to be applied to verify GR conclusions in strong fields (Will, 2001). For GR experimental verification we should go beyond its application limits, consider it in a wider context, and compare it with alternative theories. -
Matrix Calculus
Appendix D Matrix Calculus From too much study, and from extreme passion, cometh madnesse. Isaac Newton [205, §5] − D.1 Gradient, Directional derivative, Taylor series D.1.1 Gradients Gradient of a differentiable real function f(x) : RK R with respect to its vector argument is defined uniquely in terms of partial derivatives→ ∂f(x) ∂x1 ∂f(x) , ∂x2 RK f(x) . (2053) ∇ . ∈ . ∂f(x) ∂xK while the second-order gradient of the twice differentiable real function with respect to its vector argument is traditionally called the Hessian; 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 ∂x1 ∂x1∂x2 ··· ∂x1∂xK 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 2 K f(x) , ∂x2∂x1 ∂x2 ··· ∂x2∂xK S (2054) ∇ . ∈ . .. 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 ∂xK ∂x1 ∂xK ∂x2 ∂x ··· K interpreted ∂f(x) ∂f(x) 2 ∂ ∂ 2 ∂ f(x) ∂x1 ∂x2 ∂ f(x) = = = (2055) ∂x1∂x2 ³∂x2 ´ ³∂x1 ´ ∂x2∂x1 Dattorro, Convex Optimization Euclidean Distance Geometry, Mεβoo, 2005, v2020.02.29. 599 600 APPENDIX D. MATRIX CALCULUS The gradient of vector-valued function v(x) : R RN on real domain is a row vector → v(x) , ∂v1(x) ∂v2(x) ∂vN (x) RN (2056) ∇ ∂x ∂x ··· ∂x ∈ h i while the second-order gradient is 2 2 2 2 , ∂ v1(x) ∂ v2(x) ∂ vN (x) RN v(x) 2 2 2 (2057) ∇ ∂x ∂x ··· ∂x ∈ h i Gradient of vector-valued function h(x) : RK RN on vector domain is → ∂h1(x) ∂h2(x) ∂hN (x) ∂x1 ∂x1 ··· ∂x1 ∂h1(x) ∂h2(x) ∂hN (x) h(x) , ∂x2 ∂x2 ··· ∂x2 ∇ . -
Matrix Calculus
D Matrix Calculus D–1 Appendix D: MATRIX CALCULUS D–2 In this Appendix we collect some useful formulas of matrix calculus that often appear in finite element derivations. §D.1 THE DERIVATIVES OF VECTOR FUNCTIONS Let x and y be vectors of orders n and m respectively: x1 y1 x2 y2 x = . , y = . ,(D.1) . xn ym where each component yi may be a function of all the xj , a fact represented by saying that y is a function of x,or y = y(x). (D.2) If n = 1, x reduces to a scalar, which we call x.Ifm = 1, y reduces to a scalar, which we call y. Various applications are studied in the following subsections. §D.1.1 Derivative of Vector with Respect to Vector The derivative of the vector y with respect to vector x is the n × m matrix ∂y1 ∂y2 ∂ym ∂ ∂ ··· ∂ x1 x1 x1 ∂ ∂ ∂ ∂ y1 y2 ··· ym y def ∂x ∂x ∂x = 2 2 2 (D.3) ∂x . . .. ∂ ∂ ∂ y1 y2 ··· ym ∂xn ∂xn ∂xn §D.1.2 Derivative of a Scalar with Respect to Vector If y is a scalar, ∂y ∂x1 ∂ ∂ y y def ∂ = x2 .(D.4) ∂x . . ∂y ∂xn §D.1.3 Derivative of Vector with Respect to Scalar If x is a scalar, ∂y def ∂ ∂ ∂ = y1 y2 ... ym (D.5) ∂x ∂x ∂x ∂x D–2 D–3 §D.1 THE DERIVATIVES OF VECTOR FUNCTIONS REMARK D.1 Many authors, notably in statistics and economics, define the derivatives as the transposes of those given above.1 This has the advantage of better agreement of matrix products with composition schemes such as the chain rule. -
On the Visualization of Geometric Properties of Particular Spacetimes
Technische Universität Berlin Diploma Thesis On the Visualization of Geometric Properties of Particular Spacetimes Torsten Schönfeld January 14, 2009 reviewed by Prof. Dr. H.-H. von Borzeszkowski Department of Theoretical Physics Faculty II, Technische Universität Berlin and Priv.-Doz. Dr. M. Scherfner Department of Mathematics Faculty II, Technische Universität Berlin Zusammenfassung auf Deutsch Ziel und Aufgabenstellung dieser Diplomarbeit ist es, Möglichkeiten der Visualisierung in der Allgemeinen Relativitätstheorie zu finden. Das Hauptaugenmerk liegt dabei auf der Visualisierung geometrischer Eigenschaften einiger akausaler Raumzeiten, d.h. Raumzeiten, die geschlossene zeitartige Kurven erlauben. Die benutzten und untersuchten Techniken umfassen neben den gängigen Möglichkeiten (Vektorfelder, Hyperflächen) vor allem das Darstellen von Geodäten und Lichtkegeln. Um den Einfluss der Raumzeitgeometrie auf das Verhalten von kräftefreien Teilchen zu untersuchen, werden in der Diplomarbeit mehrere Geodäten mit unterschiedlichen Anfangsbedinungen abgebildet. Dies erlaubt es zum Beispiel, die Bildung von Teilchen- horizonten oder Kaustiken zu analysieren. Die Darstellung von Lichtkegeln wiederum ermöglicht es, eine Vorstellung von der kausalen Struktur einer Raumzeit zu erlangen. Ein „Umkippen“ der Lichtkegel deutet beispielsweise oft auf signifikante Änderungen in der Raumzeit hin, z.B. auf die Möglichkeit von geschlossenen zeitartigen Kurven. Zur Implementierung dieser Techniken wurde im Rahmen der Diplomarbeit ein Ma- thematica-Paket namens -
