DOCSLIB.ORG

The Vector Algebra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Vector Algebra 1 The vector algebra war: a historical perspective James M. Chappell, Azhar Iqbal, John G. Hartnett, and Derek Abbott, Fellow, IEEE Abstract—There are a wide variety of different vector for- that this system indeed provides the most natural vector system malisms currently utilized in engineering and physics. For for three-dimensional space. example, Gibbs’ three-vectors, Minkowski four-vectors, complex spinors in quantum mechanics, quaternions used to describe rigid body rotations and vectors defined in Clifford geometric algebra. II. ANALYSIS With such a range of vector formalisms in use, it thus appears The concept of vectorial quantities actually appears to that there is as yet no general agreement on a vector formalism suitable for science as a whole. This is surprising, in that, one of be quite an ancient one, with the parallelogram law for the primary goals of nineteenth century science was to suitably the addition of vectors well known to Aristotelian science describe vectors in three-dimensional space. This situation has from the fourth century B.C.E. [2] and later repeated in also had the unfortunate consequence of fragmenting knowledge Newton’s Principia. Descartes, in 1637 however, proposed a across many disciplines, and requiring a significant amount of much more radical view of vectors as quantities such that time and effort in learning the various formalisms. We thus historically review the development of our various vector systems ‘Just as arithmetic consists of only four or five operations, and conclude that Clifford’s multivectors best fulfills the goal of namely, addition, subtraction, multiplication, division and the describing vectorial quantities in three dimensions and providing extraction of roots....so in geometry, to find required lines a unified vector system for science. it is merely necessary to add or subtract lines.’ [3] This Index Terms—Vectors, Gibbs, Hamilton, Clifford, Multivectors revolutionary idea was indeed successfully formulated alge- braically in the nineteenth century by Wessel, Argand and Gauss. It was achieved through the use of complex numbers I. INTRODUCTION z = a + ib, where a,b are real numbers and i = √ 1 ∈ ℜ − ENERALLY speaking, the concept of a vector has been is the unit imaginary. The number z was then interpreted as G an extremely useful one with nearly all branches of representing a point in the plane located at the coordinates physical science now described in the language of vectors [1]. [a,b]. This point could also be viewed as representing a Despite its great value as a concept there is nevertheless vector extending out from the origin to this point. The ability a plethora of different vector formalisms currently in use. to use a single number ‘z’ to describe a two-dimensional Listed in the approximate order of their creation are: complex planar point means that we now can undertake geometrical numbers (planar vectors), quaternionic vectors, Gibbs’ vectors, analysis in a coordinate free manner. It is also compatible Minkowski four-vectors, complex spinors, Dirac matrix four- with intuition in having a single number ‘z’ to refer to a vectors and finally vectors defined using Clifford algebra, as single point, rather then needing to always refer to two separate well as several other vector-type formalisms. coordinates. For example, if we have two vectors represented This fact is surprising as one of the main goals of nineteenth by the complex numbers z1 and z2 then vectorial addition is century science was to find a suitable vector system for simply z1 + z2. That is, if z1 = a1 + ib1 and z2 = a2 + ib2 three-dimensional Euclidean space. This objective was initially then z1 + z2 = (a1 + a2)+ i(b1 + b2) and so satisfies the led by Hamilton who produced the quaternionic vectors. parallelogram law for adding vectorial quantities, as required. Unfortunately Hamilton’s system failed to live up to the We thus have extended real numbers to a more general type arXiv:1509.00501v2 [physics.hist-ph] 22 Apr 2016 initial high expectations and following an intense debate over of number with the addition of the imaginary component with several years, it was replaced by the Gibbs vector system in all fundamental arithmetic operations essentially unchanged. mainstream use today. We firstly identify why the quaternions Indeed, complex numbers are a division algebra and so satisfy fail to produce a suitable description of Cartesian vectors but Descartes vision of vectorial quantities being amenable to also importantly identify what their natural role is. We then the four common arithmetic operations. This principle of show the serious failings of the Gibbs vector system before Descartes was also consistent with a later principle by Hankel demonstrating a reconciliation of these two rival systems for extending mathematical concepts, of the the principle of 3 the permanence of the rules of calculation [4]. within Clifford geometric algebra Cℓ( ). We then conclude ℜ One defect of this approach to representing a Cartesian James M. Chappell, Azhar Iqbal and Derek Abbott are with the School vector by a complex number is that we are setting up a real and of Electrical and Electronic Engineering, University of Adelaide, SA 5005, an imaginary axis for the plane that is clearly not isotropic1 Australia e-mail: [email protected]. John G. Hartnett is with the Institute for Photonics & Advanced Sensing and so somewhat inconsistent with the principles of relativity. (IPAS), and the School of Physical Sciences, University of Adelaide, SA 5005 Australia. 