An Introduction to Tensor Analysis

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This is a short introduction to the topic of Tensor Analysis. A tensor is an entity which is represented in any coordinate system by an array of numbers called its components. The components change from coordinate system to coordinate in a systematic way described by rules. The arrays of numbers are...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Koranga, Bipin Singh
Otros Autores: Padaliya, Sanjay Kumar
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Aalborg : River Publishers, 2021.
Colección:River Publishers Series in Mathematical and Engineering Sciences Ser.
Temas:
Tensor algebra.
Algèbre tensorielle.
Tensor algebra
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Intro
  • AN INTRODUCTION TOTENSOR ANALYSIS
  • Preface
  • Syllabus
  • Contents
  • 1 Introduction
  • 1.1 Symbols Multi-Suffix
  • 1.2 Summation Convention
  • References
  • 2 Cartesian Tensor
  • 2.1 Introduction
  • 2.2 Transformation of Coordinates
  • 2.3 Relations Between the Direction Cosines of Three Mutually Perpendicular Straight Lines
  • 2.4 Transformation of Velocity Components
  • 2.5 First-Order Tensors
  • 2.6 Second-Order Tensors
  • 2.7 Notation for Tensors
  • 2.8 Algebraic Operations on Tensors
  • 2.8.1 Sum and Difference of Tensors
  • 2.8.2 Product of Tensors
  • 2.9 Quotient Law of Tensors
  • 2.10 Contraction Theorem
  • 2.11 Symmetric and Skew-Symmetric Tensor
  • 2.12 Alternate Tensor
  • 2.13 Kronecker Tensor
  • 2.14 Relation Between Alternate and Kronecker Tensors
  • 2.15 Matrices and Tensors of First and Second Orders
  • 2.16 Product of Two Matrices
  • 2.17 Scalar and Vector Inner Product
  • 2.17.1 Two Vectors
  • 2.17.2 Scalar Product
  • 2.17.3 Vector Product
  • 2.18 Tensor Fields
  • 2.18.1 Gradient of Tensor Field
  • 2.18.2 Divergence of Vector Point Function
  • 2.18.3 Curl of Vector Point Function
  • 2.19 Tensorial Formulation of Gauss's Theorem
  • 2.20 Tensorial Formulation of Stoke's Theorem
  • 2.21 Exercise
  • References
  • 3 Tensor in Physics
  • 3.1 Kinematics of Single Particle
  • 3.1.1 Momentum
  • 3.1.2 Acceleration
  • 3.1.3 Force
  • 3.2 Kinetic Energy and Potential Energy
  • 3.3 Work Function and Potential Energy
  • 3.4 Momentum and Angular Momentum
  • 3.5 Moment of Inertia
  • 3.6 Strain Tensor at Any Point
  • 3.7 Stress Tensor at any Point P
  • 3.7.1 Normal Stress
  • 3.7.2 Simple Stress
  • 3.7.3 Shearing Stress
  • 3.8 Generalised Hooke's Law
  • 3.9 Isotropic Tensor
  • 3.10 Exercises
  • References
  • 4 Tensor in Analytic Solid Geometry
  • 4.1 Vector as Directed Line Segments
  • 4.2 Geometrical Interpretation of the Sum of two Vectors
  • 4.3 Length and Angle between Two Vectors
  • 4.4 Geometrical Interpretation of Scalar and Vector Products
  • 4.4.1 Scalar Triple Product
  • 4.4.2 Vector Triple Products
  • 4.5 Tensor Formulation of Analytical Solid Geometry
  • 4.5.1 Distance Between Two Points P(xi) and Q(yi)
  • 4.5.2 Angle Between Two Lines with Direction Cosines
  • 4.5.3 The Equation of Plane
  • 4.5.4 Condition for Two Line Coplanar
  • 4.6 Exercises
  • References
  • 5 General Tensor
  • 5.1 Curvilinear Coordinates
  • 5.2 Coordinate Transformation Equation
  • 5.3 Contravariant and Covariant Tensor
  • 5.4 Contravariant Vector or Contravariant Tensor of Order-One
  • 5.5 Covariant Vector or Covariant Tensor of Order-One
  • 5.6 Mixed Second-Order Tensor
  • 5.7 General Tensor of Any Order
  • 5.8 Metric Tensor
  • 5.9 Associate Contravariant Metric Tensor
  • 5.10 Associate Metric Tensor
  • 5.11 Christoffel Symbols of the First and Second-Kind
  • 5.12 Covariant Derivative of a Covariant Vector
  • 5.13 Covariant Derivative of a Contravariant Vector
  • 5.14 Exercises
  • References

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