3.2 Arc Length and Surface in Curvilinear Coordinates.

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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.

This book comprehensively presents topics, such as Dirac notation, tensor analysis, elementary differential geometry of moving surfaces, and k-differential forms. Additionally, two new chapters of Cartan differential forms and Dirac and tensor notations in quantum mechanics are added to this second edition. The reader is provided with hands-on calculations and worked-out examples at which he will learn how to handle the bra-ket notation, tensors, differential geometry, and differential forms; and to apply them to the physical and engineering world. Many methods and applications are given in CFD, continuum mechanics, electrodynamics in special relativity, cosmology in the Minkowski four-dimensional spacetime, and relativistic and non-relativistic quantum mechanics. Tensors, differential geometry, differential forms, and Dirac notation are very useful advanced mathematical tools in many fields of modern physics and computational engineering. They are involved in special and general relativity physics, quantum mechanics, cosmology, electrodynamics, computational fluid dynamics (CFD), and continuum mechanics. The target audience of this all-in-one book primarily comprises graduate students in mathematics, physics, engineering, research scientists, and engineers. .