Arxiv:Gr-Qc/0309008V2 9 Feb 2004
The Cotton tensor in Riemannian spacetimes Alberto A. Garc´ıa∗ Departamento de F´ısica, CINVESTAV–IPN, Apartado Postal 14–740, C.P. 07000, M´exico, D.F., M´exico Friedrich W. Hehl† Institute for Theoretical Physics, University of Cologne, D–50923 K¨oln, Germany, and Department of Physics and Astronomy, University of Missouri-Columbia, Columbia, MO 65211, USA Christian Heinicke‡ Institute for Theoretical Physics, University of Cologne, D–50923 K¨oln, Germany Alfredo Mac´ıas§ Departamento de F´ısica, Universidad Aut´onoma Metropolitana–Iztapalapa Apartado Postal 55–534, C.P. 09340, M´exico, D.F., M´exico (Dated: 20 January 2004) arXiv:gr-qc/0309008v2 9 Feb 2004 1 Abstract Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension n. We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components as n(n2 4)/3 for the first time. Subsequently, we exhibit its characteristic properties − and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einstein’s field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw, and Templeton. For each class examples are given. Finally we investigate the relation between the Cotton tensor and the energy-momentum in Einstein’s theory and derive a conformally flat perfect fluid solution of Einstein’s field equations in three dimensions. -
EXCALC: a System for Doing Calculations in the Calculus of Modern Differential Geometry
EXCALC: A System for Doing Calculations in the Calculus of Modern Differential Geometry Eberhard Schr¨ufer Institute SCAI.Alg German National Research Center for Information Technology (GMD) Schloss Birlinghoven D-53754 Sankt Augustin Germany Email: [email protected] March 20, 2004 Acknowledgments This program was developed over several years. I would like to express my deep gratitude to Dr. Anthony Hearn for his continuous interest in this work, and especially for his hospitality and support during a visit in 1984/85 at the RAND Corporation, where substantial progress on this package could be achieved. The Heinrich Hertz-Stiftung supported this visit. Many thanks are also due to Drs. F.W. Hehl, University of Cologne, and J.D. McCrea, University College Dublin, for their suggestions and work on testing this program. 1 Introduction EXCALC is designed for easy use by all who are familiar with the calculus of Modern Differential Geometry. Its syntax is kept as close as possible to standard textbook notations. Therefore, no great experience in writing computer algebra programs is required. It is almost possible to input to the computer the same as what would have been written down for a hand- 1 2 DECLARATIONS 2 calculation. For example, the statement f*x^y + u _| (y^z^x) would be recognized by the program as a formula involving exterior products and an inner product. The program is currently able to handle scalar-valued exterior forms, vectors and operations between them, as well as non-scalar valued forms (indexed forms). With this, it should be an ideal tool for studying differential equations, doing calculations in general relativity and field theories, or doing such simple things as calculating the Laplacian of a tensor field for an arbitrary given frame. -
Higher Order Derivatives 1 Chapter 3 Higher Order Derivatives
Higher order derivatives 1 Chapter 3 Higher order derivatives You certainly realize from single-variable calculus how very important it is to use derivatives of orders greater than one. The same is of course true for multivariable calculus. In particular, we shall de¯nitely want a \second derivative test" for critical points. A. Partial derivatives First we need to clarify just what sort of domains we wish to consider for our functions. So we give a couple of de¯nitions. n DEFINITION. Suppose x0 2 A ½ R . We say that x0 is an interior point of A if there exists r > 0 such that B(x0; r) ½ A. A DEFINITION. A set A ½ Rn is said to be open if every point of A is an interior point of A. EXAMPLE. An open ball is an open set. (So we are fortunate in our terminology!) To see this suppose A = B(x; r). Then if y 2 A, kx ¡ yk < r and we can let ² = r ¡ kx ¡ yk: Then B(y; ²) ½ A. For if z 2 B(y; ²), then the triangle inequality implies kz ¡ xk · kz ¡ yk + ky ¡ xk < ² + ky ¡ xk = r; so that z 2 A. Thus B(y; ²) ½ B(x; r) (cf. Problem 1{34). This proves that y is an interior point of A. As y is an arbitrary point of A, we conclude that A is open. PROBLEM 3{1. Explain why the empty set is an open set and why this is actually a problem about the logic of language rather than a problem about mathematics. -
