Cover -- Foreword -- Preface -- Contents -- Chapter 1 Preliminaries -- 1.1 n-Dimensional Space -- 1.2 Superscript and Subscript -- 1.3 The Einstein's Summation Convention -- 1.4 Dummy Index -- 1.5 Free Index -- 1.6 Kronecker Delta -- Exercises -- Chapter 2 Tensor Algebra -- 2.1 Introduction -- 2.2 Transformation of Coordinates -- 2.3 Covariant and Contravariant Vectors (Tensor of Rank One) -- 2.4 Contravariant Tensor of Rank Two -- 2.5 Covariant Tensor of Rank Two -- 2.6 Mixed Tensor of Rank Two -- 2.7 Tensor of Higher Order -- 2.8 Scalar or Invariant -- 2.9 Addition and Subtraction of Tensors -- 2.10 Multiplication of Tensors (Outer Product of Tensor) -- 2.11 Contraction of a Tensor -- 2.12 Inner Product of Two Tensors -- 2.13 Symmetric Tensors -- 2.14 Skew-Symmetric Tensor -- 2.15 Quotient Law -- 2.16 Conjugate (or Reciprocal) Symmetric Tensor -- 2.17 Relative Tensor -- Miscellaneous Examples -- Exercises -- Chapter 3 Metric Tensor and Riemannian Metric -- 3.1 The Metric Tensor -- 3.2 Conjugate Metric Tensor: (Contravariant Tensor) -- 3.3 Length of a Curve -- 3.4 Associated Tensor -- 3.5 Magnitude of Vector -- 3.6 Scalar Product of Two Vectors -- 3.7 Angle Between Two Vectors -- 3.8 Angle Between Two Coordinate Curves -- 3.9 Hypersurface -- 3.10 Angle Between Two Coordinate Hypersurface -- 3.11 n-Ply Orthogonal System of Hypersurfaces -- 3.12 Congruence of Curves -- 3.13 Orthogonal Ennuple -- Miscellaneous Examples -- Exercises -- Chapter 4 Christoffel's Symbols and Covariant Differentiation -- 4.1 Christoffel's Symbols -- 4.2 Transformation of Christoffel's Symbols -- 4.3 Covariant Differentiation of a Covariant Vector -- 4.4 Covariant Differentiation of a Contravariant Vector -- 4.5 Covariant Differentiation of Tensors -- 4.6 Ricci's Theorem -- 4.7 Gradient, Divergence and Curl -- 4.8 The Laplacian Operator -- Exercises.

Chapter 5 Riemann-Christoffel Tensor -- 5.1 Riemann-Christoffel Tensor -- 5.2 Ricci Tensor -- 5.3 Covariant Riemann-Christoffel Tensor -- 5.4 Properties of Riemann-Christoffel Tensors of First Kind Rijkl -- 5.5 Bianchi Identity -- 5.6 Einstein Tensor -- 5.7 Riemann Curvature of a Vn -- 5.8 Formula for Riemannian Curvature in the Terms of Covariant curvature Tensor of Vn -- 5.9 Schur's Theorem -- 5.10 Mean Curvature -- 5.11 Ricci's Principal Directions -- 5.12 Einstein Space -- 5.13 Weyl Tensor or Projective Curvature Tensor -- Exercises -- Chapter 6 The e-Systems and the Generalized Kronecker Deltas -- 6.1 Completely Symmetric -- 6.2 Completely Skew-Symmetric -- 6.3 e-System -- 6.4 Generalised Krönecker Delta -- 6.5 Contraction of i jk -- Exercises -- Chapter 7 Geometry -- 7.1 Length of Arc -- 7.2 Curvilinear Coordinates in E3 -- 7.3 Reciprocal Base Systems -- 7.4 On the Meaning of Covariant Derivatives -- 7.5 Intrinsic Differentiation -- 7.6 Parallel Vector Fields -- 7.7 Geometry of Space Curves -- 7.8 Serret-Frenet Formula -- 7.9 Equations of a Straight Line -- Exercises -- Chapter 8 Analytical Mechanics -- 8.1 Introduction -- 8.2 Newtonian Laws -- 8.3 Equations of Motion of a Particle -- 8.4 Conservative Force Field -- 8.5 Lagrangean Equations of Motion -- 8.6 Applications of Lagrangean Equations -- 8.7 Hamilton's Principle -- 8.8 Integral of Energy -- 8.9 Principle of Least Action -- 8.10 Generalized Coordinates -- 8.11 Lagrangean Equations in Generalized Coordinates -- 8.12 Divergence Theorem, Green's Theorem, Laplacian Operator And stoke's Theorem in Tensor Notation -- 8.13 Gauss's Theorem -- 8.14 Poisson's Equation -- 8.15 Solution of Poisson's Equation -- Exercises -- Chapter 9 Curvature of Curve, Geodesic -- 9.1 Curvature of Curve: Principal Normal -- 9.2 Geodesics -- 9.3 Euler's Condition -- 9.4 Differential Equations of Geodesics.

9.5 Geodesic Coordinates -- 9.6 Riemannian Coordinates -- 9.7 Geodesic form of a Line Element -- 9.8 Geodesics in Euclidean Space -- Examples -- Exercises -- Chapter 10 Parallelism of Vectors -- 10.1 Parallelism of a Vector of Constant Magnitude (Levi-Civita's Concept) -- 10.2 Parallelism of a Vector of Variable Magnitude -- 10.3 Subspaces of a Riemannian Manifold -- 10.4 Parallelism in a Subspace -- 10.5 The Fundamental Theorem of Riemannian Geometry Statement -- Exercises -- Chapter 11 Ricci's Coefficients of Rotation and Congruence -- 11.1 Ricci's Coefficients of Rotation -- 11.2 Reason for The Name "Coefficients of Rotation" -- 11.3 Curvature of Congruence -- 11.4 Geodesic Congruence -- 11.5 Normal Congruence -- 11.6 Curl of Congruence -- 11.7 Canonical Congruence -- Examples -- Exercises -- Chapte 12 Hypersurfaces -- 12.1 Introduction -- 12.2 Generalised Covariant Differentiation -- 12.3 Laws of Tensor Differentiation -- 12.4 Gauss's Formula -- 12.5 Curvature of a Curve in a Hypersurface and Normal Curvature -- 12.6 Definitions -- 12.7 Euler's Theorem -- 12.8 Conjugate Directions and Asymptotic Directions in a Hypersurface -- 12.9 Tensor Derivative of the Unit Normal -- 12.10 The Equation of Gauss and Codazzi -- 12.11 Hypersurfaces With Indeterminate Lines of Curvature -- 12.12 Central Quadratic Hypersurface -- 12.13 Polar Hyperplane -- 12.14 Evolute of a Hypersurface in Euclidean Space -- 12.15 Hypersphere -- Exercises -- Index.