DOCSLIB.ORG

CLASS of ELECTROMAGNETIC SQ-MEDIA I. V. Lindell Department of Radio Science and Engineering School of Science and Technology, Aa

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
CLASS of ELECTROMAGNETIC SQ-MEDIA I. V. Lindell Department of Radio Science and Engineering School of Science and Technology, Aa Progress In Electromagnetics Research, Vol. 110, 371{382, 2010 CLASS OF ELECTROMAGNETIC SQ-MEDIA I. V. Lindell Department of Radio Science and Engineering School of Science and Technology, Aalto University P. O. Box 13000, Espoo 00076 AALTO, Finland Abstract|A novel class of electromagnetic media called that of SQ- media is de¯ned in terms of compact four-dimensional di®erential-form formalism. The medium class lies between two known classes, that of Q-media and SD-media (also called self-dual media). Eigen¯elds for the de¯ned medium dyadic are derived and shown to be uncoupled in a homogeneous medium. However, energy transport requires their interaction. The medium shares the nonbirefringence property of the Q-media (not shared by the SD media) and the eigen¯eld decomposition property of the SD media (not shared by the Q-media). Comparison of the three medium classes is made in terms of their three-dimensional medium dyadics. 1. INTRODUCTION The most general linear electromagnetic medium (bi-anisotropic medium) can be expressed in terms of four medium dyadics in the three-dimensional Gibbsian vector representation as [1, 2] µ ¶ µ ¶ µ ¶ D ¹² ¹» E = ¢ ; (1) B ³¹ ¹ H and the maximum number of free parameters is 4 £ 9 = 36. The four- dimensional di®erential-form representation of electromagnetic ¯elds as two-forms [3{5], elements of the space F2 © = B + E ^ d¿; ª = D ¡ H ^ d¿; (2) allows one to write the constitutive Equation (1) in the more compact form [5] ª = M¹ j©; (3) Received 6 October 2010, Accepted 8 November 2010, Scheduled 25 November 2010 Corresponding author: Ismo Veikko Lindell (ismo.lindell@tkk.¯). 372 Lindell or ¹ eN bª = M¹ gj©: (4) Here, eN = e1234 2 E4 denotes the quadrivector in the basis of vectors ei 2 E1 and b denotes the contraction operation. The reciprocal basis one-forms "j 2 F1 satisfy eij"j = ±ij and "N = "1234, where "4 = d¿ corresponds to the temporal one-form. The medium dyadic M¹ 2 F2E2 maps two-forms to two-forms and the modi¯ed medium dyadic ¹ ¹ M¹ g = eN bM¹ 2 E2E2 maps two-forms to bivectors. Basis expansions of both dyadics correspond to the same 6 £ 6 matrix. De¯nitions and operational rules for di®erential forms, multivectors and dyadics applied in this study have been summarized in the Appendices of [6, 7] and, more extensively, in the book [5]. The most general medium dyadic can be uniquely decomposed in three components as introduced by Hehl and Obukhov [4], ¹ ¹ ¹ ¹ M¹ = M¹ 1 + M¹ 2 + M¹ 3; (5) called principal (1), skewon (2) and axion (3) parts of M.