DOCSLIB.ORG

Wolfram Neutsch Coordinates

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Wolfram Neutsch Coordinates Wolfram Neutsch Coordinates w DE G Walter de Gruyter • Berlin • New York 1996 Volume 1: Theory 0. Introduction. 1 A. Foundations - 1. Historical development of the coordinate concept. 9 1.1. Geography 9 1. 2. Aff ine and protective geometry 15 1.3. Differential geometry 20 1. 4. Geodesy and cartography 24 1. 5. Special coordinates 34 2. Notation and conventions 38 2/1. Sets and topological spaces : 38 2. 2. Groups 42 2. 3. Matrices 48 2. 4. Multilinear forms and tensors 56 2. 5. Polynomials. 65 B. Geometry 3. Manifolds 70 3. 1. Continuous coordinate systems 70 3.2. Smooth manifolds 73 3.3. Curves and tangent spaces 76 3. 4. Vector bundles .80 3.5. Differential and exterior derivation 85 3.6. Partition of unity . 94 3.7. Oriented manifolds 102 3.8. Integration of differential forms 106 3.9. Stokes' theorem ; . 110 4. Riemannian spaces 113 4. 1. The metric tensor. 113 4. 2. Christoffel symbols and covariant derivation 117 4.3. Normal coordinates 123 4.4. Curvature 128 4. 5. Volume 136 VIII 4.6. Duality 145 4. 7. Classical vector analysis 149 5. Applications to physics 160 5. 1. Mechanics 160 5. 2. Hydrodynamics 166 5.3. Relativity 172 5. 4. Electromagnet ism 175 5.5. Optics 182 5.6. Quantum mechanics 187 6. Complex analysis 190 6. 1. Elementary properties of complex numbers 190 6. 2. Convergence of function series 193 6. 3. Power series 198 6.4. Analytical continuation 205 6.5. Holomorphic functions 211 6. 6. Angle-preserving transformations 225 6. 7. Lie groups 233 6.8. Lie algebras 237 7. Projective Geometry 246 7.1. Affine and projective coordinates 246 7.2. Project ivities 255 7.3. The cross ratio 260 7.4. Bezout's theorem 268 7. 5. Planar algebraic curves . 279 7. 6. Stereographic projection 287 7. 7. Higher dimensional spaces 293 C. Rotations 8. Orthogonal groups 295 8. 1. Isometries and Euclidean transformations 295 8. 2. The exponential mapping 301 8. 3. Rational parametrization. 311 9. Linear transformations of complex spaces 318 9. 1. Pauli matrices 318 9. 2. Cayley-Klein parameters. 322 9. 3. The angular momentum algebra 327 9.4. Gaussian mutations of space 336 IX 9. 5. Euler angles 339 9.6. Elementary geometry of the Riemann number sphere 342 10. Quaternions 353 10. 1. The skew field of quaternions 353 10.2. Left and right multiplication 359 10. 3. Rotations of the quaternion algebra 362 10. 4. Representation by complex matrices •. 367 10. 5. Finite groups of quaternions. 371 11. Octaves .379 11.1. Doubling method of Cayley and Dickson.. .: 379 11.2. Alternative division algebras 389 11.3. The theorem of Hurwitz ; 404 11.4. Quadratic algebras 411 11.5. Parallelizability and regular vielbeine. 425 12. Hopf mappings 429 12. 1. Homotopy groups of spheres 429 12. 2. Homotopy invariants. 433 12.3. Duplication of angles and classical Hopf fibration 438 12.4. Generalizations 444 12. 5. Geometric peculiarities 448 13. Spinors 452 13.1. Schur extensions of the symmetric groups 452 13. 2. Spin groups 466 13. 3. GraBmann algebras 470 13. 4. Clifford algebras 476 13.5. Dirac matrices . .,486 14. Lorentz transformations 492 14. 1. The Poincare group 492 14. 2. Boosts 499 14. 3. Complex Lorentz matrices 508 14. 4. Description by quaternions 516 14. 