1Isotropy implies that for an isolated physical system experimental out- Manuscript received November 13, 2015; revised: XX YY, 2015. comes are independent of its orientation in space. 2 Complex numbers actually describe the algebra of rotations It thus appeared at first that quaternions may indeed be for the plane, which is two dimensional. Hence this has the the ideal algebra for three-dimensional physical space that same dimension as a two-dimensional vector space. This is had been sought. Unfortunately there was one cloud on the why complex numbers can doubly serve as rotation operators horizon, the fact that a vector quaternion squares to the nega- as well as vectors for the plane. That is, a rotation of a vector tive Pythagorean length. Indeed, Maxwell commented on this given by a complex number z can be written unusual fact, noting that the kinetic energy, which involves the ′ square of the velocity vector, would therefore be negative [9]. z =eiθz. (1) Maxwell, despite these reservations, formulated the equations In this case both the rotation operator eiθ and the vector z are of electromagnetism in quaternionic form. Maxwell, however, represented by complex numbers. backed away from a complete endorsement of the quaternions, However, following the generally successful use of complex recommending in his treatise on electricity and magnetism ‘the numbers in describing vectors in the plane, researchers of the introduction of the ideas, as distinguished from the operations nineteenth century then turned their attention to the generaliza- and methods of Quaternions’ [10]. tion of complex numbers to three-dimensional space in order The difficulties with quaternions led to a breakaway for- to naturally describe these vectors. malism of the Gibbs vector system. Gibbs considered that the Hamilton led this program, and in 1843 he succeeded in most useful function of the quaternions was in their forming generalizing the complex number algebra to the quaternion of the dot product and the cross product operations. That algebra [5]. A quaternion can be written is, for two vector quaternions v = v1i + v2j + v3k and w = w1i+w2j+w3k we find through expanding the brackets, q = a + v1i + v2j + v3k, (2) that where the three basis vectors have a negative square i2 = 2 2 vw i j k i j k j = k = 1 and are anticommuting with each other. = (v1 + v2 + v3 )(w1 + w2 + w3 ) (4) − Hamilton’s quaternions form a four-dimensional associative = v1w1 v2w2 v3w3 + (v2w3 v3w2)i H − − − − division algebra over the real numbers represented by . Once +(v3w1 v1w3)j + (v1w2 v2w1)k − − again being a division algebra, like complex numbers, they = v w + v w, are amenable to all the common arithmetic operations. The − · × sequence of algebras , C and H are constructed to be division using the negative square and the anticommuting properties of algebras, that is, theyℜ are closed with an inverse operation. the basis vectors. We can also see how the vector quaternion Indeed, the required algebraic rules for quaternions follow squared v2 = v v is thus the negative of the Pythagorean from this closure property [6]. These properties also, in fact, length as noted− previously.· We can see though how indeed make them naturally suited to describe rotations in space as the dot and cross products naturally arise from the product of they also have the closure property. two vector quaternions. Gibbs then considered that adopting Quaternions are also isotropic, as required, with a three the separate operations of the dot and cross products acting dimensional ‘vector’ represented as v = v1i + v2j + v3k. on three-vectors could thus form the basis for a more efficient Hamilton then claimed that being a generalization of complex and straightforward vector system. numbers to three dimensions it would therefore logically be This led to an intense and lengthy debate over several years the appropriate algebra to describe three-dimensional space. between the followers of Gibbs and the followers of Hamilton, Indeed he proposed, many years before Einstein or Minkowski, beginning in 1890, over the most efficient vectorial system to that if the scalar a, in Eq. (2), was identified with time then be adopted in mathematical physics [1]. The supporters of the four-dimensional quaternion can be a representation for Hamilton were able to claim that quaternions being general- a unified spacetime framework [7].
