General Theory of Irregular Curves

1989
ISBN: 978-90-277-2811-1
Verlag: Springer Netherlands

Buch, Englisch, Band 29, 288 Seiten, HC runder Rücken kaschiert, Format (B × H): 160 mm x 241 mm, Gewicht: 1320 g

Reihe: Mathematics and its Applications

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 labc\led 'discarded non­ The series is divergent; therefore we may be sense'. Eric T. 8c\l 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.

Zielgruppe

Research

Autoren/Hrsg.

Reshetnyak, Yu. G.

Alexandrov, V.V.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Elementare Analysis und Allgemeine Begriffe
• Mathematik | Informatik Mathematik Geometrie Elementare Geometrie: Allgemeines

Inhaltsverzeichnis

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.