Tensor Calculus

By: J. L. SYNGE

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Paperback | 1 July 1978

At a Glance

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324 Pages

Edition Type
New edition

Dimensions(cm)
20.9 x 14.2 x 1.91

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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.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Calculus

Spaces and Tensors
The generalized idea of a space
Transformation of coordinates
Summation convention
Contravariant vectors and tensors
Invariants
Covariant vectors and tensors
Mixed tensors
Addition, multiplication, and contraction of tensors
Tests for tensor character
Compressed notation
Summary I
Exercises I
Basic Operations in Riemannian Space
The metric tensor and the line element
The conjugate tensor
Lowering and raising suffixes
Magnitude of a vector
Angle between vectors
Geodesics and geodesic null lines
Christoffel symbols
Derivatives of tensors
Special coordinate systems
Frenet formulae
Summary II
Exercises II
Curvature of Space
The curvature tensor
The Ricci tensor, the curvature invariant, and the Einstein tensor
Geodesic deviation
Riemannian curvature
Parallel propagation
Summary III
Exercises III
Special Types of Space
Space of constant curvature
Flat space
Cartesian tensors
A space of constant curvature regarded as a sphere in a flat space
Summary IV
Exercises IV
Applications to Classical Dynamics
Physical components of tensors
Dynamics of a particle
Dynamics of a rigid body
Moving frames of reference
General dynamical systems
Summary V
Exercises V
Applications to hydrodynamics, elasticity, and electromagnetic radiation
Hydrodynamics
Elasticity
Electromagnetic radiation
Summary VI
Exercises VI
Relative Tensors, Ideas of Volume, Green-Stokes Theorems
Relative tensors, generalized Kronecker delta, permutation symbol
Change of weight
Differentiation
Extension
Volume
Stokes' theorem
Green's theorem
Summary VII
Exercises VII
Non-Riemannian spaces
Absolute derivative
Spaces with a linear connection
Paths
Spaces with symmetric connection
Curvature
Weyl spaces
Riemannian spaces
Projective spaces
Summary VIII
Exercises VIII
Reduction of a Quadratic Form
Multiple integration
Bibliography
Index
Table of Contents provided by Publisher. All Rights Reserved.

Tensor Calculus

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