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