Tensor Calculus with Applications

Contents
Translator's Introduction
Preface to the Russian Edition
Chapter 1. Vector Spaces
	1.1 Concept of a Vector Space
	1.2 Linear Dependence of Vectors
	1.3 Dimension and Basis of a Vector Space
	1.4 Orthonormal Bases in a Three-Dimensional Space. The Scalar Product of Vectors
	1.5 The Vector Product and Triple Products of Vectors
	1.6 Transformations of Orthonormal Basis. Fundamental Problem of Tensor Calculus
	1.7 Some Topics in Three-Dimensional Analytic Geometry
Chapter 2. Multilinear Forms and Tensors
	2.1 Linear Forms
	2.2 Bilinear Forms
	2.3 Multilinear Forms. General Definition of a Tensor
	2.4 Algebraic Operations on Tensors
	2.5 Symmetric and Antisymmetric Tensors
Chapter 3. Linear Transformations and Second-Order Tensors
	3.1 Linear Transformations
	3.2 The Matrix of a Linear Transformation
	3.3 The Determinant of the Matrix of a Linear Transformation. The Rank of a Matrix
	3.4 Linear Transformations and Bilinear Forms
	3.5 Multiplication of Linear Transformations and Matrices
	3.6 Inverse Linear Transformations and Inverse Matrices
	3.7 The Group of Linear Transformations and Its Subgroups
Chapter 4. Reduction of the Matrix of Linear Transformation to the Simplest Form
	4.1 Eigenvectors and Eigenvalues of a Linear Transformation
	4.2 Reduction of the Matrix of Linear Transformation to the Simplest Form in the Case of Distinct Eigenvalues
	4.3 Matrix Polynomials and the Hamilton-Cayley Theorem
	4.4 Properties of Eigenvectors and Eigenvalues of a Symmetric Linear Transformation
	4.5 Diagonalization of the Matrix of a Symmetric Linear Transformation
	4.6 Reduction of a Quadratic Form to Canonical Form
	4.7 Representation of a Nonsingular Linear Transformation as the Product of a Symmetric Transformation and an Orthogonal Transformation
Chapter 5. The General Theory of Second-Degree Surfaces
	5.1 General Equation of a Second-Degree Surface
	5.2 Reduction of the General Equation of Second-Degree Surfaces to the Simplest Form
	5.3 Determination of the Type of a Second-Degree Surface by Means of Its Invariants
	5.4 Classification of Second-Degree Surfaces
	5.5 Application of the Theory of Invariants Classification of Second-Degree Surfaces
	5.6 Central and Noncentral Second-Degree Surfaces
	5.7 Examples
Chapter 6. Applications of Tensor Calculus to Some Problems of Mechanics and Physics
	6.1 The Inertia Tensor
	6.2 Properties of Crystals Connected with Second-Order Tensors
	6.3 The Stress and Strain Tensors
	6.4 Further Properties of Crystals
Chapter 7. Foundations of Tensor Analysis
	7.1 Tensor Fields and Their Differentiation
	7.2 Mechanics of a Deformable Medium
	7.3 Orthogonal Curvilinear Coordinate Systems
	7.4 Moving Frame of an Orthogonal Curvilinear Coordinate System and Tensor Fields
	7.5 Differentiation of a Tensor Field in Curvilinear Coordinates
Selected Answers and Hints
1.1-1.3
1.4
1.5
1.6
1.7
2.1-2.4
2.5-3.1
3.2
3.3
3.4
3.5
3.6-3.7
4.1
4.2
4.3-4.5
4.6
4.7-5
6.1
6.2
6.3
6.4
7.1-7.3
7.4-7.5
Bibliography
1-9
10-21
22-24
Index