5. Representation theory of the Lorentz group 522 D. Reflections 15. Coxeter groups 528 15.1. Discrete symmetries 528 15.2. Invariant theory; Molien's theorem 532 15. 3. Polynomial invariant rings 538 15.4. Reflection groups 550 15. 5. Fundamental systems 558 15. 6. The Euclidean Coxeter groups 566 16. Invariant rings of finite Weyl groups 574 16. 1. Root systems 574 16. 2. Weyl groups 589 16. 3. Elementary symmetric functions 599 17. Basic invariants 605 17. 1. Generic Weyl groups (A,B,D) 605 17. 2. Weyl groups of type E 607 17.3. Weyl groups of types F,<B,H, D 623 17.4. Basic degrees .635 Volume 2: Applications E. Lattices 18. Elliptic functions and modular forms 637 18. 1. Doubly periodic functions 637 18.2. WeierstraB functions 643 18. 3. The modular group 654 18.4. Modular forms ; . 661 18.5. Cusp forms 672 18.6. Modular functions; the J-invariant 678 19. Euclidean lattices 688 19. 1. Foundations of lattice theory 688 19.2. Theta functions 698 19.3. The Gosset lattice 712 19.4. Niemeier lattices 717 20. Linear codes 726 20. 1. Encoding of information ••••-, 726 20. 2. The Hexacode 730 20. 3. The binary Golay code 738 20. 4. Miracle octad generator 751 20. 5. Alternative constructions of the Golay code 757 ,20. 6. Steiner systems and Mathieu groups 767 XI 21. The Leech lattice 781 21. 1. Uniqueness of the Leech lattice 781 21.2. Explicit representations 789 21. 3. Automorphisms 792 21.4. Holes in the Leech lattice 799 21.5. Invariants of the Conway group 806 21.6. Lorentz lattices and sporadic groups 819 F. Spheres 22. Harmonic functions 837 22. 1. The theorems of Green 837 22. 2. Dirichlet' s principle 844 22.3. The Poisson integral ....854 22.4. Potential functions 862 23. Spherical surface functions 866 23. 1. Spherical harmonics 866 23. 2. Legendre polynomials 870 23. 3. Orthogonal functions on the 2-sphere 881 23. 4. Clebsch-Gordan coefficients 897 24. Lattice integration 913 24. 1. Integration points and weights 913 24. 2. Numerical integration after GauB 920 24. 3. Simple examples 928 24.4. Integration by means of the Gosset lattice 935 24. 5. Transfer via Hopf mappings 941 25. Spherical designs 945 25: 1. Integration methods with equal weights 945 25. 2. Tight designs. 951 25. 3. Optimal integration on the 2-sphere 956 25. 4. The hypersphere 966 25. 5. Very high precision numerical integration 976 25.6. Four-dimensional root systems and quaternions 981 G. Coordinate systems 26. Linear and reducible coordinates 985 26. 1. Cartesian and oblique linear frames 985 XII 26.2. Polar coordinates 991 26.3. Classical cylindrical coordinates 995 26. 4. Separable isothermal systems 997 27. Three-dimensional Stackel coordinates 1006 27. 1. Rotational symmetry 1006' 27. 2. Stackel spaces 1015 27. 3. Classification of Stackel-type metrics 1020 27. 4. Bipolar coordinates 1029 28. Conf ocal Coordinates 1037 28. 1. Orthogonal families of quadrics 1037 28. 2. Metric and differential operators 1045 28.3. Separation of the potential equation 1053 28.4. Lame functions ' 1058 28. 5. Geodesies on the ellipsoid 1064 29. GauB-Kriiger Coordinates 1068 29.1. Soldner's parameterization of the spheroid 1068 29. 2. Series expansions 1077 29. 3. Transversal Mercator projection 1083 29. 4. Bessel' s figure of the Earth 1095 29. 5. Recent models 1100 30. Coordinates for special applications 1104 30.1. Clairaut coordinates 1104 30.2. Roche coordinates 1109 30.3. Planar Weyl coordinates. 