Load more

Recommended publications
  • Quaternions and Cli Ord Geometric Algebras
    Quaternions and Cliord Geometric Algebras Robert Benjamin Easter First Draft Edition (v1) (c) copyright 2015, Robert Benjamin Easter, all rights reserved. Preface As a rst rough draft that has been put together very quickly, this book is likely to contain errata and disorganization. The references list and inline citations are very incompete, so the reader should search around for more references. I do not claim to be the inventor of any of the mathematics found here. However, some parts of this book may be considered new in some sense and were in small parts my own original research. Much of the contents was originally written by me as contributions to a web encyclopedia project just for fun, but for various reasons was inappropriate in an encyclopedic volume. I did not originally intend to write this book. This is not a dissertation, nor did its development receive any funding or proper peer review. I oer this free book to the public, such as it is, in the hope it could be helpful to an interested reader. June 19, 2015 - Robert B. Easter. (v1) [email protected] 3 Table of contents Preface . 3 List of gures . 9 1 Quaternion Algebra . 11 1.1 The Quaternion Formula . 11 1.2 The Scalar and Vector Parts . 15 1.3 The Quaternion Product . 16 1.4 The Dot Product . 16 1.5 The Cross Product . 17 1.6 Conjugates . 18 1.7 Tensor or Magnitude . 20 1.8 Versors . 20 1.9 Biradials . 22 1.10 Quaternion Identities . 23 1.11 The Biradial b/a .
    [Show full text]
  • Arxiv:1001.0240V1 [Math.RA]
    Fundamental representations and algebraic properties of biquaternions or complexified quaternions Stephen J. Sangwine∗ School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom. Email: [email protected] Todd A. Ell† 5620 Oak View Court, Savage, MN 55378-4695, USA. Email: [email protected] Nicolas Le Bihan GIPSA-Lab D´epartement Images et Signal 961 Rue de la Houille Blanche, Domaine Universitaire BP 46, 38402 Saint Martin d’H`eres cedex, France. Email: [email protected] October 22, 2018 Abstract The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner and outer products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically. 1 Introduction It is typical of quaternion formulae that, though they be difficult to find, once found they are immediately verifiable. J. L. Synge (1972) [43, p34] arXiv:1001.0240v1 [math.RA] 1 Jan 2010 The quaternions are relatively well-known but the quaternions with complex components (complexified quaternions, or biquaternions1) are less so. This paper aims to set out the fundamental definitions of biquaternions and some elementary results, which, although elementary, are often not trivial. The emphasis in this paper is on the biquaternions as an applied algebra – that is, a tool for the manipulation ∗This paper was started in 2005 at the Laboratoire des Images et des Signaux (now part of the GIPSA-Lab), Grenoble, France with financial support from the Royal Academy of Engineering of the United Kingdom and the Centre National de la Recherche Scientifique (CNRS).
