Tensor Calculus

A Concise Course

By: BARRY SPAIN

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Paperback | 2 May 2003 | Edition Number 3

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144 Pages

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20.9 x 13.6 x 1.27

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"This book will prove to be a good introduction, both for the physicist who wishes to make applications and for the mathematician who prefers to have a short survey before taking up one of the more voluminous textbooks on differential geometry." — MathSciNet (Mathematical Reviews on the Web), American Mathematical Society
A compact exposition of the fundamental results in the theory of tensors, this text also illustrates the power of the tensor technique by its applications to differential geometry, elasticity, and relativity. The first five chapters--comprising tensor algebra, the line element, covariant differentiation, geodesics and parallelism, and curvature tensor--develop their subjects without undue rigor. The final three chapters function independently of each other and cover Euclidean three-dimensional differential geometry, Cartesian tensors and elasticity, and the theory of relativity. Both special and general theories of relativity are reviewed, with introductory material for readers unfamiliar with the concepts.

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"This book will prove to be a good introduction, both for the physicist who wishes to make applications and for the mathematician who prefers to have a short survey before taking up one of the more voluminous textbooks on differential geometry."

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Tensor Algebra
Introduction	p. 1
N-Dimensional space	p. 2
Transformation of co-ordinates	p. 2
Indicial and summation conventions	p. 3
Contravariant vectors	p. 4
Covariant vectors	p. 6
Invariants	p. 7
Second order tensors	p. 8
Higher order tensors	p. 9
Addition, subtraction and multiplication of tensors	p. 11
Contraction	p. 12
Quotient law	p. 13
Conjugate symmetric tensors of the second order	p. 15
The Line Element
Fundamental tensor	p. 17
Length of a Curve	p. 18
Magnitude of a vector	p. 19
Associate tensors	p. 20
Angle between two vectors--orthogonality	p. 21
Principal directions	p. 22
Covariant Differentiation
Christoffel symbols	p. 26
Transformation law of Christoffel symbols	p. 28
Covariant differentiation of vectors	p. 30
Covariant differentiation of tensors	p. 33
Laws of covariant differentiation	p. 35
Intrinsic derivatives	p. 36
Geodesics - Parallelism
Geodesics	p. 38
Null-geodesics	p. 41
Geodesic coordinates	p. 41
Parallelism	p. 44
Covariant derivative	p. 46
Curvature Tensor
Riemann-Christoffel tensor	p. 49
Curvature tensor	p. 50
Ricci tensor--Curvature invariant	p. 52
Bianchi's identity	p. 53
Riemannian curvature	p. 54
Flat space	p. 55
Space of constant curvature	p. 56
Euclidean Three-Dimensional Differential Geometry
Permutation tensors	p. 58
Vector product	p. 60
Frenet formulae	p. 61
Surface--First fundamental form	p. 63
Surface vectors	p. 66
Permutation surface tensor	p. 68
Surface covariant differentiation	p. 70
Geodesic curvature	p. 71
Normal vector	p. 73
Tensor derivatives of tensors	p. 74
Second fundamental form	p. 76
Third fundamental form	p. 77
Gauss-Codazzi equations	p. 78
Normal curvature--asymptotic lines	p. 79
Principal curvatures--lines of curvature	p. 81
Cartesian Tensors - Elasticity
Orthogonal transformations	p. 83
Rotations	p. 85
Cartesian tensors	p. 88
Infinitesimal strain	p. 89
Stress	p. 92
Equations of equilibrium	p. 94
Generalised Hooke's law	p. 95
Isotropic tensors	p. 96
Homogeneous and isotropic body	p. 98
Curvilinear coordinates	p. 100
Mechanics of continuous matter	p. 103
Theory of Relativity
Special theory	p. 106
Maxwell's Equations	p. 109
General theory	p. 111
Spherically symmetrical metric	p. 113
Schwarzschild metric	p. 115
Planetary motion	p. 116
Einstein's universe	p. 118
De Sitter's universe	p. 120
Bibliography	p. 122
Index	p. 123
Table of Contents provided by Ingram. All Rights Reserved.

Tensor Calculus

A Concise Course

At a Glance

Paperback

Industry Reviews

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