Tensor Calculus
Fundamental introduction for beginning student of absolute differential calculus and for those interested in applications of tensor calculus to mathematical physics and engineering. Topics include spaces and tensors; basic operations in Riemannian space, curvature of space, special types of space, relative tensors, ideas of volume, and more.
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Fundamental introduction of absolute differential calculus and for those interested in applications of tensor calculus to mathematical physics and engineering. Topics include spaces and tensors; basic operations in Riemannian space, curvature of space, more.
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