¹ The axion ¹ ¹(2)T part M¹ 3 is a multiple of the unit dyadic ¹I mapping any two-form to itself and the other parts are trace free. The skewon part is de¯ned so ¹ that the corresponding modi¯ed medium dyadic M¹ g2 is antisymmetric, ¹ ¹ while the principal part M¹ 1 is trace free and M¹ g1 is symmetric. Media with vanishing components can be called accordingly, e.g., a medium ¹ ¹ de¯ned by M¹ = M¹ 1 is called a principal medium and one with ¹ ¹ ¹ M¹ = M¹ 2 + M¹ 3 is called a skewon-axion medium. Four-dimensional formalism allows simple de¯nition of important classes of electromagnetic media. For example, if the modi¯ed medium ¹ dyadic can be expressed in terms of some dyadic Q¹ 2 E1E1 mapping one-forms to vectors as 1 M¹ = Q¹ ^Q¹ = Q¹ (2); (6) g 2 ^ it is called a Q-medium, which has the property of being non- birefringent to propagating waves [5, 8]. Thus, media in this class can be conceived as generalizations of isotropic media. A more general medium class was called that of generalized Q-media and de¯ned by medium dyadics of the form [14] ¹ ¹ (2) M¹ g = Q¹ + AB; (7) where A; B 2 E2 are two bivectors. Such media were shown to coincide with the class of decomposable media, de¯ned in terms of Gibbsian three-dimensional dyadics [2, 9] in a more complicated way [10]. In such a medium any ¯eld can be decomposed in two noncoupling parts, Progress In Electromagnetics Research, Vol. 110, 2010 373 which is a generalization of the well-known TE/TM decomposition of ¯elds in simpler media. Other important classes of media arise from simple conditions satis¯ed by the medium condition. It is known that any medium dyadic M¹ like any 6 £ 6 matrix satis¯es an algebraic equation of the sixth order. Medium dyadics satisfying equations of lower order de¯ne certain classes of media. First-order equations, M¹ + A¹I(2)T = 0; (8) obviously de¯ne axion media. Axion media were also called by the name perfect electromagnetic conductor (PEMC), because they unify and generalize the concepts of perfect electric and magnetic conductor (PEC and PMC). Another interesting class of media is de¯ned by medium dyadics satisfying the second-order equation, M¹ 2 + AM¹ + B¹I(2)T = 0; (9) for some parameters A and B. Excluding the axion media, such a class was called that of SD-media, or self-dual media [12, 13], because one can de¯ne a linear (duality) transformation of electromagnetic ¯elds in which the medium appears invariant. It is the purpose of the present paper to introduce a class of media which appears to be in between those of Q-media and SD-media in sharing properties of both of these medium classes. 2. SQ-MEDIA 2.1. Properties of Some Dyadics The contraction dyadic transforming two-forms to bivectors à !