An Introduction to Tensor Analysis

✍ Scribed by Bipin Singh Koranga

• Year: 2021
• Tongue: English
• Leaves: 128
• Category: Library

✦ Table of Contents

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
6 Tensor in Relativity
6.1 Special Theory of Relativity
6.2 Four-Vectors in Relativity
6.3 Maxwell's Equations
6.4 General Theory of Relativity
6.5 Spherically Symmetrical Metric
6.6 Planetary Motion
6.7 Exercises
References
7 Geodesics and Its Coordinate
7.1 Families of Curves
7.2 Euler's Form
7.3 Geodesics
7.4 Geodesic Form of the Line Elements
7.5 Geodesic Coordinate
7.6 Exercise
References
Index
About the Authors