An Introduction to Tensor Analysis.
- 1st ed.
- 1 online resource (127 pages)
Cover -- Half Title -- Series Page -- Title Page -- Copyright Page -- Preface -- Syllabus -- Table of Contents -- 1: Introduction -- 1.1 Symbols Multi-Suffix -- 1.2 Summation Convention -- 2: Cartesian Tensor -- 2.1 Introduction -- 2.2 Transformation of Coordinates -- 2.3 Relations Between the Direction Cosines -- 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 -- 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 -- 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 -- 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 -- 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 -- 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 -- Index -- About the Authors.
he primary purpose of this book is the study of the invariance form of equation relative to the totally of the rectangular co-ordinate system in the three-dimensional Euclidean space.