(2)T j j X4 j X k ¹(2)T ¹(2) eN ¹I = eN ei"i = eN "ij eij = ¹I eN i i<j = e12e34 + e23e14 + e31e24 + e14e23 + e24e31 + e34e12 (10) is symmetric: ³ j ´T j k ¹(2)T ¹(2)T ¹(2) eN ¹I = eN ¹I = ¹I eN ; (11) and its inverse can be expressed as ³ j ´¡1 k j j X ¹(2)T ¹(2)T ¹(2) eN ¹I = ¹I "N = "N ¹I = "N eij "ij : (12) i<j 374 Lindell ¹ ¡1 ¹ The inverse Q¹ 2 F1F1 of a dyadic Q¹ 2 E1E1 can be expressed as [5] jj ¹ ¡1 1 ¹ (3)T Q¹ = "N "N Q¹ : (13) ¢Q This rule requires that the determinant-like quantity ¹ (4) ¢Q = "N "N jjQ¹ (14) be nonzero. In the following we assume that the Q¹ dyadic is normalized by assuming ¢Q = 1: (15) ¹ (2) The inverse of the double-wedge square dyadic Q¹ 2 E2E2, denoted by ³ ´¡1 ³ ´(2) Q¹ (2) = Q¹ ¡1 = Q¹ (¡2); (16) can then be obtained through the rule [5] jj jj ¹ (¡2) 1 ¹ (2)T ¹ (2)T Q¹ = "N "N Q¹ = "N "N Q¹ : (17) ¢Q The converse rule is jj ¹ (2) ¹ (¡2)T Q¹ = eN eN Q¹ : (18) ¹ Introducing a dyadic K¹ 2 F2E2 mapping two-forms to two-forms by j ³ j ´ ¯ ¹ ¹ (2) ¹(2) ¯ ¹ (2) K¹ = "N Q¹ = "N ¹I ¯Q¹ : (19) we have ¯ ³ j ´¡1 k j ³ jj ´ ¹ ¡1 ¹ (¡2)¯ ¹(2) ¹ (¡2) ¹ (¡2) K¹ = Q¹ ¯ "N ¹I = Q¹ eN = "N eN eN Q¹ j jj ¹ (2)T ¹ T = "N Q¹ = "N eN K¹ ; (20) which gives rise to the rule ¯ ³ j ´ ³ k ´ ¯ ³ j ´ ¯ j ¹ T ¯ ¹ ¹ T ¯ ¹ ¹ ¡1 ¯ ¹ ¹(2)T K¹ ¯ eN K¹ = K¹ eN ¯K¹ = eN K¹ ¯K¹ = eN ¹I : (21) A natural dot product of two-forms ¡1 and ¡2 yielding a scalar, de¯ned as ¯ ¯ ¡ ¥ ¢ ¡1 ¢ ¡2 = eN ¯ (¡1 ^ ¡2) = ¡1¯ eN ¡2 = ¡2 ¢ ¡1; (22) satis¯es ³ ¯ ´ ¯ j ³ ¯ ´ ¯ ¯ ³ j ´ ¯ ¯ ¡ ¥ ¢ ¹ ¯ ¯ ¹ ¯ ¯ ¹ T ¯ ¹ K¹ ¯¡1 ¯eN K¹ ¯¡2 = ¡1¯K¹ ¯ eN K¹ ¯¡2 = ¡1¯ eN ¡2 ; (23) or ³ ¯ ´ ³ ¯ ´ ¹ ¯ ¹ ¯ K¹ ¯¡1 ¢ K¹ ¯¡2 = ¡1 ¢ ¡2: (24) Progress In Electromagnetics Research, Vol. 110, 2010 375 Thus, mapping by the dyadic K¹ does not change the dot product of two-forms. In this, it resembles a rotation or reflection operation. In particular, a simple¯ two-form ¡ satisfying ¡ ¢ ¡ = 0, is mapped to a simple two-form K¹ ¯¡. 2.2. De¯nition of SQ-media Let us now assume that Q¹ (2) is a symmetric dyadic. In terms of ¹ ¹ ¹ symmetric and antisymmetric parts, Q = Qs + Qa, this requires that ¹ ^ ¹ ¹ (2) the condition Qs^Qa = 0 be satis¯ed. For symmetric Q we have jj jj j j ¹ T ¹ (2) T ¹ (2) ¹ "N eN K¹ = "N eN ("N Q¹ ) = "N Q¹ = K¹ ; (25) whence (20) becomes K¹ ¡1 = K¹ ; ) K¹ 2 = ¹I(2)T : (26) Thus, the dyadic K¹ acts as a square root of the unit dyadic. Of course, there are other square roots as well, like the unit dyadic itself. Let us consider an extension to Q-media with symmetric Q¹ (2) by adding a multiple of the unit dyadic in the medium dyadic, i.e., de¯ning M¹ = ®¹I(2)T + ¯K¹ : (27) Here, we exclude the axion medium special case by assuming ¯ 6= 0. Any medium de¯ned by a medium dyadic of the form (27), based by symmetric Q¹ (2), will now called an SQ-medium for brevity. The product of two SQ-medium dyadics satis¯es ³ ´ ¯ ³ ´ ¹ ¹ ¹(2)T ¹ ¯ ¹(2)T ¹ M¹ 1jM¹ 2 = ®1¹I + ¯1K¹ ¯ ®2¹I + ¯2K¹ ¹(2)T ¹ ¹ ¹ = (®1®2 + ¯1¯2)¹I + (®1¯2 + ®2¯1) K¹ = M¹ 2jM¹ 1: (28) De¯ning ®i = Mi cosh θi; ¯i = Mi sinh θi; i = 1; 2; (29) the rule can be cast in the more suggestive form ³ ´ ¹ ¹ ¹(2)T ¹ M¹ 1jM¹ 2 = M1M2 cosh(θ1 + θ2)¹I + sinh(θ1 + θ2)K¹ : (30) Thus, the medium dyadic of any SQ-medium acts as a hyperbolic rotation dyadic multiplied by a magnitude coe±cient. Actually, we can write more compactly ¹ M¹ = MeθK; (31) 376 Lindell when the expotential function of a dyadic is understood in terms of its power series.