1113 30. 4. Weyl coordinates in three-dimensional space 1118 30. 5. Higher dimensional Weyl coordinates 1126 H. Tables Calculation and organization of the tables 1132 Coordinates in R2 1136 Cartesian coordinates. 1136 Polar coordinates 1139 Parabolic coordinates 1142 Elliptic coordinates 1145 Conf ocal coordinates 1148 Bipolar coordinates 1152 Digonal coordinates 1156 XIII Trigonal coordinates 1159 Tetragonal coordinates 1162 Pentagonal coordinates '. 1165 Hexagonal coordinates 1168 Coordinates in R3 1171 Cartesian coordinates 1171 Cylindrical coordinates 1174 ' Polar coordinates 1178 Geographic coordinates. 1182 Coordinates of the parabolic cylinder 1186 Coordinates of the elliptic cylinder 1190 Confocal cylindrical coordinates. 1194 Coordinates of the circular paraboloid 1198 Coordinates of the elliptic paraboloid 1202 Ellipsoidal coordinates (prolate) 1207 Ellipsoidal coordinates (oblate) 1211 Spheroidal coordinates (prolate) 1215 Spheroidal coordinates (oblate) 1219 Conical coordinates 1223 Confocal coordinates 1228 Bicylindrical coordinates 1234 Bispherical coordinates 1238 Torus coordinates 1243 Tetrahedral coordinates 1248 Octahedral coordinates 1253 Icosahedral coordinates 1259 Coordinates in R4 1265 ' Cartesian coordinates 1265 Cylindrical coordinates of type (1,3) 1268 Cylindrical coordinates of type (2,2) 1272 Polar coordinates. ; 1276 Geographic coordinates 1280 Double polar coordinates. 1284 Confocal coordinates 1288 Appendix References 1297 Index 1322.
Load more

Recommended publications
  • Curvilinear Coordinate Systems
    Appendix A Curvilinear coordinate systems Results in the main text are given in one of the three most frequently used coordinate systems: Cartesian, cylindrical or spherical. Here, we provide the material necessary for formulation of the elasticity problems in an arbitrary curvilinear orthogonal coor- dinate system. We then specify the elasticity equations for four coordinate systems: elliptic cylindrical, bipolar cylindrical, toroidal and ellipsoidal. These coordinate systems (and a number of other, more complex systems) are discussed in detail in the book of Blokh (1964). The elasticity equations in general curvilinear coordinates are given in terms of either tensorial or physical components. Therefore, explanation of the relationship between these components and the necessary background are given in the text to follow. Curvilinear coordinates ~i (i = 1,2,3) can be specified by expressing them ~i = ~i(Xl,X2,X3) in terms of Cartesian coordinates Xi (i = 1,2,3). It is as- sumed that these functions have continuous first partial derivatives and Jacobian J = la~i/axjl =I- 0 everywhere. Surfaces ~i = const (i = 1,2,3) are called coordinate surfaces and pairs of these surfaces intersects along coordinate curves. If the coordinate surfaces intersect at an angle :rr /2, the curvilinear coordinate system is called orthogonal. The usual summation convention (summation over repeated indices from 1 to 3) is assumed, unless noted otherwise. Metric tensor In Cartesian coordinates Xi the square of the distance ds between two neighboring points in space is determined by ds 2 = dxi dXi Since we have ds2 = gkm d~k d~m where functions aXi aXi gkm = a~k a~m constitute components of the metric tensor.
