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AuthorAlexandrov, A. D. author
TitleGeneral Theory of Irregular Curves [electronic resource] / by A. D. Alexandrov, Yu. G. Reshetnyak
ImprintDordrecht : Springer Netherlands, 1989
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Descript X, 288 p. online resource


I: General Notion of a Curve -- 1.1. Definition of a Curve -- 1.2. Normal Parametrization of a Curve -- 1.3. Chains on a Curve and the Notion of an Inscribed Polygonal Line -- 1.4. Distance Between Curves and Curve Convergence -- 1.5. On a Non-Parametric Definition of the Notion of a Curve -- II: Length of a Curve -- 2.1. Definition of a Curve Length and its Basic Properties -- 2.2. Rectifiable Curves in Euclidean Spaces -- 2.3. Rectifiable Curves in Lipshitz Manifolds -- III: Tangent and the Class of One-Sidedly Smooth Curves -- 3.1. Definition and Basic Properties of One-Sidedly Smooth Curves -- 3.2. Projection Criterion of the Existence of a Tangent in the Strong Sense -- 3.3. Characterizing One-Sidedly Smooth Curves with Contingencies -- 3.4. One-Sidedly Smooth Functions -- 3.5. Notion of c-Correspondence. Indicatrix of Tangents of a Curve -- 3.6. One-Sidedly Smooth Curves in Differentiable Manifolds -- IV: Some Facts of Integral Geometry -- 4.1. Manifold Gnk of k-Dimensional Directions in Vn -- 4.2. Imbedding of Gnk into a Euclidean Space -- 4.3. Existence of Invariant Measure of Gnk -- 4.4. Invariant Measure in Gnk and Integral. Uniqueness of an Invariant Measure -- 4.5. Some Relations for Integrals Relative to the Invariant Measure in Gnk -- 4.6. Some Specific Subsets of Gnk -- 4.7. Length of a Spherical Curve as an Integral of the Function Equal to the Number of Intersection Points -- 4.8. Length of a Curve as an Integral of Lengths of its Projections -- 4.9. Generalization of Theorems on the Mean Number of the Points of Intersection and Other Problems -- V: Turn or Integral Curvature of a Curve -- 5.1. Definition of a Turn. Basic Properties of Curves of a Finite Turn -- 5.2. Definition of a Turn of a Curve by Contingencies -- 5.3. Turn of a Regular Curve -- 5.4. Analytical Criterion of Finiteness of a Curve Turn -- 5.5. Basic Integra-Geometrical Theorem on a Curve Turn -- 5.6. Some Estimates and Theorems on a Limiting Transition -- 5.7. Turn of a Curve as a Limit of the Sum of Angles Between the Secants -- 5.8. Exact Estimates of the Length of a Curve -- 5.9. Convergence with a Turn -- 5.10 Turn of a Plane Curve -- VI: Theory of a Turn on an n-Dimensional Sphere -- 6.1. Auxiliary Results -- 6.2. Integro-Geometrical Theorem on Angles and its Corrolaries -- 6.3. Definition and Basic Properties of Spherical Curves of a Finite Geodesic Turn -- 6.4. Definition of a Geodesic Turn by Means of Tangents -- 6.5. Curves on a Two-Dimensional Sphere -- VII: Osculating Planes and Class of Curves with an Osculating Plane in the Strong Sense -- 7.1. Notion of an Osculating Plane -- 7.2. Osculating Plane of a Plane Curve -- 7.3. Properties of Curves with an Osculating Plane in the Strong Sense -- VIII: Torsion of a Curve in a Three-Dimensional Euclidean Space -- 8.1. Torsion of a Plane Curve -- 8.2. Curves of a Finite Complete Torsion -- 8.3. Complete Two-Dimensional Indicatrix of a Curve of a Finite Complete Torsion -- 8.4. Continuity and Additivity of Absolute Torsion -- 8.5. Definition of an Absolute Torsion Through Triple Chains and Paratingences -- 8.6. Right-Hand and Left-Hand Indices of a Point. Complete Torsion of a Curve -- IX: Frenet Formulas and Theorems on Natural Parametrization -- 9.1. Frenet Formulas -- 9.2. Theorems on Natural Parametrization -- X: Some Additional Remarks -- References

Mathematics Mathematical analysis Analysis (Mathematics) Geometry Mathematics Analysis Geometry


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