Table of Contents
Preface to the Second Edition v
Preface to the First Edition vii
Chapter 1 Vector Algebra 1
1.1 Introduction
1.2 Definition of a Vector
1.3 Geometric Representation of a Vector
1.4 Addition and Scalar Multiplication
1.5 Some Applications in Geometry
1.6 Scalar Product
1.7 Vector Product
1.8 Lines and Planes in Space
1.9 Scalar and Vector Triple Products
Chapter 2 Differential Calculus of Vector Functions of One Variable 75
2.1 Vector Functions of a Real Variable
2.2 Algebra of Vector Functions
2.3 Limit, Continuity, and Derivatives
2.4 Space Curves and Tangent Vectors
2.5 Arc Length as a Parameter
2.6 Simple Geometry of Curves
2.7 Torsion and Frenet-Serret Formulas
2.8 Applications to Curvilinear Motions
2.9 Curvilinear Motion in Polar Coordinates
2.10 Cylindrical and Spherical Coordinates
Chapter 3 Differential Calculus of Scalar and Vector Fields 147
3.1 Scalar and Vector Fields
3.2 Algebra of Vector Fields
3.3 Directional Derivative of a Scalar Field
3.4 Gradient of a Scalar Field
3.5 Divergence of a Vector Field
3.6 Curl of a Vector Field
3.7 Other Properties of the Divergence and the Curl
3.8 Curvilinear Coordinate Systems
3.9 Gradient Divergence, and Curl in Orthogonal Curvilinear Coordinate Systems
Chapter 4 Integral Calculus of Scalar and Vector Fields 207
4.1 Line Integrals of Scalar Fields
4.2 Line Integrals of Vector Fields
4.3 Properties of Line Integrals
4.4 Line Integrals Independent of Path
4.5 Green's Theorem in the Plane
4.6 Parametric Representation of Surfaces
4.7 Surface Area
4.8 Surface Integrals
4.9 The Divergence Theorem
4.10 Applications of the Divergence Theorem
4.11 Stokes' Theorem
4.12 Some Applications of Stokes' Theorem
Chapter 5 Tensors in Rectangular Cartesian Coordinate Systems 307
5.1 Introduction
5.2 Notation and Summation Convention
5.3 Transformations of Rectangular Cartesian Coordinate Systems
5.4 Transformation Law for Vectors
5.5 Cartesian Tensors
5.6 Stress Tensor
5.7 Algebra of Cartesian Tensors
5.8 Principal Axes of Second Order Tensors
5.9 Differentiation of Cartesian Tensor Fields
5.10 Strain Tensor
Chapter 6 Tensors in General Coordinates 373
6.1 Oblique Cartesian Coordinates
6.2 Reciprocal Basis; Transformations of Oblique Coordinate Systems
6.3 Tensors in Oblique Cartesian Coordinate Systems
6.4 Algebra of Tensors in Oblique Coordinates
6.5 The Metric Tensor
6.6 Transformations of Curvilinear Coordinates
6.7 General Tensors
6.8 Covariant Derivative of a Vector
6.9 Transformation of Christoffel Symbols
6.10 Covariant Derivative of Tensors
6.11 Gradient, Divergence, Laplacian, and Curl in General Coordinates
6.12 Equations of Motion of a Particle
Solutions to Selected Problems 469
Index 495
