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,

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