Tensor and Vector Analysis With Applications to Differential Geometry

by Springer, C. E.

• ISBN13: 9780486498010
• ISBN10: 0486498018
• Format: Paperback
• Copyright: 2012-11-21
• Publisher: Dover Publications

Summary

Assuming only a knowledge of basic calculus, this text presents an elementary and gradual development of tensor theory. From this treatment, the traditional material of courses on vector analysis is deduced as a particular case. In addition, the book forms an introduction to metric differential geometry. 1962 edition.

Author Biography

C. E. Springer was Professor of Mathematics at the University of Oklahoma.

Table of Contents

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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