Load more

Recommended publications
  • S-96.510 Advanced Field Theory Course for Graduate Students Lecture Viewgraphs, Fall Term 2004
    S-96.510 Advanced Field Theory Course for graduate students Lecture viewgraphs, fall term 2004 I.V.Lindell Helsinki University of Technology Electromagnetics Laboratory Espoo, Finland I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 00.01 1 Contents [01] Complex Vectors and Dyadics [02] Dyadic Algebra [03] Basic Electromagnetic Equations [04] Conditions for Fields and Media [05] Duality Transformation [06] Affine Transformation [07] Electromagnetic Field Solutions [08] Singularities and Complex Sources [09] Plane Waves [10] Source Equivalence [11] Huygens’ Principle [12] Field Decompositions Vector Formulas, Dyadic Identites as an appendix I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 00.02 2 Foreword This lecture material contains all viewgraphs associated with the gradu- ate course S-96.510 Advanced Field Theory given at the Department of Electrical and Communications Engineering, fall 2004. The course is based on Chapters 1–6 of the book Methods for Electromagnetic Field Analysis (Oxford University Press 1992, 2nd edition IEEE Press, 1995, 3rd printing Wiley 2002) by this author. The figures drawn by hand on the blackboard could not be added to the present material. Otaniemi, September 13 2004 I.V. Lindell I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 00.03 3 I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 00.04 4 S-96.510 Advanced Field Theory 1. Complex Vectors and Dyadics I.V.Lindell I.V.Lindell: Advanced Field Theory, 2004
    [Show full text]
  • Concept of a Dyad and Dyadic: Consider Two Vectors a and B Dyad: It Consists of a Pair of Vectors a B for Two Vectors a a N D B
    1/11/2010 CHAPTER 1 Introductory Concepts • Elements of Vector Analysis • Newton’s Laws • Units • The basis of Newtonian Mechanics • D’Alembert’s Principle 1 Science of Mechanics: It is concerned with the motion of material bodies. • Bodies have different scales: Microscropic, macroscopic and astronomic scales. In mechanics - mostly macroscopic bodies are considered. • Speed of motion - serves as another important variable - small and high (approaching speed of light). 2 1 1/11/2010 • In Newtonian mechanics - study motion of bodies much bigger than particles at atomic scale, and moving at relative motions (speeds) much smaller than the speed of light. • Two general approaches: – Vectorial dynamics: uses Newton’s laws to write the equations of motion of a system, motion is described in physical coordinates and their derivatives; – Analytical dynamics: uses energy like quantities to define the equations of motion, uses the generalized coordinates to describe motion. 3 1.1 Vector Analysis: • Scalars, vectors, tensors: – Scalar: It is a quantity expressible by a single real number. Examples include: mass, time, temperature, energy, etc. – Vector: It is a quantity which needs both direction and magnitude for complete specification. – Actually (mathematically), it must also have certain transformation properties. 4 2 1/11/2010 These properties are: vector magnitude remains unchanged under rotation of axes. ex: force, moment of a force, velocity, acceleration, etc. – geometrically, vectors are shown or depicted as directed line segments of proper magnitude and direction. 5 e (unit vector) A A = A e – if we use a coordinate system, we define a basis set (iˆ , ˆj , k ˆ ): we can write A = Axi + Ay j + Azk Z or, we can also use the A three components and Y define X T {A}={Ax,Ay,Az} 6 3 1/11/2010 – The three components Ax , Ay , Az can be used as 3-dimensional vector elements to specify the vector.
    [Show full text]
  • Magnetic Potential Green's Dyadics of Multilayered
    Progress In Electromagnetics Research, PIER 38, 125–146, 2002 MAGNETIC POTENTIAL GREEN’S DYADICS OF MULTILAYERED WAVEGUIDE FOR SPATIAL POWER COMBINING APPLICATIONS M. V. Lukic and A. B. Yakovlev Department of Electrical Engineering The University of Mississippi University, MS 38677-1848, USA Abstract—Integral equation formulation and magnetic potential Green’s dyadics for multilayered rectangular waveguide are presented for modeling interacting printed antenna arrays used in waveguide- based spatial power combiners. Dyadic Green’s functions are obtained as a partial eigenfunction expansion in the form of a double series over the complete system of eigenfunctions of transverse Laplacian operator. In this expansion, one-dimensional characteristic Green’s functions along a multilayered waveguide are derived in closed form as the solution of a Sturm-Liouville boundary value problem with appropriate boundary and continuity conditions. A method introduced here is based on the transmission matrix approach, wherein the amplitude coefficients of forward and backward traveling waves in the scattered Green’s function in different dielectric layers are obtained as a product of transmission matrices of corresponding layers. Convergence of Green’s function components in the source region is illustrated for a specific example of a two-layered, terminated rectangular waveguide. 1 Introduction 2 Integral Equation Formulation 3 Magnetic Potential Green’s Dyadics for a Semi-Infinite Multilayered Rectangular Waveguide 4 Numerical Results and Discussion 5 Conclusion Appendix A. Magnetic Potential Green’s Dyadics for a Semi-infinite Multilayered Rectangular Waveguide Used in the Magnetic-field Integral Equation Formu- lation 126 Lukic and Yakovlev Appendix B. Characteristic Green’s Functions for a Two- layered, Terminated Rectangular Waveguide References 1.
