Preface to the Second Edition
Preface to the First Edition
CHAPTER Ⅰ Introduction: Vectors and Tensors
Three-Dimensional Euclidean Space
Directed Line Segments
Addition of Two Vectors
Multiplication of a Vector v by a Scalar
Things That Vectors May Represent
Cartesian Coordinates
The Dot Product
Cartesian Base Vectors
The Interpretation of Vector Addition
The Cross Product
Alternative Interpretation of the Dot and Cross Product. Tensors
Definitions
The Cartesian Components of a Second Order Tensor
The Cartesian Basis for Second Order Tensors
Exercises
CHAPTER Ⅱ General Bases and Tensor Notation
General Bases
The Jacobian of a Basis Is Nonzero
The Summation Convention
Computing the Dot Product in a General Basis
Reciprocal Base Vectors
The Roof (Contravariant) and Cellar (Covariant) Components of a Vector
Simplification of the Component Form of the Dot Product in a General Basis
Computing the Cross Product in a General Basis
A Second Order Tensor Has Four Sets of Components in General
Change of Basis
Exercises
CHAPTER Ⅲ Newton’’s Law and Tensor Calculus
Rigid Bodies
New Conservation Laws
Nomenclature
Newton’’s Law in Cartesian Components
Newton’’s Law in Plane Polar Coordinates
The Physical Components of a Vector
The Christoffel Symbols
General Three-Dimensional Coordinates
Newton’’s Law in General Coordinates
Computation of the Christoffel Symbols
An Alternative Formula for Computing the Christoffel Symbols
A Change of Coordinates
Transformation of the Christoffel Symbols
Exercises
CHAPTER Ⅳ The Gradient, the Del Operator, Covariant Differentiation, and the Divergence Theorem
The Gradient
Linear and Nonlinear Eigenvalue Problems
The Del Operator
The Divergence, Curl, and Gradient of a Vector Field
The lnvariance of V. v, V x v, and Vv
The Covariant Derivative
The Component Forms of V- v, V x v, and Vv
The Kinematics of Continuum Mechanics
The Divergence Theorem
Differential Geometry
Exercises
Index