    [Show full text]
  • Analytical Results Regarding Electrostatic Resonances of Surface Phonon/Plasmon Polaritons: Separation of Variables with a Twist
    Analytical results regarding electrostatic resonances of surface phonon/plasmon polaritons: separation of variables with a twist R. C. Voicu1 and T. Sandu1 1Research Centre for Integrated Systems, Nanotechnologies, and Carbon Based Materials, National Institute for Research and Development in Microtechnologies-IMT, 126A, Erou Iancu Nicolae Street, Bucharest, ROMANIA∗ (Dated: February 16, 2017) Abstract The boundary integral equation method ascertains explicit relations between localized surface phonon and plasmon polariton resonances and the eigenvalues of its associated electrostatic opera- tor. We show that group-theoretical analysis of Laplace equation can be used to calculate the full set of eigenvalues and eigenfunctions of the electrostatic operator for shapes and shells described by separable coordinate systems. These results not only unify and generalize many existing studies but also offer the opportunity to expand the study of phenomena like cloaking by anomalous localized resonance. For that reason we calculate the eigenvalues and eigenfunctions of elliptic and circular cylinders. We illustrate the benefits of using the boundary integral equation method to interpret recent experiments involving localized surface phonon polariton resonances and the size scaling of plasmon resonances in graphene nano-disks. Finally, symmetry-based operator analysis can be extended from electrostatic to full-wave regime. Thus, bound states of light in the continuum can be studied for shapes beyond spherical configurations. PACS numbers: 02.20.Sv,02.30.Em,02.30.Uu,41.20.Cv,63.22.-m,78.67.Bf arXiv:1702.04655v1 [cond-mat.mes-hall] 15 Feb 2017 ∗Electronic address: [email protected] 1 I. INTRODUCTION Materials with negative permittivity allow light confinement to sub-diffraction limit and field enhancement at the interface with ordinary dielectrics [1].
    [Show full text]
  • Computer Facilitated Generalized Coordinate Transformations of Partial Differential Equations with Engineering Applications
    Computer Facilitated Generalized Coordinate Transformations of Partial Differential Equations With Engineering Applications A. ELKAMEL,1 F.H. BELLAMINE,1,2 V.R. SUBRAMANIAN3 1Department of Chemical Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1 2National Institute of Applied Science and Technology in Tunis, Centre Urbain Nord, B.P. No. 676, 1080 Tunis Cedex, Tunisia 3Department of Chemical Engineering, Tennessee Technological University, Cookeville, Tennessee 38505 Received 16 February 2008; accepted 2 December 2008 ABSTRACT: Partial differential equations (PDEs) play an important role in describing many physical, industrial, and biological processes. Their solutions could be considerably facilitated by using appropriate coordinate transformations. There are many coordinate systems besides the well-known Cartesian, polar, and spherical coordinates. In this article, we illustrate how to make such transformations using Maple. Such a use has the advantage of easing the manipulation and derivation of analytical expressions. We illustrate this by considering a number of engineering problems governed by PDEs in different coordinate systems such as the bipolar, elliptic cylindrical, and prolate spheroidal. In our opinion, the use of Maple or similar computer algebraic systems (e.g. Mathematica, Reduce, etc.) will help researchers and students use uncommon transformations more frequently at the very least for situations where the transformations provide smarter and easier solutions. ß2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 19: 365À376, 2011; View this article online at wileyonlinelibrary.com; DOI 10.1002/cae.20318 Keywords: partial differential equations; symbolic computation; Maple; coordinate transformations INTRODUCTION usual Cartesian, polar, and spherical coordinates. For example, Figure 1 shows two identical pipes imbedded in a concrete slab.
