Table of Contents
1 Coordinate Transformations and Mappings 3
Two Aspects 3
A Change of Notation 6
Rotations in Three Dimensions 7
The Kronecker Delta 10
2 Loci in Three-Space 12
One-Dimensional Extent 12
Two-Dimensional Extent 13
Some Differential Geometry of Space Curves 15
Some Differential Geometry of Surfaces 21
3 Transformation of Coordinates in Space; Differentiation 26
Linear Transformation 26
Transformation to Curvilinear Coordinates 26
Partial Differentiation 28
Derivative of a Determinant 30
Cramer's Rule 32
Product of Determinants 33
4 Tensor Algebra 36
Cogredience and Contragredience 36
First View of a Tensor 38
Operations of Tensor Algebra 40
Transitivity, Symmetry, Skew-Symmetry 42
5 Tensor Analysis 45
The Fundamental Quadratic Form 45
Covariant and Contravariant Tensors of the First Order 48
A Quadratic Form from a Tensor Product 51
Definition of a General Tensor 52
Inner Product of Two Vectors 52
Associate Tensors 54
6 Vector Analysis 57
Length of a Vector 57
Angle Between Two Vectors; Orthogonal Vectors 58
Some Applications 61
Geometric Meaning of Contravariant and Covariant Components of a Vector 65
Alternative (Reciprocal) Geometrical Interpretation of Contravariant and Covariant Components of a Vector 68
7 Vector Algebra 72
Base Vectors 72
Products of Vectors 74
Linear Dependence 79
Vector Equation of a Line 80
Applications in Mechanics 82
Vector Methods in Geometry 84
8 Differentiation of Vectors 88
Vector Functions of a Scalar Variable 88
Frenet Formulas for Space Curves 90
Application in Mechanics 93
Motion in a Plane 93
Law of Transformation for Velocity Components 98
Vector Functions of Two Scalar Parameters 100
Riemannian Metric 101
Extrinsic and Intrinsic Geometry 103
Surface Normal and Tangent Plane 105
Local and Global Geometry 105
9 Differentiation of Tensors 109
Equivalence of Forms; Christoffel Symbols 109
The Riemann-Christoffel Tensor 112
Covariant Differentiation; Parallelism of Vectors 116
Covariant Derivative of Covariant Tensors 118
Covariant Derivative of a General Tensor 120
Tensors Which Behave as Constants 121
10 Scalar and Vector Fields 124
Fields 124
Divergence of a Vector Fields; the Laplacian 125
The Curl of a Vector Field 128
Physical Components 130
Some Vector Identities Involving Divergence and Curl 131
Frenet Formulas in General Coordinates 132
The Acceleration Vector 134
Equations of Motion 135
The Lagrange Form of the Equations of Motion 137
11 Integration of Vectors 147
Line Integrals 147
Vector Form of Line Integrals 153
Surface and Volume Integrals 156
Green's Theorem in the Plane 162
Simply and Multiply Connected Regions 167
Independence of the Path of Integration 174
Test for Independence of Path 176
Green's Theorem in Three-Space (The Divergence Theorem) 181
An Application of the Divergence Theorem 186
The Theorem of Stokes (The Curl Theorem) 189
Applications of the Curl Theorem 194
12 Geodesic and Union Curves 200
Two-Dimensional Curved Space 200
Geodesics as Curves of Shortest Distance 201
The Second Fundamental Form of a Surface 207
Normal Curvature of a Surface 212
Curvature Formulas 214
Geodesic Curvature 218
Geodesics as Auto-Parallel Curves 221
A Generalization of the Theorem of Meusnier 227
Union Curves on a Surface 230
Union Curves and Dynamical Trajectories 233
Index 239
