Tensor and Vector Analysis: With Applications to Differential Geometry

Overview


Concise and user-friendly, this college-level text assumes only a knowledge of basic calculus in its elementary and gradual development of tensor theory. The introductory approach bridges the gap between mere manipulation and a genuine understanding of an important aspect of both pure and applied mathematics.
Beginning with a consideration of coordinate transformations and mappings, the treatment examines loci in three-space, transformation of coordinates in space and ...
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Tensor and Vector Analysis

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Overview


Concise and user-friendly, this college-level text assumes only a knowledge of basic calculus in its elementary and gradual development of tensor theory. The introductory approach bridges the gap between mere manipulation and a genuine understanding of an important aspect of both pure and applied mathematics.
Beginning with a consideration of coordinate transformations and mappings, the treatment examines loci in three-space, transformation of coordinates in space and differentiation, tensor algebra and analysis, and vector analysis and algebra. Additional topics include differentiation of vectors and tensors, scalar and vector fields, and integration of vectors. The concluding chapter employs tensor theory to develop the differential equations of geodesics on a surface in several different ways to illustrate further differential geometry.
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Product Details

  • ISBN-13: 9780486498010
  • Publisher: Dover Publications
  • Publication date: 11/21/2012
  • Series: Dover Books on Mathematics Series
  • Pages: 256
  • Sales rank: 813,525
  • Product dimensions: 8.80 (w) x 5.90 (h) x 0.60 (d)

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

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