    [Show full text]
  • Conversion of Latitude and Longitude to UTM Coordinates
    Paper 410, CCG Annual Report 11 , 2009 (© 2009) Conversion of Latitude and Longitude to UTM Coordinates John G. Manchuk The frame of reference is an important aspect of natural resource modeling. Two principal coordinate systems are encountered in resource analysis: latitude and longitude and universal transverse Mercator or UTM. Operations can also have their own local coordinate system that is defined within a specific lease area; however, these are typically just translated and/or rotated UTM coordinates. This paper reviews some of the basics behind these two coordinate systems and describes a program for conversion. Introduction Map projections are useful for presentation purposes and to simplify calculations of distances, areas, and volumes. In the earth’s coordinate system, which is ellipsoidal, these computations can be cumbersome. The two coordinate systems that are explained here are the ellipsoidal coordinates defining the earth having longitude, latitude, and height axes, and universal transverse Mercator (UTM) coordinates which is a map projection to a cylindrical coordinate system that is discretized into a set of zones, each being an approximate Cartesian system with East and North coordinates. Of course, coordinate systems require a point of reference or datum. Defining latitude and longitude from an ellipsoidal model of the earth is only possible by defining a point of reference on the ellipsoid. For the World Geodetic System of 1984 (WGS 84) defined principally for the global positioning system (GPS), the reference is a series of monitoring stations positioned on the earth with known coordinates. This provides an ellipsoid that fits the earth, or geoid, with minimal error in height between them.
    [Show full text]
  • Geodetic Computations on Triaxial Ellipsoid
    International Journal of Mining Science (IJMS) Volume 1, Issue 1, June 2015, PP 25-34 www.arcjournals.org Geodetic Computations on Triaxial Ellipsoid Sebahattin Bektaş Ondokuz Mayis University, Faculty of Engineering, Geomatics Engineering, Samsun, [email protected] Abstract: Rotational ellipsoid generally used in geodetic computations. Triaxial ellipsoid surface although a more general so far has not been used in geodetic applications and , the reason for this is not provided as a practical benefit in the calculations. We think this traditional thoughts ought to be revised again. Today increasing GPS and satellite measurement precision will allow us to determine more realistic earth ellipsoid. Geodetic research has traditionally been motivated by the need to continually improve approximations of physical reality. Several studies have shown that the Earth, other planets, natural satellites,asteroids and comets can be modeled as triaxial ellipsoids. In this paper we study on the computational differences in results,fitting ellipsoid, use of biaxial ellipsoid instead of triaxial elipsoid, and transformation Cartesian ( Geocentric ,Rectangular) coordinates to Geodetic coodinates or vice versa on Triaxial ellipsoid. Keywords: Reference Surface, Triaxial ellipsoid, Coordinate transformation,Cartesian, Geodetic, Ellipsoidal coordinates 1. INTRODUCTION Although Triaxial ellipsoid equation is quite simple and smooth but geodetic computations are quite difficult on the Triaxial ellipsoid. The main reason for this difficulty is the lack of symmetry. Triaxial ellipsoid generally not used in geodetic applications. Rotational ellipsoid (ellipsoid revolution ,biaxial ellipsoid , spheroid) is frequently used in geodetic applications . Triaxial ellipsoid is although a more general surface so far but has not been used in geodetic applications. The reason for this is not provided as a practical benefit in the calculations.
    [Show full text]
  • Path Integral Discussion for Smorodinsky-Winternitz Potentials: I
    best; sq V 0// 8 qw Fw ro DESY 94-018 February 1994 Path Integral Discussion for Smorodinsky-Winternitz Potentials: I. Two- and Three Dimensional Euclidean Space C. Grosche II. Institut hir Theoretische Physik, Universitat Hamburg G. S. Pogosyan, A. N. Sissakian Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia naml\\\\\\\\\\\ Lraanexas. 1\\\l\\ll\\\ smava ISSN 0418-9833 OCR Output OCR OutputDESY 94 - 018 ISSN 0418 - 9833 February 1994 hep-th/9402121 PATH INTEGRAL DISCUSSION FOR SMORODINSKY-WINTERNITZ POTENTIALS: I. TWO- AND THREE DIMENSIONAL EUCLIDEAN SPACE C. Grosche* II. Institut fur Theoretische Physik Universitat Hamburg, Luruper Chaussee 149 22761 Hamburg, Germany G. S. Pog0syan** and A. N. Sissakian* Laboratory of Theoretical Physics Joint Institute for Nuclear Research (Dubna} 141980 Dubna, Moscow Region, Russia Abstract Path integral formulations for the Smorodinsky-Winternitz potentials in two- and three dimensional Euclidean space are presented. We mention all coordinate systems which sep arate the Smorodinsky-Winternitz potentials and state the corresponding path integral for mulations. Whereas in many coordinate systems an explicit path integral formulation is not possible, we list in all soluble cases the path integral evaluations explicitly in terms of the propagators and the spectral expansions into the wave-functions. Supported by Deutsche Forschungsgemeinschaft under contract number GR 1031/2-1. Supported by Heisenberg-Landau program. OCR Output OCR Output1. Introduction. In the study of the Kepler problem and therharmonic oscillator it turns out that they possess properties making them of special interest, for instance, all finite classical trajectories are closed and all energy eigen-values are multiply degenerated.