    [Show full text]
  • Vectors and Beyond: Geometric Algebra and Its Philosophical
    dialectica Vol. 63, N° 4 (2010), pp. 381–395 DOI: 10.1111/j.1746-8361.2009.01214.x Vectors and Beyond: Geometric Algebra and its Philosophical Significancedltc_1214 381..396 Peter Simons† In mathematics, like in everything else, it is the Darwinian struggle for life of ideas that leads to the survival of the concepts which we actually use, often believed to have come to us fully armed with goodness from some mysterious Platonic repository of truths. Simon Altmann 1. Introduction The purpose of this paper is to draw the attention of philosophers and others interested in the applicability of mathematics to a quiet revolution that is taking place around the theory of vectors and their application. It is not that new math- ematics is being invented – on the contrary, the basic concepts are over a century old – but rather that this old theory, having languished for many decades as a quaint backwater, is being rediscovered and properly applied for the first time. The philosophical importance of this quiet revolution is not that new applications for old mathematics are being found. That presumably happens much of the time. Rather it is that this new range of applications affords us a novel insight into the reasons why vectors and their mathematical kin find application at all in the real world. Indirectly, it tells us something general but interesting about the nature of the spatiotemporal world we all inhabit, and that is of philosophical significance. Quite what this significance amounts to is not yet clear. I should stress that nothing here is original: the history is quite accessible from several sources, and the mathematics is commonplace to those who know it.
    [Show full text]
  • Recent Books on Vector Analysis. 463
    1921.] RECENT BOOKS ON VECTOR ANALYSIS. 463 RECENT BOOKS ON VECTOR ANALYSIS. Einführung in die Vehtoranalysis, mit Anwendungen auf die mathematische Physik. By R. Gans. 4th edition. Leip­ zig and Berlin, Teubner, 1921. 118 pp. Elements of Vector Algebra. By L. Silberstein. New York, Longmans, Green, and Company, 1919. 42 pp. Vehtor analysis. By C. Runge. Vol. I. Die Vektoranalysis des dreidimensionalen Raumes. Leipzig, Hirzel, 1919. viii + 195 pp. Précis de Calcul géométrique. By R. Leveugle. Preface by H. Fehr. Paris, Gauthier-Villars, 1920. lvi + 400 pp. The first of these books is for the most part a reprint of the third edition, which has been reviewed in this BULLETIN (vol. 21 (1915), p. 360). Some small condensation of results has been accomplished by making deductions by purely vector methods. The author still clings to the rather common idea that a vector is a certain triple on a certain set of coordinate axes, so that his development is rather that of a shorthand than of a study of expressions which are not dependent upon axes. The second book is a quite brief exposition of the bare fundamentals of vector algebra, the notation being that of Heaviside, which Silberstein has used before. It is designed primarily to satisfy the needs of those studying geometric optics. However the author goes so far as to mention a linear vector operator and the notion of dyad. The exposition is very simple and could be followed by high school students who had had trigonometry. The third of the books mentioned is an elementary exposition of vectors from the standpoint of Grassmann.
    [Show full text]
  • Geometric-Algebra Adaptive Filters Wilder B
    1 Geometric-Algebra Adaptive Filters Wilder B. Lopes∗, Member, IEEE, Cassio G. Lopesy, Senior Member, IEEE Abstract—This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are Faces generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well- Edges suited for the description of geometric transformations. Also, (directed lines) differently from standard adaptive-filtering theory, Geometric Calculus (the extension of GA to differential calculus) allows Fig. 1. A polyhedron (3-dimensional polytope) can be completely described for applying the same derivation techniques regardless of the by the geometric multiplication of its edges (oriented lines, vectors), which type (subalgebra) of the data, i.e., real, complex numbers, generate the faces and hypersurfaces (in the case of a general n-dimensional quaternions, etc. Relying on those characteristics (among others), polytope). a deterministic quadratic cost function is posed, from which the GAAFs are devised, providing a generalization of regular adaptive filters to subalgebras of GA. From the obtained update rule, it is shown how to recover the following least-mean squares perform calculus with hypercomplex quantities, i.e., elements (LMS) adaptive filter variants: real-entries LMS, complex LMS, that generalize complex numbers for higher dimensions [2]– and quaternions LMS. Mean-square analysis and simulations in [10]. a system identification scenario are provided, showing very good agreement for different levels of measurement noise. GA-based AFs were first introduced in [11], [12], where they were successfully employed to estimate the geometric Index Terms—Adaptive filtering, geometric algebra, quater- transformation (rotation and translation) that aligns a pair of nions.
