Modern Geometry - Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields (Google eBook)
Springer Science & Business Media, 1992 - Language Arts & Disciplines - 470 pages
This is the first volume of a three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This material is explained in as simple and concrete a language as possible, in a terminology acceptable to physicists. The text for the second edition has been substantially revised.
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The Behaviour of Tensors Under Mappings
The Differential Calculus of Tensors
SkewSymmetric Tensors and the Theory of Integration
Differential Forms on Complex Spaces
Covariant Differentiation and the Metric
The Second Fundamental Form
The Metric on the Sphere
Spacelike Surfaces in PseudoEuclidean Space
The Language of Complex Numbers in Geometry
The Conformal Form of the Metric on a Surface
Transformation Groups as Surfaces in NDimensional Space
Conformal Transformations of Euclidean and PseudoEuclidean Spaces of Several Dimensions
Tensors The Algebraic Theory
The General Definition of a Tensor
Tensors of Type 0 k
Tensors in Riemannian and PseudoRiemannian Spaces
The Crystallographic Groups and the Finite Subgroups of the Rotation Group of Euclidean 3Space Examples of Invariant Tensors
Rank 2 Tensors in PseudoEuclidean Space and Their Eigenvalues
The Curvature Tensor
The Elements of the Calculus of Variations
The Geometrical Theory of Phase Space
The Second Variation for the Equation of the Geodesies
The Calculus of Variations in Several Dimensions Fields and Their Geometric Invariants
Examples of Lagrangians
The Simplest Concepts of the General Theory of Relativity
The Spinor Representations of the Groups SO3 and O3 1 Diracs Equation and Its Properties
Covariant Differentiation of Fields with Arbitrary Symmetry
Examples of GaugeInvariant Functionals Maxwells Equations and the YangMills Equation Functionals with Identically Zero Variational Derivative C...
3-dimensional analytic arbitrary Cartesian co-ordinate change co-ordinate system co-ordinates x1 co-ordinatized commutator completes the proof complex components connexion consider const constant corresponding covector curvature tensor curve defined definition denote derivative electromagnetic field element Euclidean 3-space Euclidean co-ordinates Euler-Lagrange equations example Exercise follows function Gaussian curvature geodesic geometry given GL(n gradient gravitational Hamiltonian Hence identity integral invariant isometries isomorphic Jacobian Lagrangian lattice Lemma length Lie algebra linear transformation Lorentz matrix metric gu Minkowski Minkowski metric Minkowski space momentum neighbourhood non-zero notation Note obtain operation orthogonal parameter particle particular phase space pseudo-Euclidean quadratic form region relative respect Riemannian Riemannian metric right-hand side rotation satisfying scalar product skew-symmetric skew-symmetric tensor smooth sphere spinor Stokes formula subgroup Suppose symmetric tangent space tangent vector tensor of type Theorem theory translations underlying space vector field whence zero
Page 462 - Electrodynamics of Continuous Media, Course of Theoretical Physics, Vol. 8, Pergamon Press, Oxford, UK, 1982.
Page vi - Riemannian geometry and basic topology were not included, even by departments or faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in...
Page vi - The task of designing a modernized course in geometry was begun in 1971 in the mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University.