    [Show full text]
  • Symmetry and Separation of Variables for the Helmholtz and Laplace Equations
    C. P. Boyer, E. G. Kalnins, and W. Miller, Jr. Nagoya Math. J. Vol. 60 (1976), 35-80 SYMMETRY AND SEPARATION OF VARIABLES FOR THE HELMHOLTZ AND LAPLACE EQUATIONS C. P. BOYER, E. G. KALNINS, AND W. MILLER, JR. Introduction. This paper is one of a series relating the symmetry groups of the principal linear partial differential equations of mathematical physics and the coordinate systems in which variables separate for these equations. In particular, we mention [1] and paper [2] which is a survey of and introduction to the series. Here we apply group-theoretic methods to study the separable coordinate systems for the Helmholtz equation. 2 (4* + ω )Ψ(x) = 0 , x = (xlf x29 x3) , 4* = 3ii + d22 + d33 , ω > 0 , and the Laplace equation (0.2) ΔzW{x) = 0 . It is well-known that (0.1) separates in eleven coordinate systems, see [3], Chapter 5, and references contained therein. Moreover, in [4] it is shown that these systems correspond to commuting pairs of second order symmetric operators in the enveloping algebra of <?(3), the sym- metry algebra of (0.1). However, we show here for the first time how one can systematically make use of the representation theory of the Euclidean symmetry group £7(3) of the Helmholtz equation to derive identities relating the different separable solutions. As we will point out, some of these identities are new. It is also known that there are 17 types of cyclidic coordinate systems which permit i?-separation of variables in the Laplace equation and these appear to be the only such separable systems for (0.2), [5].
    [Show full text]
  • Triaxial Coordinate Systems and Their Geometrical Interpretation
    Triaxial coordinate systems and their geometrical interpretation G. Panou, R. Korakitis, D. Delikaraoglou Department of Surveying Engineering, ational Technical University of Athens, Zografou Campus, 15780 Athens, Greece Abstract: Coordinate systems on a triaxial ellipsoid are summarized and presented, to- gether with their geometrical interpretation. Firstly, the geodetic coordinate system is a generalization of the geodetic system on a biaxial ellipsoid. Then, the ellipsoidal coordinate system is a triply orthogonal system, related to the theory of potential. Finally, the geomet- ric coordinate system, which is based on the previous systems, is interpreted. The three coordinate systems on a biaxial ellipsoid are obtained as degenerate cases. In addition, a method to compute the three curvilinear coordinates of each system is described. 1. Introduction It is generally accepted in the geodetic community that a triaxial ellipsoid could better approximate the geoid than the biaxial ellipsoid. Furthermore, several non- spherical celestial bodies such as planets, natural satellites, asteroids and comets are modeled by a triaxial ellipsoid. Also, present day accuracy requirements and modern computational capabilities encourage the study of the triaxial ellipsoid as a geometrical and a physical model in geodesy and related sciences. From a geometrical viewpoint, Shebl and Farag (2007) and Panou (2013) presented methods for the solution of the geodesic problem on a triaxial ellipsoid. Also, Klein (2012) gave a solution to the problem of the intersection of an ellipsoid and a plane. Furthermore, various map projections have been developed on a triaxial el- lipsoid (Weightman 1961, Snyder 1985, Grafarend and Krumm 2006, Fleis et al. 2013, Nyrtsov 2014). The geodetic (planetographic) coordinate system on a triaxial ellipsoid has been presented by Grafarend and Krumm (2006) and recently by Fel- tens (2009), Ligas (2012a, b) and Bektaş (2014).