    [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]
  • Methods of Theoretical Physics
    METHODS OF THEORETICAL PHYSICS Philip M. Morse PROFESSOR OF PHYSICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY Herman Feshbach PROFESSOR OF PHYSICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY PART I: CHAPTERS 1 TO 8 FESHBACH PUBLISHING, LLC Minneapolis Contents PREFACE PART I CHAPTER 1 Types of Fields 1 1.1 Scalar Fields 4 Isotimic Surfaces. The Laplacian. 1.2 Vector Fields 8 Multiplication of Vectors. Axial Vectors. Lines of Flow. Potential Sur­ faces. Surface Integrals. Source Point. Line Integrals. Vortex Line. Singularities of Fields. 1.3 Curvilinear Coordinates 21 Direction Cosines. Scale Factors. Curvature of Coordinate Lines. The Volume Element and Other Formulas. Rotation of Axes. Law of Trans­ formation of Vectors. Contravariant and Covariant Vectors. 1.4 The Differential Operator v 31 The Gradient. Directional Derivative. Infinitesimal Rotation. The Divergence. Gauss' Theorem. A Solution of Poisson's Equation. The Curl. Vorticity Lines. Stokes' Theorem. The Vector Operator V. 1.5 Vector and Tensor Formalism 44 Covariant and Contravariant Vectors. Axial Vectors. Christoffel Sym­ bols. Covariant Derivative. Tensor Notation for Divergence and Curl. Other Differential Operators. Other Second-order Operators. Vector as a Sum of Gradient and Curl. 1.6 Dyadics and Other Vector Operators 54 Dyadics. Dyadics as Vector Operators. Symmetric and Antisymmetric Dyadics. Rotation of Axes and Unitary Dyadics. Dyadic Fields. Deformation of Elastic Bodies. Types of Strain. Stresses in an Elastic Medium. Static Stress-Strain Relations for an Isotropic Elastic Body. Dyadic Operators. Complex Numbers and Quaternions as Operators. Abstract Vector Spaces. Eigenvectors and Eigenvalues. Operators in Quantum Theory. Direction Cosines and Probabilities. Probabilities and Uncertainties. Complex Vector Space. Generalized Dyadics. Hermitian ix X Contents Operators.
    [Show full text]
  • Quaternions: a History of Complex Noncommutative Rotation Groups in Theoretical Physics
    QUATERNIONS: A HISTORY OF COMPLEX NONCOMMUTATIVE ROTATION GROUPS IN THEORETICAL PHYSICS by Johannes C. Familton A thesis submitted in partial fulfillment of the requirements for the degree of Ph.D Columbia University 2015 Approved by ______________________________________________________________________ Chairperson of Supervisory Committee _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ Program Authorized to Offer Degree ___________________________________________________________________ Date _______________________________________________________________________________ COLUMBIA UNIVERSITY QUATERNIONS: A HISTORY OF COMPLEX NONCOMMUTATIVE ROTATION GROUPS IN THEORETICAL PHYSICS By Johannes C. Familton Chairperson of the Supervisory Committee: Dr. Bruce Vogeli and Dr Henry O. Pollak Department of Mathematics Education TABLE OF CONTENTS List of Figures......................................................................................................iv List of Tables .......................................................................................................vi Acknowledgements .......................................................................................... vii Chapter I: Introduction ......................................................................................... 1 A. Need for Study ........................................................................................
