Tensor Calculus and Applications: Simplified Tools and Techniques

Cover......Page 1
Half Title......Page 2
Series Page......Page 3
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 10
About the Book......Page 12
Author......Page 14
Part I: Formalism of Tensor Calculus......Page 16
  1.2 Curvilinear Coordinates and Contravariant and Covariant Components of a Vector (the Entity)......Page 18
  1.3 Quadratic Forms, Properties, and Classifications......Page 22
  1.4 Quadratic Differential Forms and Metric of a Space in the Form of Quadratic Differentials......Page 24
  Exercises......Page 26
  2.1 Some Useful Definitions......Page 28
  2.2 Transformation of Coordinates......Page 29
  2.3 Second and Higher Order Tensors......Page 32
  2.4 Operations on Tensors......Page 33
  2.5 Symmetric and Antisymmetric (or Skew-Symmetric) Tensors......Page 35
  2.6 Quotient Law......Page 40
  Exercises......Page 41
  3.2 Cartesian Coordinate System and Orthogonal Coordinate System......Page 44
  3.4 The Metric Functions g[sub(ij)] Are Second-Order Covariant Symmetric Tensors......Page 45
  3.5 The Function g[sub(ij)] Is a Contravariant Second-Order Symmetric Tensor......Page 48
  3.7 Angle Between Two Vectors and Orthogonal Condition......Page 53
  Exercises......Page 54
  4.1 Christoffel Symbols (or Brackets) of the First and Second Kinds......Page 56
  4.2 Two Standard Applicable Results of Christoffel Symbols......Page 57
  4.3 Evolutionary Basis of Christoffel Symbols (Brackets)......Page 58
  4.4 Use of Symmetry Condition for the Ultimate Result......Page 64
   4.5.1 Transformation of the First Kind ┌[sub(ij, k)]......Page 65
   4.5.2 Transformation of the Second Kind ┌[sup(ij)][sub(k)]......Page 66
  4.6 Covariant Derivative of Covariant Tensor of Rank One......Page 70
  4.7 Covariant Derivative of Contravariant Tensor of Rank One......Page 71
  4.8 Covariant Derivative of Covariant Tensor of Rank Two......Page 72
  4.9 Covariant Derivative of Contravariant Tensor of Rank Two......Page 74
  4.10 Covariant Derivative of Mixed Tensor of Rank Two......Page 75
   4.10.1 Generalization......Page 76
  4.11 Covariant Derivatives of g[sub(ij)] g[sup(ij)] and also g[sub(i)][sub(j)]......Page 77
  4.12 Covariant Differentiations of Sum (or Difference) and Product of Tensors......Page 78
  4.13 Gradient of an Invariant Function......Page 81
  4.14 Curl of a Vector......Page 82
  4.15 Divergence of a Vector......Page 83
  4.16 Laplacian of a Scalar Invariant......Page 84
  4.17 Intrinsic Derivative or Derived Vector of v......Page 86
   4.18.1 When Magnitude Is Constant......Page 87
   4.18.2 Parallel Displacement When a Vector Is of Variable Magnitude......Page 88
  Exercises......Page 91
  5.1 The First Curvature of a Curve......Page 92
  5.3 Derivation of Differential Equations of Geodesics......Page 93
  5.4 Aliter: Differential Equations of Geodesics as Stationary Length......Page 95
  5.5 Geodesic Is an Autoparallel Curve......Page 97
  5.6 Integral Curve of Geodesic Equations......Page 100
  5.7 Riemannian and Geodesic Coordinates, and Conditions for Riemannian and Geodesic Coordinates......Page 101
   5.7.1 Another Form of Condition for Geodesic Coordinates......Page 103
  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)......Page 104
  Exercises......Page 106
  6.2 Riemannian Tensors (Curvature Tensors)......Page 108
  6.3 Derivation of the Transformation Law of Riemannian Tensor R[sup(α)][sub(abc)]......Page 110
  6.4 Properties of the Curvature Tensor RR[sup(α)][sub(ijk)]......Page 112
  6.5 Covariant Curvature Tensor......Page 114
  6.6 Properties of the Curvature Tensor R[sub(hijk)] of the First Kind......Page 115
  6.7 Bianchi Identity......Page 116
  6.8 Einstein Tensor Is Divergence Free......Page 117
  6.10 Three-Dimensional Orthogonal Cartesian Coordinate Metric and Two-Dimensional Curvilinear Coordinate Surface Metric Imbedded in It......Page 118
  6.11 Gaussian Curvature of the Surface S immersed in E[sub(3)]......Page 119
  Exercises......Page 123
Part II: Application of Tensors......Page 126
  7.1 Introduction......Page 128
  7.2 Curvature of a Riemannian Space......Page 129
  7.4 Covariant Differential of a Vector......Page 133
  7.5 Motion of Free Particle in a Curvilinear Co-Ordinate System for Curved Space......Page 134
  7.6 Necessity of Ricci Tensor in Einstein’s Gravitational Field Equation......Page 135
  8.2 Mathematical Tools Required for Continuum Mechanics......Page 138
  8.3 Stress at a Point and the Stress Tensor......Page 140
  8.4 Deformation and Displacement Gradients......Page 141
  8.5 Deformation Tensors and Finite Strain Tensors......Page 142
  8.6 Linear Rotation Tensor and Rotation Vector in Relation to Relative Displacement......Page 145
  9.1 Introduction......Page 148
  9.2 Equation for the Determination of Shearing Stresses on Any Plane Surface......Page 150
  9.3 General Transformation and Maximum and Minimum Longitudinal Strains......Page 152
  9.4 Determination of the Two Principal Strains in a Plane......Page 155
  10.2 Equations of Motion for Newtonian Fluid......Page 158
  10.3 Navier–Stokes Equations for the Motion of Viscous Fluids......Page 160
Appendix......Page 164
Remarks......Page 168
Bibliography......Page 170
Index......Page 172