    [Show full text]
  • Wgs 84 Implementation Manual
    Version 2.4 February 12, 1998 WGS 84 IMPLEMENTATION MANUAL Prepared by EUROCONTROL European Organization for the Safety of Air Navigation Brussels, Belgium and IfEN Institute of Geodesy and Navigation (IfEN) University FAF Munich, Germany Foreword i FOREWORD 1. The Council of the International Civil Aviation Organization (ICAO), at the thirteenth meeting of its 126th Session on 3 March 1989, approved Recommendation 3.2/1 of the fourth meeting of the Special Committee on Future Air Navigation Systems (FANS/4) concerning the adoption of the World Geodetic System - 1984 (WGS 84) as the standard geodetic reference system for future navigation with respect to international civil aviation. FANS/4 Recommendation 3.2/1 reads: "Recommendation 3.2/1 - Adoption of WGS 84 That ICAO adopts, as a standard, the geodetic reference system WGS 84 and develops appropriate ICAO material, particularly in respect of Annexes 4 and 15, in order to ensure a rapid and comprehensive implementation of the WGS 84 geodetic reference system. " 2. The ICAO Council at the ninth meeting of its 141st Session, on 28 February 1994, adopted amendment 28 to Annex 15. Consequential amendments to Annexes 4, 11 and 14, Volume I and II will be adopted by the Council in due course. The Standards and Recommended Practices (SARPS) in Annexes 11 and 14, Volumes I and II govern the determination and reporting of the geographical coordinates in terms of WGS 84 geodetic reference system. Annexes 4 and 15 SARPS govern the publication of information in graphic and textual form. The States' aeronautical information service department will publish in Aeronautical Information Publications (AIP), on charts and in electronic data base where applicable, geographical coordinate values based on WGS 84 which are supplied by the other State aeronautical services, i.e.
    [Show full text]
  • 14.2.2 Covariant and Contravariant Coordinates
    14. Vector Algebra and Vector Analysis 2016-07-07 $Version 10.0 for Mac OS X x86 (64-bit) (December 4, 2014) Mathematica 9: The commands for vector analysis are contained in the kernel in contrast to earlier versions. Mathematica 7 or 8: The package must be loaded by the command << "VectorAnalysis`" . Some of the commands differ from those given in version 9, so in these lecture notes. In particular, the arguments of the operators do not contain the list of the independent variables as arguments. Ph. Boylard: A Guide to Standard Mathematica Packages. Mathematica Technical Report (1991). E. Martin: Mathematica 3.0 Standard Add-On Packages. Cambridge University Press. pp.74-82. 14.1 Rectangular, Cylindrical and Spherical Coordinates Rectangular (as well as cylindrical and spherical) coordinates have been treated at the end of Chap.3. Knowledge of these subjects is pressuposed in this chapter. 14.2 Curvilinear Orthogonal Coordinates 14.2.1 Literature Murray R. Spiegel, Schaum's Mathemtical Handbook of Formulas and Tables, McGraw-Hill, New York 1968, pp.116-130 Philip M. Morse, Herman Feshbach: Methods of Theoretical Physics, McGraw-Hill, New York, 1953. (MF), 656ff. George B. Arken, Mathematical Methods for Pysicists, 3 rd ed., Academic Press, New York, 1985. B. Schnizer, Analytical methods in applied theoretrical physics. Lecture notes (in German). Chaps. 6 and 7. http://itp.tugraz.at/~schnizer/AnalyticalMethods/ P. Moon, D.E. Spencer, Field Theory Handbook. 2 nd ed., Springer, 1988. (MS) G. Wendt, Statische Felder und Stationäre Ströme. Hb. d. Physik XVI Springer, 1958, S.1 ff. W.R. Smith, Static and Dynamic Electricity, McGraw-Hill, 1950 14.2.2 Covariant and contravariant coordinates A general non-singular transformation from Cartesian coordinates x1, x2, x3 to more general curvilinear coordinates u1, u2, u3 is given by: xi = xi (u1, u2, u3), uk = uk (x1, x2, x3); i, k = 1, 2, 3.
