Full Description
Tensor algebra and tensor analysis were developed by Riemann, Christo?el, Ricci, Levi-Civita and others in the nineteenth century. The special theory of relativity, as propounded by Einstein in 1905, was elegantly expressed by Minkowski in terms of tensor ?elds in a ?at space-time. In 1915, Einstein formulated the general theory of relativity, in which the space-time manifold is curved. The theory is aesthetically and intellectually satisfying. The general theory of relativity involves tensor analysis in a pseudo- Riemannian manifold from the outset. Later, it was realized that even the pre-relativistic particle mechanics and continuum mechanics can be elegantly formulated in terms of tensor analysis in the three-dimensional Euclidean space. In recent decades, relativistic quantum ?eld theories, gauge ?eld theories, and various uni?ed ?eld theories have all used tensor algebra analysis exhaustively. This book develops from abstract tensor algebra to tensor analysis in va- ous di?erentiable manifolds in a mathematically rigorous and logically coherent manner. The material is intended mainly for students at the fourth-year and ?fth-year university levels and is appropriate for students majoring in either mathematical physics or applied mathematics.
Contents
Finite- Dimensional Vector Spaces and Linear Mappings.- Fields.- Finite-Dimensional Vector Spaces.- Linear Mappings of a Vector Space.- Dual or Covariant Vector Space.- Tensor Algebra.- The Second Order Tensors.- Higher Order Tensors.- Exterior or Grassmann Algebra.- Inner Product Vector Spaces and the Metric Tensor.- Tensor Analysis on a Differentiable Manifold.- Differentiable Manifolds.- Vectors and Curves.- Tensor Fields over Differentiable Manifolds.- Differential Forms and Exterior Derivatives.- Differentiable Manifolds with Connections.- The Affine Connection and Covariant Derivative.- Covariant Derivatives of Tensors along a Curve.- Lie Bracket, Torsion, and Curvature Tensor.- Riemannian and Pseudo-Riemannian Manifolds.- Metric, Christoffel, Ricci Rotation.- Covariant Derivatives.- Curves, Frenet-Serret Formulas, and Geodesics.- Special Coordinate Charts.- Speical Riemannian and Pseudo-Riemannian Manifolds.- Flat Manifolds.- The Space of Constant Curvature.- Extrinsic Curvature.