    [Show full text]
  • On Mathematical and Physical Principles of Transformations
    1 On mathematical and physical principles of transformations of the coherent radar backscatter matrix David Bebbington and Laura Carrea Abstract The congruential rule advanced by Graves for polarization basis transformation of the radar backscat- ter matrix is now often misinterpreted as an example of consimilarity transformation. However, consim- ilarity transformations imply a physically unrealistic antilinear time-reversal operation. This is just one of the approaches found in literature to the description of transformations where the role of conjugation has been misunderstood. In this paper, the different approaches are examined in particular in respect to the role of conjugation. In order to justify and correctly derive the congruential rule for polarization basis transformation and properly place the role of conjugation, the origin of the problem is traced back to the derivation of the antenna hight from the transmitted field. In fact, careful consideration of the role played by the Green’s dyadic operator relating the antenna height to the transmitted field shows that, under general unitary basis transformation, it is not justified to assume a scalar relationship between them. Invariance of the voltage equation shows that antenna states and wave states must in fact lie in dual spaces, a distinction not captured in conventional Jones vector formalism. Introducing spinor formalism, and with the use of an alternate spin frame for the transmitted field a mathematically consistent implementation of the directional wave formalism is obtained. Examples are given comparing the wider generality of the congruential rule in both active and passive transformations with the consimilarity rule. arXiv:1308.6758v1 [physics.optics] 29 Aug 2013 Index Terms polarimetry, backscatter, unitary bases, spinors.
    [Show full text]
  • On Mathematical and Physical Principles of Transformations of the Coherent Radar Backscatter Matrix
    On mathematical and physical principles of transformations of the coherent radar backscatter matrix Article Accepted Version Bebbington, D. and Carrea, L. (2012) On mathematical and physical principles of transformations of the coherent radar backscatter matrix. IEEE Transactions on Geoscience and Remote Sensing, 50 (11). pp. 4657-4669. ISSN 0196-2892 doi: https://doi.org/10.1109/TGRS.2012.2191294 Available at http://centaur.reading.ac.uk/54612/ It is advisable to refer to the publisher’s version if you intend to cite from the work. See Guidance on citing . To link to this article DOI: http://dx.doi.org/10.1109/TGRS.2012.2191294 Publisher: IEEE All outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders. Terms and conditions for use of this material are defined in the End User Agreement . www.reading.ac.uk/centaur CentAUR Central Archive at the University of Reading Reading’s research outputs online 1 On mathematical and physical principles of transformations of the coherent radar backscatter matrix David Bebbington and Laura Carrea Abstract— The congruential rule advanced by Graves for exhibits a coherent mathematical framework. In particular, our polarization basis transformation of the radar backscatter matrix purpose here is not a general review, but to focus instead on is now often misinterpreted as an example of consimilarity a small number of key contributions to the literature where transformation. However, consimilarity transformations imply a physically unrealistic antilinear time-reversal operation. This we can most clearly demonstrate that ambiguities have been is just one of the approaches found in literature to the de- introduced in the process of justifying the current formulation scription of transformations where the role of conjugation has of polarimetry.
    [Show full text]
  • Simple Matrices
    Appendix B Simple matrices Mathematicians also attempted to develop algebra of vectors but there was no natural definition of the product of two vectors that held in arbitrary dimensions. The first vector algebra that involved a noncommutative vector product (that is, v w need not equal w v) was proposed by Hermann Grassmann in his book× Ausdehnungslehre× (1844 ). Grassmann’s text also introduced the product of a column matrix and a row matrix, which resulted in what is now called a simple or a rank-one matrix. In the late 19th century the American mathematical physicist Willard Gibbs published his famous treatise on vector analysis. In that treatise Gibbs represented general matrices, which he called dyadics, as sums of simple matrices, which Gibbs called dyads. Later the physicist P. A. M. Dirac introduced the term “bra-ket” for what we now call the scalar product of a “bra” (row) vector times a “ket” (column) vector and the term “ket-bra” for the product of a ket times a bra, resulting in what we now call a simple matrix, as above. Our convention of identifying column matrices and vectors was introduced by physicists in the 20th century. Suddhendu Biswas [52, p.2] − B.1 Rank-1 matrix (dyad) Any matrix formed from the unsigned outer product of two vectors, T M N Ψ = uv R × (1783) ∈ where u RM and v RN , is rank-1 and called dyad. Conversely, any rank-1 matrix must have the∈ form Ψ . [233∈, prob.1.4.1] Product uvT is a negative dyad. For matrix products ABT, in general, we have − (ABT) (A) , (ABT) (BT) (1784) R ⊆ R N ⊇ N B.1 § with equality when B =A [374, §3.3, 3.6] or respectively when B is invertible and (A)= 0.
