A Brief on Tensor Analysis

Cover......Page 1
Preface to the Second Edition......Page 8
Preface to the First Edition......Page 9
Contents......Page 12
CHAPTER I Introduction: Vectors and Tensors......Page 14
Directed Line Segments......Page 16
Addition of Two Vectors......Page 17
Things That Vectors May Represent......Page 18
Cartesian Coordinates......Page 19
The Dot Product......Page 20
The Interpretation of Vector Addition......Page 23
The Cross Product......Page 24
Alternative Interpretation of the Dot and Cross Product. Tensors......Page 28
Definitions......Page 29
The Cartesian Components of a Second Order Tensor......Page 30
The Cartesian Basis for Second Order Tensors......Page 32
Exercises......Page 33
General Bases......Page 38
The Summation Convention......Page 40
Reciprocal Base Vectors......Page 41
The Roof (Contravariant) and Cellar (Covariant) Components of a Vector......Page 43
Simplification of the Component Form of the Dot Product in a General Basis......Page 44
Computing the Cross Product in a General Basis......Page 45
A Second Order Tensor Has Four Sets of Components in General......Page 47
Change of Basis......Page 49
Exercises......Page 51
Rigid Bodies......Page 58
New Conservation Laws......Page 59
Nomenclature......Page 60
Newton\'s Law in Cartesian Components......Page 62
Newton\'s Law in Plane Polar Coordinates......Page 63
The Physical Components of a Vector......Page 64
The Christoffel Symbols......Page 65
General Three-Dimensional Coordinates......Page 67
Newton\'s Law in General Coordinates......Page 68
Computation of the Christoffel Symbols......Page 71
An Alternative Formula for Computing the Christoffel Symbols......Page 72
A Change of Coordinates......Page 75
Transformation of the Christoffel Symbols......Page 77
Exercises......Page 78
The Gradient......Page 83
Linear and Nonlinear Eigenvalue Problems......Page 87
The Del Operator......Page 88
The Divergence, Curl, and Gradient of a Vector Field......Page 89
The Invariance of div v, curl v, and grad v......Page 90
The Covariant Derivative......Page 91
The Component Forms of div v, curl v, and grad v......Page 92
The Kinematics of Continuum Mechanics......Page 93
The Divergence Theorem......Page 95
Differential Geometry......Page 99
Exercises......Page 109
Index......Page 118