General Theory of Irregular Curves

ISBN-13: 9789401076715 / Angielski / Miękka / 2011 / 288 str.

V. V. Alexandrov; Yu G. Reshetnyak

Opis

One service mathematics has rendered the "Et moi, ...si j'a'ait su comment en revenir, human race. It has put common sense back je n'y scrais point alit: Jules Verne where it belongs, on the topmost shelf next to the dusty canister labcled 'discarded non- The series is divergent; therefore we may be sense'. Eric T. 8cl able to do something with it. O. Hcaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Spis treści

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.