张量分析简论

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 Newtons Law and Tensor Calculus

Rigid Bodies

New Conservation Laws

Nomenclature

Newtons Law in Cartesian Components

Newtons Law in Plane Polar Coordinates

The Physical Components of a Vector

The Christoffel Symbols

General Three-Dimensional Coordinates

Newtons 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

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