    [Show full text]
  • Multiforms, Dyadics, and Electromagnetic Media IEEE Press 445 Hoes Lane Piscataway, NJ 08854
    Multiforms, Dyadics, and Electromagnetic Media IEEE Press 445 Hoes Lane Piscataway, NJ 08854 IEEE Press Editorial Board Tariq Samad, Editor in Chief George W. Arnold Vladimir Lumelsky Linda Shafer Dmitry Goldgof Pui-In Mak Zidong Wang Ekram Hossain Jeffrey Nanzer MengChu Zhou Mary Lanzerotti Ray Perez George Zobrist Kenneth Moore, Director of IEEE Book and Information Services (BIS) Multiforms, Dyadics, and Electromagnetic Media Ismo V. Lindell IEEE Antennas and Propagation Society, Sponsor The IEEE Press Series on Electromagnetic Wave Theory Andreas C. Cangellaris, Series Editor Copyright © 2015 by The Institute of Electrical and Electronics Engineers, Inc. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. All rights reserved. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose.
    [Show full text]
  • Vector, Dyadic, and Integral Relations
    A Vector, Dyadic, and Integral Relations A.1 Vector Identities A·B=B·A A x B = -B x A A· (B x C) = B· (C x A) = C· (A x B) A x (B x C) = B (A· C) - C (A· B) V (qn/J) = ¢ V1jJ + 'ljJ\l ¢ V· (1jJA) = A· V1fJ +~'V . A Vx (1jJA) = V1jJ x A+'ljJ\l x A V· (A x B) = B . V x A - A· V x B Vx (A x B) = (B· V) A - (A· V) B + A V . B - BV . A V (A· B) = (A· V) B + (B· V) A + A x V x B + B x V x A V x V x A = VV . A - v2 A V . V'ljJ = V21jJ V x V?/J = 0 V· V x A =0 605 606 A. Vector, Dyadic, and Integral Relations A.2 Dyadic Identities (AB) . (CD) = A (B . C) D (AB) x C = A (B x C) A· (C· B) = (A .C). B = A· C· B (A. C) x B = A· (C x B) = A· C x B (A x C) . B = A x (C· B) (A x C) x B = Ax (C x B) = AxC x B (A· C) . D = A· (C . D) = A· C . D (C· D) . A = C· (D . A) = C . D· A (C . D) x A = C . (D x A) = C . D x A (A x C)· D = Ax (C· D) = A x C· D (C x A) . D = C . (A x D) A- (B x C) = -B· (A x C) = (A x B) . C (C x A)· B = C· A x B = - (C x B)· A A x (B x C) = B (A· C) - (A· B) C A· C = (C) T .
    [Show full text]
  • Introduction to Tensors and Dyadics
    Cambridge University Press 0521817307 - Elastic Wave Propagation and Generation in Seismology Jose Pujol Excerpt More information 1 Introduction to tensors and dyadics 1.1 Introduction Tensors play a fundamental role in theoretical physics. The reason for this is that physical laws written in tensor form are independent of the coordinate system used (Morse and Feshbach, 1953). Before elaborating on this point, consider a simple example, based on Segel (1977). Newton’s second law is f = ma, where f and a are vectors representing the force and acceleration of an object of mass m. This basic law does not have a coordinate system attached to it. To apply the law in a particular situation it will be convenient to select a coordinate system that simplifies the mathematics, but there is no question that any other system will be equally acceptable. Now consider an example from elasticity, discussed in Chapter 3. The stress vector T (force/area) across a surface element in an elastic solid is related to the vector n normal to the same surface via the stress tensor. The derivation of this relation is carried out using a tetrahedron with faces along the three coordinate planes in a Cartesian coordinate system. Therefore, it is reasonable to ask whether the same result would have been obtained if a different Cartesian coordinate system had been used, or if a spherical, or cylindrical, or any other curvilinear system, had been used. Take another example. The elastic wave equation will be derived in a Cartesian coordinate system. As discussed in Chapter 4, two equations will be found, one in component form and one in vector form in terms of a combination of gradient, divergence, and curl.
    [Show full text]