Tensor Calculus and Applications: Simplified Tools and Techniques

Description

Theoretically, the subject of "Tensor calculus" is critical for students to under understand for the complex nature of uses of subscripts and superscripts. Besides the lack of knowledge about the fields of application, this poses rather more difficulty to learn the concepts. The elegant nature of description of the theory with specific style of changing the suffixes and prefixes and reasons to recover meaningful results of the subsequent fields will inspire the readers for the book. Special techniques are adopted in the book to remove confusion from the minds (as inferred from practical experience) of students/readers for pure knowledge. For absolute understanding of the subject clear indication on where to change the suffix in terms and to what, are given with directions (hints) in respective terms. Moreover, this book distinguishes different classes of tensors; pictorial representations are furnished as well. Without identification of non-isotropic media where application of tensors is inevitable, researchers /readers cannot use it in true sense. For this purpose, the Chapter 7 is included for step-by-step evolution of the branch. To amplify the uses of tensors in gravitationally distorted space-time of General Theory of Relativity and in non-isotropic media like viscous fluids and physical bodies with elasticity including Structural Geology, Chapter 7 is developed to infuse real interest in to the minds of readers. Features Provides a clear indication and understanding of the subject on how to change suffixes Describes the original evolution of symbols necessary for tensors Offers a pictorial representation of referential systems required for different kinds of tensors for physical problems Presents the correlation between critical concepts Cover Half Title Series Page Title Page Copyright Page Contents Preface About the Book Author Part I: Formalism of Tensor Calculus 1. Prerequisites for Tensors 1.1 Ideas of Coordinate Systems 1.2 Curvilinear Coordinates and Contravariant and Covariant Components of a Vector (the Entity) 1.3 Quadratic Forms, Properties, and Classifications 1.4 Quadratic Differential Forms and Metric of a Space in the Form of Quadratic Differentials Exercises 2. Concept of Tensors 2.1 Some Useful Definitions 2.2 Transformation of Coordinates 2.3 Second and Higher Order Tensors 2.4 Operations on Tensors 2.5 Symmetric and Antisymmetric (or Skew-Symmetric) Tensors 2.6 Quotient Law Exercises 3. Riemannian Metric and Fundamental Tensors 3.1 Riemannian Metric 3.2 Cartesian Coordinate System and Orthogonal Coordinate System 3.3 Euclidean Space of n Dimensions, Euclidean Co-Ordinates, and Euclidean Geometry 3.4 The Metric Functions g[sub(ij)] Are Second-Order Covariant Symmetric Tensors 3.5 The Function g[sub(ij)] Is a Contravariant Second-Order Symmetric Tensor 3.6 Scalar Product and Magnitude of Vectors 3.7 Angle Between Two Vectors and Orthogonal Condition Exercises 4. Christoffel Three-Index Symbols (Brackets) and Covariant Differentiation 4.1 Christoffel Symbols (or Brackets) of the First and Second Kinds 4.2 Two Standard Applicable Results of Christoffel Symbols 4.3 Evolutionary Basis of Christoffel Symbols (Brackets) 4.4 Use of Symmetry Condition for the Ultimate Result 4.5 Coordinate Transformations of Christoffel Symbols 4.5.1 Transformation of the First Kind ┌[sub(ij, k)] 4.5.2 Transformation of the Second Kind ┌[sup(ij)][sub(k)] 4.6 Covariant Derivative of Covariant Tensor of Rank One 4.7 Covariant Derivative of Contravariant Tensor of Rank One 4.8 Covariant Derivative of Covariant Tensor of Rank Two 4.9 Covariant Derivative of Contravariant Tensor of Rank Two 4.10 Covariant Derivative of Mixed Tensor of Rank Two 4.10.1 Generalization 4.11 Covariant Derivatives of g[sub(ij)] g[sup(ij)] and also g[sub(i)][sub(j)] 4.12 Covariant Differentiations of Sum (or Difference) and Product of Tensors 4.13 Gradient of an Invariant Function 4.14 Curl of a Vector 4.15 Divergence of a Vector 4.16 Laplacian of a Scalar Invariant 4.17 Intrinsic Derivative or Derived Vector of v 4.18 Definition: Parallel Displacement of Vectors 4.18.1 When Magnitude Is Constant 4.18.2 Parallel Displacement When a Vector Is of Variable Magnitude Exercises 5. Properties of Curves in V[sub(n)] and Geodesics 5.1 The First Curvature of a Curve 5.2 Geodesics 5.3 Derivation of Differential Equations of Geodesics 5.4 Aliter: Differential Equations of Geodesics as Stationary Length 5.5 Geodesic Is an Autoparallel Curve 5.6 Integral Curve of Geodesic Equations 5.7 Riemannian and Geodesic Coordinates, and Conditions for Riemannian and Geodesic Coordinates 5.7.1 Another Form of Condition for Geodesic Coordinates 5.8 If a Curve Is a Geodesic of a Space (V[sub(m)]), It Is also a Geodesic of Any Space V[sub(n)] in Which It Lies (V[sub(n)] a Subspace) Exercises 6. Riemann Symbols (Curvature Tensors) 6.1 Introduction 6.2 Riemannian Tensors (Curvature Tensors) 6.3 Derivation of the Transformation Law of Riemannian Tensor R[sup(α)][sub(abc)] 6.4 Properties of the Curvature Tensor RR[sup(α)][sub(ijk)] 6.5 Covariant Curvature Tensor 6.6 Properties of the Curvature Tensor R[sub(hijk)] of the First Kind 6.7 Bianchi Identity 6.8 Einstein Tensor Is Divergence Free 6.9 Isometric Surfaces 6.10 Three-Dimensional Orthogonal Cartesian Coordinate Metric and Two-Dimensional Curvilinear Coordinate Surface Metric Imbedded in It 6.11 Gaussian Curvature of the Surface S immersed in E[sub(3)] Exercises Part II: Application of Tensors 7. Application of Tensors in General Theory of Relativity 7.1 Introduction 7.2 Curvature of a Riemannian Space 7.3 Flat Space and Condition for Flat Space 7.4 Covariant Differential of a Vector 7.5 Motion of Free Particle in a Curvilinear Co-Ordinate System for Curved Space 7.6 Necessity of Ricci Tensor in Einstein’s Gravitational Field Equation 8. Tensors in Continuum Mechanics 8.1 Continuum Concept 8.2 Mathematical Tools Required for Continuum Mechanics 8.3 Stress at a Point and the Stress Tensor 8.4 Deformation and Displacement Gradients 8.5 Deformation Tensors and Finite Strain Tensors 8.6 Linear Rotation Tensor and Rotation Vector in Relation to Relative Displacement 9. Tensors in Geology 9.1 Introduction 9.2 Equation for the Determination of Shearing Stresses on Any Plane Surface 9.3 General Transformation and Maximum and Minimum Longitudinal Strains 9.4 Determination of the Two Principal Strains in a Plane 10. Tensors in Fluid Dynamics 10.1 Introduction 10.2 Equations of Motion for Newtonian Fluid 10.3 Navier–Stokes Equations for the Motion of Viscous Fluids Appendix Remarks Bibliography Index