    [Show full text]
  • Coordinate Systems and Coordinate Transformations
    © Copyright 2004 2 Coordinate Systems and Coordinate Transformations The field of mathematics known as topology describes space in a very general sort of way. Many spaces are exotic and have no counterpart in the physical world. Indeed, in the hierarchy of spaces defined within topology, those that can be described by a coordinate system are among the more sophisticated. These are the spaces of interest to the physical scientist because they are reminiscent of the physical space in which we exist. The utility of such spaces is derived from the presence of a coordinate system which allows one to describe phenomena that take place within the space. However, the spaces of interest need not simply be the physical space of the real world. One can imagine the temperature-pressure-density space of thermodynamics or many of the other spaces where the dimensions are physical variables. One of the most important of these spaces for mechanics is phase space. This is a multi-dimensional space that contains the position and momentum coordinates for a collection of particles. Thus physical spaces can have many forms. However, they all have one thing in common. They are described by some coordinate system or frame of reference. Imagine a set of rigid rods or vectors all connected at a point. Such a set of 'rods' is called a frame of reference. If every point in the space can uniquely be projected onto the rods so that a unique collection of rod-points identify the point in space, the reference frame is said to span the space.
    [Show full text]
  • Closed-Form Transformation Between Geodetic and Ellipsoidal Coordinates
    Closed-form transformation between geodetic and ellipsoidal coordinates W.E. Featherstone (corresponding author) Western Australian Centre for Geodesy & The Institute for Geoscience Research, Curtin Uni- versity of Technology, GPO Box U1987, Perth, WA 6845, Australia; Phone: +61 8 9266 2734; Fax: +61 8 9266 2703; Email: [email protected] S.J. Claessens Western Australian Centre for Geodesy & The Institute for Geoscience Research, Curtin Uni- versity of Technology, GPO Box U1987, Perth, WA 6845, Australia; Phone: +61 8 9266 3505; Fax: +61 8 9266 2703; Email: [email protected] Abstract: We present formulas for direct closed-form transformation between geodetic co- ordinates (Á; ¸; h) and ellipsoidal coordinates (¯; ¸; u) for any oblate ellipsoid of revolution. These will be useful for those dealing with ellipsoidal representations of the Earth’s gravity field or other oblate ellipsoidal figures. The numerical stability of the transformations for near- polar and near-equatorial regions is also considered. Keywords: Geodetic coordinates, ellipsoidal coordinates, coordinate transformations, ellip- soidal geodesy Generic notation a semi-major axis length of an ellipsoid b = a(1 ¡ f) semi-minor axis length of an ellipsoid f = (a ¡ b)=a flattening of an ellipsoid [dimensionless] p e = 2f ¡ f 2 first numerical eccentricity of an ellipsoid [dimensionless] E = ae linear eccentricity (or focal length) of an ellipsoid (Á; ¸; h) geodetic coordinates: latitude, longitude and height (¯; ¸; u) ellipsoidal coordinates: ellipsoidal co-latitude, longitude and ellipsoidal parameter (X; Y; Z) 3D Cartesian coordinates aligned with the ellipsoids’ axes 1 1 Introduction 1.1 Geodetic coordinates (Á; ¸; h) Transformation between geodetic coordinates (geodetic latitude Á, longitude ¸ and ellipsoidal/geodetic height h) and 3D Cartesian coordinates (X; Y; Z) has been studied repeatedly over the years (Table 1).
    [Show full text]