    [Show full text]
  • Arithmetic of Hyperbolic 3-Manifolds
    Arithmetic of Hyperbolic 3-Manifolds Alan Reid Rice University Hyperbolic 3-Manifolds and Discrete Groups Hyperbolic 3-space can be defined as 3 H = f(z; t) 2 C × R : t > 0g dsE and equipped with the metric ds = t . Geodesics are vertical lines perpendicular to C or semi-circles perpendicular to C. Codimension 1 geodesic submanifolds are Euclidean planes in 3 H orthogonal to C or hemispheres centered on C. 3 The full group of orientation-preserving isometries of H can be identified with PSL(2; C). Abuse of notation We will often view Kleinian groups as groups of matrices! A discrete subgroup of PSL(2; C) is called a Kleinian group. 3 Such a group acts properly discontinuously on H . If Γ is torsion-free (does not contain elements of finite order) then Γ acts freely. 3 In this latter case we get a quotient manifold H =Γ and a 3 3 covering map H ! H =Γ which is a local isometry. 3 We will be interested in the case when H =Γ has finite volume. Note: If Γ is a finitely generated subgroup of PSL(2; C) there is always a finite index subgroup that is torsion-free. Examples 1. Let d be a square-freep positive integer, and Od the ring of algebraic integers in Q( −d). Then PSL(2; Od) is a Kleinian group. 3 Indeed H =PSL(2; Od) has finite volume but is non-compact. These are known as the Bianchi groups. d = 1 (picture by Jos Leys) 2. Reflection groups. Tessellation by all right dodecahedra (from the Not Knot video) 3.Knot Complements The Figure-Eight Knot Thurston's hyperbolization theorem shows that many knots have complements that are hyperbolic 3-manifolds of finite volume.
    [Show full text]
  • Simple Circuit Theory and the Solution of Two Electricity Problems from The
    Simple circuit theory and the solution of two electricity problems from the Victorian Age A C Tort ∗ Departamento de F´ısica Te´orica - Instituto de F´ısica Universidade Federal do Rio de Janeiro Caixa Postal 68.528; CEP 21941-972 Rio de Janeiro, Brazil May 22, 2018 Abstract Two problems from the Victorian Age, the subdivision of light and the determination of the leakage point in an undersea telegraphic cable are discussed and suggested as a concrete illustrations of the relationships between textbook physics and the real world. Ohm’s law and simple algebra are the only tools we need to discuss them in the classroom. arXiv:0811.0954v1 [physics.pop-ph] 6 Nov 2008 ∗e-mail: [email protected]. 1 1 Introduction Some time ago, the present author had the opportunity of reading Paul J. Nahin’s [1] fascinating biog- raphy of the Victorian physicist and electrician Oliver Heaviside (1850-1925). Heaviside’s scientific life unrolls against a background of theoretical and technical challenges that the scientific and technological developments fostered by the Industrial Revolution presented to engineers and physicists of those times. It is a time where electromagnetic theory as formulated by James Clerk Maxwell (1831-1879) was un- derstood by only a small group of men, Lodge, FitzGerald and Heaviside, among others, that had the mathematical sophistication and imagination to grasp the meaning and take part in the great Maxwellian synthesis. Almost all of the electrical engineers, or electricians as they were called at the time, considered themselves as “practical men”, which effectively meant that most of them had a working knowledge of the electromagnetic phenomena spiced up with bits of electrical theory, to wit, Ohm’s law and the Joule effect.