Fundamentals of Tensor Calculus for Engineers with a Primer on Smooth Manifolds Solid Mechanics and Its Applications
Solid Mechanics and Its Applications Uwe Mühlich Fundamentals of Tensor Calculus for Engineers with a Primer on Smooth Manifolds Solid Mechanics and Its Applications Volume 230 Series editors J.R. Barber, Ann Arbor, USA Anders Klarbring, Linköping, Sweden Founding editor G.M.L. Gladwell, Waterloo, ON, Canada Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is to the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. More information about this series at http://www.springer.com/series/6557 Uwe Mühlich Fundamentals of Tensor Calculus for Engineers with a Primer on Smooth Manifolds 123 Uwe Mühlich Faculty of Applied Engineering University of Antwerp Antwerp Belgium ISSN 0925-0042 ISSN 2214-7764 (electronic) Solid Mechanics and Its Applications ISBN 978-3-319-56263-6 ISBN 978-3-319-56264-3 (eBook) DOI 10.1007/978-3-319-56264-3 Library of Congress Control Number: 2017935015 © Springer International Publishing AG 2017 This work is subject to copyright. -
Kronecker Products and Matrix Calculus in System Theory
772 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-25, NO. 9, ISEFTEMBER 1978 Kronecker Products and Matrix Calculus in System Theory JOHN W. BREWER I Absfrucr-The paper begins with a review of the algebras related to and are based on the developments of Section IV. Also, a Kronecker products. These algebras have several applications in system novel and interesting operator notation will ble introduced theory inclluding the analysis of stochastic steady state. The calculus of matrk valued functions of matrices is reviewed in the second part of the in Section VI. paper. This cakuhs is then used to develop an interesting new method for Concluding comments are placed in Section VII. the identification of parameters of linear time-invariant system models. II. NOTATION I. INTRODUCTION Matrices will be denoted by upper case boldface (e.g., HE ART of differentiation has many practical ap- A) and column matrices (vectors) will be denoted by T plications in system analysis. Differentiation is used lower case boldface (e.g., x). The kth row of a matrix such. to obtain necessaryand sufficient conditions for optimiza- as A will be denoted A,. and the kth colmnn will be tion in a.nalytical studies or it is used to obtain gradients denoted A.,. The ik element of A will be denoted ujk. and Hessians in numerical optimization studies. Sensitiv- The n x n unit matrix is denoted I,,. The q-dimensional ity analysis is yet another application area for. differentia- vector which is “I” in the kth and zero elsewhereis called tion formulas. In turn, sensitivity analysis is applied to the the unit vector and is denoted design of feedback and adaptive feedback control sys- tems. -
On Rotor Calculus, I 403 1
ON ROTOR CALCULUS I H. A. BUCHDAHL (Received 25 October 1965) Summary It is known that to every proper homogeneous Lorentz transformation there corresponds a unique proper complex rotation in a three-dimensional complex linear vector space, the elements of which are here called "rotors". Equivalently one has a one-one correspondence between rotors and self- dual bi-vectors in space-time (w-space). Rotor calculus fully exploits this correspondence, just as spinor calculus exploits the correspondence between real world vectors and hermitian spinors; and its formal starting point is the definition of certain covariant connecting quantities rAkl which trans- form as vectors under transformations in rotor space (r-space) and as tensors of valence 2 under transformations in w-space. In the present paper, the first of two, w-space is taken to be flat. The general properties of the rA*i are established in detail, without recourse to any special representation. Corresponding to any proper Lorentz transformation there exists an image in r-space, i.e. an ^-transformation such that the two transformations carried out jointly leave the xAkl numerically unchanged. Nevertheless, all relations are written in such a way that they are invariant under arbitrary ^-transformations and arbitrary r-transformations, which may be carried out independently of one another. For this reason the metric tensor a.AB in r-space may be chosen arbitrarily, except that it shall be non-singular and symmetric. A large number of identities involving the basic quan- tities of the calculus is presented, including some which relate to com- plex conjugated rotors rAkl, xAg.