INTRODUCTION TO TENSOR ANALYSIS

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The subject of Tensor Analysis deals with the problem of the formulation of the relation between various entities in forms which remain invariant when we pass from one system of coordinates to another. The invariant form of equation is necessarily related to the possible system of coordinates with r...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autores principales: Koranga, Bipin Singh (Autor), Padaliya, Sanjay Kumar (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: [S.l.] : RIVER PUBLISHERS, 2020.
Colección:River Publishers Series in Mathematical and Engineering Sciences
Temas:
Calculus of tensors.
Tensor products.
Tensor algebra.
Calcul tensoriel.
Produits tensoriels.
Algèbre tensorielle.
SCIENCE / Energy
Calculus of tensors
Tensor products
Electronic books.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Preface v
  • Syllabus vii
  • 1 Introduction 1
  • 1.1 Symbols Multi-Suffix 2
  • 1.2 Summation Convention 3
  • 2 Cartesian Tensor 7
  • 2.1 Introduction 7
  • 2.2 Transformation of Coordinates 8
  • 2.3 Relations Between the Direction Cosines 10
  • 2.4 Transformation of Velocity Components 11
  • 2.5 First-Order Tensors 12
  • 2.6 Second-Order Tensors 13
  • 2.7 Notation for Tensors 14
  • 2.8 Algebraic Operations on Tensors 14
  • 2.8.1 Sum and Difference of Tensors 15
  • 2.8.2 Product of Tensors 16
  • 2.9 Quotient Law of Tensors 17
  • 2.10 Contraction Theorem 19
  • 2.11 Symmetric and Skew-Symmetric Tensor 21
  • 2.12 Alternate Tensor 23
  • 2.13 Kronecker Tensor 24
  • 2.14 Relation Between Alternate and Kronecker Tensors 25
  • 2.15 Matrices and Tensors of First and Second Orders 26
  • 2.16 Product of Two Matrices 28
  • 2.17 Scalar and Vector Inner Product 31
  • 2.17.1 Two Vectors 31
  • 2.17.2 Scalar Product 31
  • 2.17.3 Vector Product 31
  • 2.18 Tensor Fields 32
  • 2.18.1 Gradient of Tensor Field 32
  • 2.18.2 Divergence of Vector Point Function 34
  • 2.18.3 Curl of Vector Point Function 34
  • 2.19 Tensorial Formulation of Gauss's Theorem 35
  • 2.20 Tensorial Formulation of Stoke's Theorem 35
  • 2.21 Exercise 36
  • 3 Tensor in Physics 39
  • 3.1 Kinematics of Single Particle 39
  • 3.1.1 Momentum 40
  • 3.1.2 Acceleration 40
  • 3.1.3 Force 40
  • 3.2 Kinetic Energy and Potential Energy 41
  • 3.3 Work Function and Potential Energy 41
  • 3.4 Momentum and Angular Momentum 43
  • 3.5 Moment of Inertia 44
  • 3.6 Strain Tensor at Any Point 46
  • 3.7 Stress Tensor at any Point P 49
  • 3.7.1 Normal Stress 50
  • 3.7.2 Simple Stress 50
  • 3.7.3 Shearing Stress 50
  • 3.8 Generalised Hooke's Law 50
  • 3.9 Isotropic Tensor 51
  • 3.10 Exercises 52
  • 4 Tensor in Analytic Solid Geometry 55
  • 4.1 Vector as Directed Line Segments 55
  • 4.2 Geometrical Interpretation of the Sum of two Vectors 57
  • 4.3 Length and Angle between Two Vectors 57
  • 4.4 Geometrical Interpretation of Scalar and Vector Products 58.
  • 5 General Tensor 67
  • 5.1 Curvilinear Coordinates 68
  • 5.2 Coordinate Transformation Equation 68
  • 5.3 Contravariant and Covariant Tensor 69
  • 5.4 Contravariant Vector or Contravariant Tensor of Order-One 71
  • 5.5 Covariant Vector or Covariant Tensor of Order-One 71
  • 5.6 Mixed Second-Order Tensor 72
  • 5.7 General Tensor of Any Order 72
  • 5.8 Metric Tensor 73
  • 5.9 Associate Contravariant Metric Tensor 74
  • 5.10 Associate Metric Tensor 75
  • 5.11 Christoffel Symbols of the First and Second -Kind 76
  • 5.12 Covariant Derivative of a Covariant Vector 79
  • 5.13 Covariant Derivative of a Contravariant Vector 80
  • 5.14 Exercises 81
  • 6 Tensor in Relativity 85
  • 6.1 Special Theory of Relativity 85
  • 6.2 Four-Vectors in Relativity 88
  • 6.3 Maxwell's Equations 91
  • 6.4 General Theory of Relativity 94
  • 6.5 Spherically Symmetrical Metric 95
  • 6.6 Planetary Motion 96
  • 6.7 Exercises 97
  • 7 Geodesics and Its Coordinate 99
  • 7.1 Families of Curves 99
  • 7.2 Euler's Form 100
  • 7.3 Geodesics 101
  • 7.4 Geodesic Form of the Line Elements 103
  • 7.5 Geodesic Coordinate 105
  • 7.6 Exercise 107
  • Index 109
  • About the Authors 111.
  • 4.4.1 Scalar Triple Product 60
  • 4.4.2 Vector Triple Products 60
  • 4.5 Tensor Formulation of Analytical Solid Geometry 61
  • 4.5.1 Distance Between Two Points P(xi) and Q(yi) 61
  • 4.5.2 Angle Between Two Lines with Direction Cosines 61
  • 4.5.3 The Equation of Plane 62
  • 4.5.4 Condition for Two Line Coplanar 63
  • 4.6 Exercises 64.

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