    [Show full text]
  • The Concept of Field in the History of Electromagnetism
    The concept of field in the history of electromagnetism Giovanni Miano Department of Electrical Engineering University of Naples Federico II ET2011-XXVII Riunione Annuale dei Ricercatori di Elettrotecnica Bologna 16-17 giugno 2011 Celebration of the 150th Birthday of Maxwell’s Equations 150 years ago (on March 1861) a young Maxwell (30 years old) published the first part of the paper On physical lines of force in which he wrote down the equations that, by bringing together the physics of electricity and magnetism, laid the foundations for electromagnetism and modern physics. Statue of Maxwell with its dog Toby. Plaque on E-side of the statue. Edinburgh, George Street. Talk Outline ! A brief survey of the birth of the electromagnetism: a long and intriguing story ! A rapid comparison of Weber’s electrodynamics and Maxwell’s theory: “direct action at distance” and “field theory” General References E. T. Wittaker, Theories of Aether and Electricity, Longam, Green and Co., London, 1910. O. Darrigol, Electrodynamics from Ampère to Einste in, Oxford University Press, 2000. O. M. Bucci, The Genesis of Maxwell’s Equations, in “History of Wireless”, T. K. Sarkar et al. Eds., Wiley-Interscience, 2006. Magnetism and Electricity In 1600 Gilbert published the “De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure” (On the Magnet and Magnetic Bodies, and on That Great Magnet the Earth). ! The Earth is magnetic ()*+(,-.*, Magnesia ad Sipylum) and this is why a compass points north. ! In a quite large class of bodies (glass, sulphur, …) the friction induces the same effect observed in the amber (!"#$%&'(, Elektron). Gilbert gave to it the name “electricus”.
    [Show full text]
  • The Geometry and Topology of Arithmetic Hyperbolic 3-Manifolds
    The geometry and topology of arithmetic hyperbolic 3-manifolds Alan W. Reid∗ May 12, 2007 1 Introduction This paper is based on three lectures given by the author at the RIMS Symposium, “Topology, Com- plex Analysis and Arithmetic of Hyperbolic Spaces” held at the Research Institute for Mathematical Sciences, Kyoto University, in December 2006. The goal of the lectures was to describe recent work on understanding topological and geometric properties of arithmetic hyperbolic 3-manifolds, and the connections with number theory. This is the theme of the paper. Our discussion of topological aspects of arithmetic hyperbolic 3-orbifolds is motivated by the following central conjectures from 3-manifold topology: Conjecture 1.1. Let M be a closed hyperbolic 3-manifold, then M is virtually Haken; ie M has a finite sheeted cover N which contains embedded incompressible surface (necessarily of genus at least 2). Conjecture 1.2. Let M be a closed hyperbolic 3-manifold, then M has a finite sheeted cover N for which b1(N) > 0 (where b1(N) is the first Betti number of N). We can also rephrase Conjecture 1.2 to say that vb1(M) > 0 where vb1(M) is defined by vb1(M) = sup{b1(N) : N is a finite cover of M}, and is called the virtual first Betti number of M. Conjecture 1.3. Let M be a closed hyperbolic 3-manifold, then vb1(M) = ∞. Conjecture 1.4. Let M be a closed hyperbolic 3-manifold, then π1(M) is large; that is to say, some finite index subgroup of π1(M) admits a surjective homomorphism onto a non-abelian free group.
