The geometry of vector fields / Yu. Aminov.

Format:

Book

Author/Creator:

Aminov, I︠U︡. A. (I︠U︡riĭ Akhmetovich)

Standardized Title:

Geometrii͡a vektornogo poli͡a. English

Language:

English

Russian

Subjects (All):

Vector fields.

Geometry.

Local Subjects:

Geometry.

Vector fields.

Physical Description:

xiii, 172 pages : illustrations ; 26 cm

Place of Publication:

Amsterdam, the Netherlands : Gordon and Breach Science Publishers, [2000]

Summary:

A specialist in classical differential geometry, Aminov (Ukrainian Academy of Sciences, Kharkov) presents a classical approach to the foundations and development of the geometry of vector fields, describing vector fields in three-dimensional Euclidean space and triply-orthogonal systems. He also discusses Pfaffian forms and systems in n-dimensional space, foliations and their Godbillon-Vey invariant, and applications in mechanics. Annotation copyrighted by Book News, Inc., Portland, OR

Contents:

1 Vector Fields in Three-Dimensional Euclidean Space 1

1.1 The Non-Holonomicity Value of a Vector Field 1

1.2 Normal Curvature of a Vector Field and Principal Normal Curvatures of the First Kind 8

1.3 The Streamline of Vector Field 12

1.4 The Straightest and the Shortest Lines 14

1.5 The Total Curvature of the Second Kind 22

1.6 The Asymptotic Lines 29

1.7 The First Divergent Form of Total Curvature of the Second Kind 32

1.8 The Second Divergent Representation of Total Curvature of the Second Kind 34

1.9 The Interrelation of Two Divergent Representations of the Total Curvatures of the Second Kind 37

1.10 The Generalization of the Gauss-Bonnet Formula for the Closed Surface 39

1.11 The Gauss-Bonnet Formula for the Case of a Surface with a Boundary 42

1.12 The Extremal Values of Geodesic Torsion 49

1.13 The Singularities as the Sources of Curvature of a Vector Field 52

1.14 The Mutual Restriction of the Fundamental Invariants of a Vector Field and the Size of Domain of Definition 55

1.15 The Behavior of Vector Field Streamlines in a Neighborhood of a Closed Streamline 60

1.16 The Complex Non-Holonomicity 66

1.17 The Analogues of Gauss-Weingarten Decompositions and the Bonnet Theorem Analogue 69

1.18 Triorthogonal Family of Surfaces 71

1.19 Triorthogonal Bianchi System 79

1.20 Geometrical Properties of the Velocity Field of an Ideal Incompressible Liquid 82

1.21 The Caratheodory-Rashevski Theorem 87

1.22 Parallel Transport on the Non-Holonomic Manifold and the Vagner Vector 92

2 Vector Fields and Differential Forms in Many-Dimensional Euclidean and Riemannian Spaces 99

2.1 The Unit Vector Field in Many-Dimensional Euclidean Space 99

2.2 The Regular Vector Field Defined in a Whole Space 101

2.3 The Many-Dimensional Generalization of the Gauss-Bonnet Formula to the Case of a Vector Field 104

2.4 The Family of Parallel Hypersurfaces of Riemannian Space 109

2.5 The Constant Vector Fields and the Killing Fields 111

2.6 On Symmetric Polynomials of Principal Curvatures of a Vector Field on Riemannian Space 114

2.7 The System of Pfaff Equations 116

2.8 An Example from the Mechanics of Non-Holonomic Constraints 123

2.9 The Exterior Differential Forms 124

2.10 The Exterior Codifferential 129

2.11 Some Formulas for the Exterior Differential 131

2.12 Simplex, the Simplex Orientation and the Induced Orientation of a Simplex Boundary 133

2.13 The Simplicial Complex, the Incidence Coefficients 134

2.14 The Integration of Exterior Forms 135

2.15 Homology and Cohomology Groups 139

2.16 Foliations on the Manifolds and the Reeb's Example 141

2.17 The Godbillon-Vey Invariant for the Foliation on a Manifold 143

2.18 The Expression for the Hopf Invariant in Terms of the Integral of the Field Non-Holonomicity Value 147

2.19 Vector Fields Tangent to Spheres 150

2.20 On the Family of Surfaces which Fills a Ball 159.

Notes:

Translation of: Geometrii͡a vektornogo poli͡a.

Includes bibliographical references (pages 163-168) and indexes.

ISBN:

9056992015

OCLC:

42407495