Tensor Calculus

Cover......Page 1
Preface......Page 4
Contents......Page 6
1.2 Repeated Indices in Sums......Page 10
1.4 Substitutions......Page 11
1.5 Kronecker Delta and Algebraic Manipulations......Page 12
2.2 Tensor Notation for Matrices, Vectors, and Determinants......Page 17
2.4 Matrix Expressions for Linear Systems and Quadratic Forms......Page 19
2.5 Linear Transformations......Page 20
2.6 General Coordinate Transformations......Page 21
2.7 The Chain Rule for Partial Derivatives......Page 22
3.1 Coordinate Transformations......Page 32
3.2 First-Order Tensors......Page 35
3.3 Invariants......Page 37
3.5 The Stress Tensor......Page 38
3.6 Cartesian Tensors......Page 40
4.1 Fundamental Operations......Page 52
4.3 Tensor Equations......Page 54
5.2 Arc Length in Euclidean Space......Page 60
5.3 Generalized Metrics; The Metric Tensor......Page 61
5.5 Generalized Inner-Product Spaces......Page 64
5.6 Concepts of Length and Angle......Page 65
6.2 Christoffel Symbols of the First Kind......Page 77
6.3 Christoffel Symbols of the Second Kind......Page 79
6.4 Covariant Differentiation......Page 80
6.5 Absolute Differentiation along a Curve......Page 81
6.6 Rules for Tensor Differentiation......Page 83
7.2 Length and Angle under an Indefinite Metric......Page 92
7.3 Null Curves......Page 93
7.4 Regular Curves: Unit Tangent Vector......Page 94
7.5 Regular Curves: Unit Principal Normal and Curvature......Page 95
7.6 Geodesies as Shortest Arcs......Page 97
8.2 Properties of the Riemann Tensor......Page 110
8.3 Riemannian Curvature......Page 112
8.4 The Ricci Tensor......Page 114
9.1 Zero Curvature and the Euclidean Metric......Page 123
9.2 Flat Riemannian Spaces......Page 125
9.3 Normal Coordinates......Page 126
9.5 The Einstein Tensor......Page 128
10.2 Curve Theory; The Moving Frame......Page 136
10.4 Regular Surfaces......Page 139
10.5 Parametric Lines; Tangent Space......Page 141
10.6 First Fundamental Form......Page 142
10.7 Geodesies on a Surface......Page 144
10.8 Second Fundamental Form......Page 145
10.9 Structure Formulas for Surfaces......Page 146
10.10 Isometries......Page 147
11.2 Particle Kinematics in Rectangular Coordinates......Page 163
11.3 Particle Kinematics in Curvilinear Coordinates......Page 164
11.4 Newton\'s Second Law in Curvilinear Coordinates......Page 165
11.5 Divergence, Laplacian, Curl......Page 166
12.2 Event Space......Page 173
12.3 The Lorentz Group and the Metric of SR......Page 175
12.4 Simple Lorentz Matrices......Page 176
12.6 Relativistic Kinematics......Page 178
12.7 Relativistic Mass, Force, and Energy......Page 180
12.8 Maxwell\'s Equations in SR......Page 181
13.2 Abstract Vector Spaces and the Group Concept......Page 198
13.3 Important Concepts for Vector Spaces......Page 199
13.4 The Algebraic Dual of a Vector Space......Page 200
13.5 Tensors on Vector Spaces......Page 202
13.6 Theory of Manifolds......Page 203
13.7 Tangent Space; Vector Fields on Manifolds......Page 206
13.8 Tensor Fields on Manifolds......Page 208
ANSWERS TO SUPPLEMENTARY PROBLEMS......Page 222
INDEX......Page 232