    [Show full text]
  • Josiah Willard Gibbs
    GENERAL ARTICLE Josiah Willard Gibbs V Kumaran The foundations of classical thermodynamics, as taught in V Kumaran is a professor textbooks today, were laid down in nearly complete form by of chemical engineering at the Indian Institute of Josiah Willard Gibbs more than a century ago. This article Science, Bangalore. His presentsaportraitofGibbs,aquietandmodestmanwhowas research interests include responsible for some of the most important advances in the statistical mechanics and history of science. fluid mechanics. Thermodynamics, the science of the interconversion of heat and work, originated from the necessity of designing efficient engines in the late 18th and early 19th centuries. Engines are machines that convert heat energy obtained by combustion of coal, wood or other types of fuel into useful work for running trains, ships, etc. The efficiency of an engine is determined by the amount of useful work obtained for a given amount of heat input. There are two laws related to the efficiency of an engine. The first law of thermodynamics states that heat and work are inter-convertible, and it is not possible to obtain more work than the amount of heat input into the machine. The formulation of this law can be traced back to the work of Leibniz, Dalton, Joule, Clausius, and a host of other scientists in the late 17th and early 18th century. The more subtle second law of thermodynamics states that it is not possible to convert all heat into work; all engines have to ‘waste’ some of the heat input by transferring it to a heat sink. The second law also established the minimum amount of heat that has to be wasted based on the absolute temperatures of the heat source and the heat sink.
    [Show full text]
  • 1 Vectors & Tensors
    1 Vectors & Tensors The mathematical modeling of the physical world requires knowledge of quite a few different mathematics subjects, such as Calculus, Differential Equations and Linear Algebra. These topics are usually encountered in fundamental mathematics courses. However, in a more thorough and in-depth treatment of mechanics, it is essential to describe the physical world using the concept of the tensor, and so we begin this book with a comprehensive chapter on the tensor. The chapter is divided into three parts. The first part covers vectors (§1.1-1.7). The second part is concerned with second, and higher-order, tensors (§1.8-1.15). The second part covers much of the same ground as done in the first part, mainly generalizing the vector concepts and expressions to tensors. The final part (§1.16-1.19) (not required in the vast majority of applications) is concerned with generalizing the earlier work to curvilinear coordinate systems. The first part comprises basic vector algebra, such as the dot product and the cross product; the mathematics of how the components of a vector transform between different coordinate systems; the symbolic, index and matrix notations for vectors; the differentiation of vectors, including the gradient, the divergence and the curl; the integration of vectors, including line, double, surface and volume integrals, and the integral theorems. The second part comprises the definition of the tensor (and a re-definition of the vector); dyads and dyadics; the manipulation of tensors; properties of tensors, such as the trace, transpose, norm, determinant and principal values; special tensors, such as the spherical, identity and orthogonal tensors; the transformation of tensor components between different coordinate systems; the calculus of tensors, including the gradient of vectors and higher order tensors and the divergence of higher order tensors and special fourth order tensors.
    [Show full text]
  • On the Tensor Product of Two Composition Algebras
    On The Tensor Product of Two Composition Algebras Patrick J. Morandi S. Pumplun¨ 1 Introduction Let C1 F C2 be the tensor product of two composition algebras over a field F with char(F ) =¡ 2. R. Brauer [7] and A. A. Albert [1], [2], [3] seemed to be the first math- ematicians who investigated the tensor product of two quaternion algebras. Later their results were generalized to this more general situation by B. N. Allison [4], [5], [6] and to biquaternion algebras over rings by Knus [12]. In the second section we give some new results on the Albert form of these algebras. We also investigate the F -quadric defined by this Albert form, generalizing a result of Knus ([13]). £ £ Since Allison regarded the involution ¢ = 1 2 as an essential part of the algebra C = C1 C2, he only studied automorphisms of C which are compatible with ¢ . In the last section we show that any automorphism of C that preserves a certain biquaternion subalgebra also is compatible with ¢ . As a consequence, if C is the tensor product of two octonion algebras, we show that C does not satisfy the Skolem-Noether Theorem. Let F be a field and C a unital, nonassociative F -algebra. Then C is a composition algebra if there exists a nondegenerate quadratic form n : C ¤ F such that n(x · y) = n(x)n(y) for all x, y ¥ C. The form n is uniquely determined by these conditions and is called the norm of C. We will write n = nC . Composition algebras only exist in ranks 1, 2, 4 or 8 (see [10]).
    [Show full text]