Foreword -- Preface -- Contents -- Chapter 1 Preliminaries -- 1.1 The Vector Concept Revisited -- 1.2 A First Look at Tensors -- 1.3 Assumed Background -- 1.4 More on the Notion of a Vector -- Chapter 2 Transformations and Vectors -- 2.1 Change of Basis -- 2.2 Dual Bases -- 2.3 Transformation to the Reciprocal Frame -- 2.4 Transformation Between General Frames -- 2.5 Covariant and Contravariant Components -- 2.6 The Cross Product in Index Notation -- 2.7 Closing Remarks -- Chapter 3 Tensors -- 3.1 Dyadic Quantities and Tensors -- 3.2 Tensors from an Operator Viewpoint -- 3.3 Dyadic Components Under Transformation -- 3.4 More Dyadic Operations -- 3.5 Properties of Second Rank Tensors -- The tensor transpose -- Tensors raised to powers -- Symmetric and antisymmetric tensors -- Eigenvalues and eigenvectors -- The Cayley-Hamilton theorem -- Tensors of rotation -- Polar decomposition -- Isotropic tensors and isotropic scalar functions -- Deviator and ball tensor representation -- 3.6 Extending the Dyad Idea -- 3.7 Tensors of the Fourth and Higher Ranks -- Chapter 4 Tensor Fields -- 4.1 Vector Fields -- 4.2 Differentials and the Nabla Operator -- 4.3 Differentiation of a Vector Function -- 4.4 Derivatives of the Frame Vectors -- 4.5 Christoffel Coefficients and their Properties -- 4.6 Covariant Differentiation -- 4.7 Covariant Derivative of a Second Rank Tensor -- 4.8 Differential Operations -- 4.9 Orthogonal Coordinate Systems -- 4.10 Some Formulas of Integration -- 4.11 Norms on Spaces of Vectors and Tensors -- Normed spaces -- Chapter 5 Elements of Differential Geometry -- 5.1 Elementary Facts from the Theory of Curves -- Curvature -- The moving trihedron -- Curves in the plane -- 5.2 The Torsion of a Curve -- 5.3 Serret-Frenet Equations -- 5.4 Elements of the Theory of Surfaces -- The first fundamental form -- Geodesics.

5.5 The Second Fundamental Form of a Surface -- The normal curvature of the surface -- 5.6 Derivation Formulas -- Some useful formulas -- 5.7 Implicit Representation of a Curve -- Contact of Curves -- Contact of curves -- Contact of a curve with a circle -- evolutes -- Contact of nth order between a curve and a surface -- 5.8 Osculating Paraboloid -- 5.9 The Principal Curvatures of a Surface -- 5.10 Surfaces of Revolution -- 5.11 Natural Equations of a Curve -- 5.12 A Word About Rigor -- 5.13 Conclusion -- Appendix A Formulary -- Chapter 2 -- Reciprocal (dual) basis -- Frame transformation -- Miscellaneous -- Chapter 3 -- Dyad product -- Tensors from operator viewpoint -- Dyadic components under transformation -- More dyadic operations -- Second rank tensor topics -- Chapter 4 -- Vector fields -- Differentials and the nabla operator -- Differentiation of a vector function -- Covariant differentiation of second-rank tensor -- Differential operations -- Orthogonal coordinate systems -- Integration formulas -- Chapter 5 -- Elementary theory of curves -- Serret-Frenet equations -- Theory of surfaces -- Appendix B Hints and Answers -- Chapter 1 -- Chapter 2 -- Chapter 3 -- Chapter 4 -- Chapter 5 -- Bibliography -- Index.