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Table of Contents
Preface to the Second Edition; Preface to the First Edition; Contents; About the Authors; Chapter 1: General Basis and Bra-Ket Notation; 1.1 Introduction to General Basis and Tensor Types; 1.2 General Basis in Curvilinear Coordinates; 1.2.1 Orthogonal Cylindrical Coordinates; 1.2.2 Orthogonal Spherical Coordinates; 1.3 Eigenvalue Problem of a Linear Coupled Oscillator; 1.4 Notation of Bra and Ket; 1.5 Properties of Kets; 1.6 Analysis of Bra and Ket; 1.6.1 Bra and Ket Bases; 1.6.2 Gram-Schmidt Scheme of Basis Orthonormalization; 1.6.3 Cauchy-Schwarz and Triangle Inequalities.
1.6.4 Computing Ket and Bra Components1.6.5 Inner Product of Bra and Ket; 1.6.6 Outer Product of Bra and Ket; 1.6.7 Ket and Bra Projection Components on the Bases; 1.6.8 Linear Transformation of Kets; 1.6.9 Coordinate Transformations; 1.6.10 Hermitian Operator; 1.7 Applying Bra and Ket Analysis to Eigenvalue Problems; References; Chapter 2: Tensor Analysis; 2.1 Introduction to Tensors; 2.2 Definition of Tensors; 2.2.1 An Example of a Second-Order Covariant Tensor; 2.3 Tensor Algebra; 2.3.1 General Bases in General Curvilinear Coordinates; 2.3.1.1 Orthogonal Cylindrical Coordinates.
2.3.1.2 Orthogonal Spherical Coordinates2.3.2 Metric Coefficients in General Curvilinear Coordinates; 2.3.3 Tensors of Second Order and Higher Orders; 2.3.4 Tensor and Cross Products of Two Vectors in General Bases; 2.3.4.1 Tensor Product; 2.3.4.2 Cross Product; 2.3.5 Rules of Tensor Calculations; 2.3.5.1 Calculation of Tensor Components; 2.3.5.2 Addition Law; 2.3.5.3 Outer Product; 2.3.5.4 Contraction Law; 2.3.5.5 Inner Product; 2.3.5.6 Indices Law; 2.3.5.7 Quotient Law; 2.3.5.8 Symmetric Tensors; 2.3.5.9 Skew-Symmetric Tensors; 2.4 Coordinate Transformations.
2.4.1 Transformation in the Orthonormal Coordinates2.4.2 Transformation of Curvilinear Coordinates in EN; 2.4.3 Examples of Coordinate Transformations; 2.4.3.1 Cylindrical Coordinates; 2.4.3.2 Spherical Coordinates; 2.4.4 Transformation of Curvilinear Coordinates in RN; 2.5 Tensor Calculus in General Curvilinear Coordinates; 2.5.1 Physical Component of Tensors; 2.5.2 Derivatives of Covariant Bases; 2.5.3 Christoffel Symbols of First and Second Kind; 2.5.4 Prove That the Christoffel Symbols Are Symmetric; 2.5.5 Examples of Computing the Christoffel Symbols.
2.5.6 Coordinate Transformations of the Christoffel Symbols2.5.7 Derivatives of Contravariant Bases; 2.5.8 Derivatives of Covariant Metric Coefficients; 2.5.9 Covariant Derivatives of Tensors; 2.5.9.1 Contravariant First-Order Tensors with Components Ti; 2.5.9.2 Covariant First-Order Tensors with Components Ti; 2.5.9.3 Second-Order Tensors; 2.5.10 Riemann-Christoffel Tensor; 2.5.11 Ricciś Lemma; 2.5.12 Derivative of the Jacobian; 2.5.13 Ricci Tensor; 2.5.14 Einstein Tensor; References; Chapter 3: Elementary Differential Geometry; 3.1 Introduction.
1.6.4 Computing Ket and Bra Components1.6.5 Inner Product of Bra and Ket; 1.6.6 Outer Product of Bra and Ket; 1.6.7 Ket and Bra Projection Components on the Bases; 1.6.8 Linear Transformation of Kets; 1.6.9 Coordinate Transformations; 1.6.10 Hermitian Operator; 1.7 Applying Bra and Ket Analysis to Eigenvalue Problems; References; Chapter 2: Tensor Analysis; 2.1 Introduction to Tensors; 2.2 Definition of Tensors; 2.2.1 An Example of a Second-Order Covariant Tensor; 2.3 Tensor Algebra; 2.3.1 General Bases in General Curvilinear Coordinates; 2.3.1.1 Orthogonal Cylindrical Coordinates.
2.3.1.2 Orthogonal Spherical Coordinates2.3.2 Metric Coefficients in General Curvilinear Coordinates; 2.3.3 Tensors of Second Order and Higher Orders; 2.3.4 Tensor and Cross Products of Two Vectors in General Bases; 2.3.4.1 Tensor Product; 2.3.4.2 Cross Product; 2.3.5 Rules of Tensor Calculations; 2.3.5.1 Calculation of Tensor Components; 2.3.5.2 Addition Law; 2.3.5.3 Outer Product; 2.3.5.4 Contraction Law; 2.3.5.5 Inner Product; 2.3.5.6 Indices Law; 2.3.5.7 Quotient Law; 2.3.5.8 Symmetric Tensors; 2.3.5.9 Skew-Symmetric Tensors; 2.4 Coordinate Transformations.
2.4.1 Transformation in the Orthonormal Coordinates2.4.2 Transformation of Curvilinear Coordinates in EN; 2.4.3 Examples of Coordinate Transformations; 2.4.3.1 Cylindrical Coordinates; 2.4.3.2 Spherical Coordinates; 2.4.4 Transformation of Curvilinear Coordinates in RN; 2.5 Tensor Calculus in General Curvilinear Coordinates; 2.5.1 Physical Component of Tensors; 2.5.2 Derivatives of Covariant Bases; 2.5.3 Christoffel Symbols of First and Second Kind; 2.5.4 Prove That the Christoffel Symbols Are Symmetric; 2.5.5 Examples of Computing the Christoffel Symbols.
2.5.6 Coordinate Transformations of the Christoffel Symbols2.5.7 Derivatives of Contravariant Bases; 2.5.8 Derivatives of Covariant Metric Coefficients; 2.5.9 Covariant Derivatives of Tensors; 2.5.9.1 Contravariant First-Order Tensors with Components Ti; 2.5.9.2 Covariant First-Order Tensors with Components Ti; 2.5.9.3 Second-Order Tensors; 2.5.10 Riemann-Christoffel Tensor; 2.5.11 Ricciś Lemma; 2.5.12 Derivative of the Jacobian; 2.5.13 Ricci Tensor; 2.5.14 Einstein Tensor; References; Chapter 3: Elementary Differential Geometry; 3.1 Introduction.