Alan F. Beardon

Indeholder "Preface", "1 Groups and permutations", " 1.1 Introduction", " 1.2 Groups", " 1.3 Permutations of a finite set", " 1.4 The sign of a permutation", " 1.5 Permutations of an arbitrary set", "2 The real numbers", " 2.1 The integers", " 2.2 The real numbers", " 2.3 Fields", " 2.4 Modular arithmetic", "3 The complexplane", " 3.1 Complex numbers", " 3.2 Polar coordinates", " 3.3 Lines and circles", " 3.4 Isometries of the plane", " 3.5 Roots of unity", " 3.6 Cubic and quartic show more equations", " 3.7 The Fundamental Theorem of Algebra", "4 Vectors in three-dimensional space", " 4.1 Vectors", " 4.2 The scalar product", " 4.3 The vector product", " 4.4 The scalar triple product", " 4.5 The vector triple product", " 4.6 Orientation and determinants", " 4.7 Applications to geometry", " 4.8 Vector equations", "5 Spherical geometry", " 5.1 Spherical distance", " 5.2 Spherical trigonometry", " 5.3 Area on the sphere", " 5.4 Euler's formula", " 5.5 Regular polyhedra", " 5.6 General polyhedra", "6 Quaternions and isometries", " 6.1 Isometries of Euclidean space", " 6.2 Quaternions", " 6.3 Reflections and rotations", "7 Vector spaces", " 7.1 Vector spaces", " 7.2 Dimension", " 7.3 Subspaces", " 7.4 The direct sum of two subspaces", " 7.5 Linear difference equations", " 7.6 The vector space of polynomials", " 7.7 Linear transformations", " 7.8 The kernel of a linear transformation", " 7.9 Isomorphisms 130", " 7.10 The space of linear maps", "8 Linear equations", " 8.1 Hyperplanes", " 8.2 Homogeneous linear equations", " 8.3 Row rank and column rank", " 8.4 Inhomogeneous linear equations", " 8.5 Determinants and linear equations", " 8.6 Determinants", "9 Matrices", " 9.1 The vector space of matrices", " 9.2 A matrix as a linear transformation", " 9.3 The matrix of a linear transformation", " 9.4 Inverse maps and matrices", " 9.5 Change of bases", " 9.6 The resultant of two polynomials", " 9.7 The number of surjections", "10 Eigenvectors", " 10.1 Eigenvalues and eigenvectors", " 10.2 Eigenvalues and matrices", " 10.3 Diagonalizable matrices", " 10.4 The Cayley–Hamilton theorem", " 10.5 Invariant planes", "11 Linear maps of Euclidean space", " 11.1 Distance in Euclidean space", " 11.2 Orthogonal maps", " 11.3 Isometries of Euclidean n-space", " 11.4 Symmetric matrices", " 11.5 The field axioms", " 11.6 Vector products in higher dimensions", "12 Groups", " 12.1 Groups", " 12.2 Subgroups and cosets", " 12.3 Lagrange's theorem", " 12.4 Isomorphisms", " 12.5 Cyclic groups", " 12.6 Applications to arithmetic", " 12.7 Product groups", " 12.8 Dihedral groups", " 12.9 Groups of small order", " 12.10 Conjugation", " 12.11 Homomorphisms", " 12.12 Quotient groups", "13 Möbius transformations", " 13.1 Möbius transformations", " 13.2 Fixed points and uniqueness", " 13.3 Circles and lines", " 13.4 Cross-ratios", " 13.5 Möbius maps and permutations", " 13.6 Complex lines", " 13.7 Fixed points and eigenvectors", " 13.8 A geometric view of infinity", " 13.9 Rotations of the sphere", "14 Group actions", " 14.1 Groups of permutations", " 14.2 Symmetries of a regular polyhedron", " 14.3 Finite rotation groups in space", " 14.4 Groups of isometries of the plane", " 14.5 Group actions", "15 Hyperbolic geometry", " 15.1 The hyperbolic plane", " 15.2 The hyperbolic distance", " 15.3 Hyperbolic circles", " 15.4 Hyperbolic trigonometry", " 15.5 Hyperbolic three-dimensional space", " 15.6 Finite Möbius groups", "Index".

Smuk opsummering af meget af det matematik, jeg har set. Forfatteren begrunder bogen med at han var kommet til at undervise et helt kursus uden at lægge mærke til at han ikke havde vist, at de objekter han talte om, faktisk eksisterede. show less