Mathematicians, theoretical physicists, and engineers unacquainted with tensor calculus are at a serious disadvantage in several fields of pure and applied mathematics. They are cut off from the study of Reimannian geometry and the general theory of relativity. Even in Euclidean geometry and Newtonian mechanics (particularly the mechanics of continua), they are compelled to work in notations which lack the compactness of tensor calculus. This classic text is a fundamental introduction to the subject for the beginning student of absolute differential calculus, and for those interested in the applications of tensor calculus to mathematical physics and engineering.
Tensor Calculus contains eight chapters. The first four deal with the basic concepts of tensors, Riemannian spaces, Riemannian curvature, and spaces of constant curvature. The next three chapters are concerned with applications to classical dynamics, hydrodynamics, elasticity, electromagnetic radiation, and the theorems of Stokes and Green. In the final chapter, an introduction is given to non-Riemannian spaces including such subjects as affine, Weyl, and projective spaces. There are two appendixes which discuss the reduction of a quadratic form and multiple integration. At the conclusion of each chapter a summary of the most important formulas and a set of exercises are given. More exercises are scattered throughout the text. The special and general theory of relativity is briefly discussed where applicable.
1. Spaces and Tensors 1.1 The generalized idea of a space 1.2 Transformation of coordinates. Summation convention 1.3 Contravariant vectors and tensors. Invariants 1.4 Covariant vectors and tensors. Mixed tensors 1.5 Addition, multiplication, and contraction of tensors 1.6 Tests for tensor character 1.7 Compressed notation Summary I, Exercises I II. Basic Operations in Riemannian Space 2.1 The metric tensor and the line element 2.2 The conjugate tensor. Lowering and raising suffixes 2.3 Magnitude of a vector. Angle between vectors 2.4 Geodesics and geodesic null lines. Christoffel symbols 2.5 Derivatives of tensors 2.6 Special coordinate systems 2.7 Frenet formulae Summary II, Exercises II III. Curvature of Space 3.1 The curvature tensor 3.2 The Ricci tensor, the curvature invariant, and the Einstein tensor 3.3 Geodesic deviation 3.4 Riemannian curvature 3.5 Parallel propagation Summary III, Exercises III IV. Special Types of Space 4.1 Space of constant curvature 4.2 Flat space 4.3 Cartesian tensors 4.4 A space of constant curvature regarded as a sphere in a flat space Summary IV, Exercises IV V. Applications to Classical Dynamics 5.1 Physical components of tensors 5.2 Dynamics of a particle 5.3 Dynamics of a rigid body 5.4 Moving frames of reference 5.5 General dynamical systems Summary V, Exercises V VI. Applications to hydrodynamics, elasticity, and electromagnetic radiation 6.1 Hydrodynamics 6.2 Elasticity 6.3 Electromagnetic radiation Summary VI, Exercises VI VII. Relative Tensors, Ideas of Volume, Green-Stokes Theorems 7.1 Relative tensors, generalized Kronecker delta, permutation symbol 7.2 Change of weight. Differentiation 7.3 Extension 7.4 Volume 7.5 Stokes' theorem 7.6 Green's theorem Summary VII, Exercises VII VIII. Non-Riemannian spaces 8.1 Absolute derivative. Spaces with a linear connection. Paths 8.2 Spaces with symmetric connection. Curvature 8.3 Weyl spaces. Riemannian spaces. Projective spaces Summary VIII, Exercises VIII Appendix A. Reduction of a Quadratic Form Appendix B. Multiple integration Bibliography, Index
