Multiplicative Differential Geometry

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Author Bio
1. Elements of the Multiplicative Euclidean Geometry
1.1 The Multiplicative Vector Space ℝ2⋆
1.2 The Multiplicative Inner Product Space ℝ2⋆
1.3 The Multiplicative Euclidean Plane E2⋆
1.4 Multiplicative Lines
1.5 Multiplicative Orthonormal Pairs
1.6 Equations of a Multiplicative Line
1.7 Perpendicular Multiplicative Lines
1.8 Parallel and Intersecting Multiplicative Lines
1.9 Multiplicative Rays and Multiplicative Angles
1.10 The Space E2⋆
1.11 The Multiplicative Cross Product
1.12 Multiplicative Orthonormal Bases
1.13 Multiplicative Planes
1.14 Advanced Practical Problems
2. Multiplicative Curves in ℝn
2.1 Multiplicative Frenet Curves in ℝn
2.2 Analytical Representations of Curves
2.2.1 Multiplicative Plane Curves
2.2.1.1 Parametric Representation
2.2.1.2 Explicit Representation
2.2.1.3 Implicit Representation
2.2.2 Multiplicative Space Curves
2.2.2.1 Parametric Representation
2.2.2.2 Explicit Representation
2.2.2.3 Implicit Representation
2.3 The Multiplicative Tangent and the Multiplicative Normal Plane. The Multiplicative Normal at a Multiplicative Plane Curve
2.4 Multiplicative Osculating Plane
2.5 Multiplicative Curvature of a Multiplicative Curve
2.6 The Multiplicative Frenet Frame
2.7 Multiplicative Oriented Space Curves. The Multiplicative Frenet Frame of an Multiplicative Oriented Space Curve
2.8 The Multiplicative Frenet Formulae. The Multiplicative Torsion
2.9 Applications of the Multiplicative Frenet Formulae
2.10 The Multiplicative General Helices
2.11 The Multiplicative Bertrand Curves
2.12 The Local Behaviour of a Multiplicative Parameterized Curve
2.13 The Multiplicative Rigid Motion
2.14 The Existence Theorem
2.15 The Uniqueness Theorem
2.16 Advanced Practical Problems
3. Multiplicative Plane Curves
3.1 Multiplicative Envelopes of Multiplicative Plane Curves
3.2 The Multiplicative Evolute
3.3 The Multiplicative Complex Structure on ℝ2⋆
3.4 Multiplicative Curvature of Multiplicative Plane Curves
3.5 Multiplicative Rotation Angle of Multiplicative Plane Curves
3.6 Advanced Practical Problems
4. General Theory of Multiplicative Surfaces
4.1 Multiplicative Parameterized Surfaces
4.2 The Multiplicative Equivalence of Multiplicative Local Representations
4.3 Multiplicative Curves on Multiplicative Surfaces
4.4 The Multiplicative Tangent Vector Space, the Multiplicative Tangnet Plane, the Multiplicative Normal to a Multiplicative Surface
4.5 Multiplicative Differentiable Maps on a Multiplicative Surface
4.6 The Multiplicative Differential of a Multiplicative Smooth Map between Two Multiplicative Surfaces
4.7 The Multiplicative Spherical Map. The Multiplicative Shape Operator
4.8 The First Multiplicative Fundamental Form of a Multiplicative Surface
4.9 Applications of the First Multiplicative Fundamental Form
4.9.1 The Multiplicative Length of a Multiplicative Segment of a Multiplicative Curve on a Multiplicative Surface
4.9.2 The Multiplicative Angle between Two Multiplicative Curves on a Multiplicative Surface
4.9.3 The Multiplicative Area of a Multiplicative Parameterized Surface
4.10 The Multiplicative Matrix of the Multiplicative Shape Operator
4.11 The Second Multiplicative Fundamental Form of a Multiplicative Surface
4.12 The Multiplicative Normal Curvature. The Multiplicative Meusnier Theorem
4.13 Multiplicative Asymptotic Directions. Multiplicative Asymptotic Lines
4.14 Multiplicative Principal Directions, Multiplicative Principal Curvatures, Multiplicative Gauss Curvature and Multiplicative Mean Curvature
4.15 The Computation of the Multiplicative Curvatures of a Multiplicative Surface
4.16 Advanced Practical Problems
5. Multiplicative Fundamental Equations of a Multiplicative Surface
5.1 Some Relations
5.2 The Multiplicative Christoffel Coefficients
5.3 The Multiplicative Weingarten Coefficients
5.4 The Multiplicative Gauss and Godazzi-Mainardi Equations
5.5 The Multiplicative Darboux Frame
5.6 The Multiplicative Geodesic Curvature. Multiplicative Geodesic Lines
5.7 The Multiplicative Geodesics of the Multiplicative Planes
5.8 The Multiplicative Geodesics of the Multiplicative Unit Sphere
5.9 Multiplicative Geodesics of Multiplicative Liouville Surfaces
6. Special Classes of Multiplicative Surfaces
6.1 Definition of Multiplicative Ruled Surfaces
6.2 The First Multiplicative Fundamental Form of Multiplicative Ruled Surfaces
6.3 The Multiplicative Tangent Plane of Multiplicative Ruled Surfaces
6.4 The Multiplicative Gaussian Curvature of Multiplicative Ruled Surfaces
6.5 Multiplicative Minimal Surfaces
7. Multiplicative Differential Forms
7.1 Algebra of Multiplicative Differential Forms
7.2 Multiplicative Exterior Differentiation
7.3 Properties of the Multiplicative Exterior Differentiation
7.4 Multiplicative Closed Differential Forms. Multiplicative Exact Differential Forms
7.5 Multiplicative Gradient, Multiplicative Curl and Multiplicative Divergence
7.6 Multiplicative Differential Forms in ℝ⋆n
7.7 Advanced Practical Problems
8. The Multiplicative Nature Connection
8.1 The Multiplicative Directional Derivative
8.2 Multiplicative Tangent Spaces
8.3 The Multiplicative Covariant Derivative
8.4 The Multiplicative Lie Brackets
8.5 Advanced Practical Problems
9. Multiplicative Riemannian Manifolds
9.1 The Notion of a Multiplicative Manifold
9.2 Multiplicative Differentiable Maps
9.3 Multiplicative Tangent Spaces
9.4 Multiplicative Riemannian Metrics
9.5 The Multiplicative Riemannian Connection
9.6 The Multiplicative Christoffel Coefficients
10. The Multiplicative Curvature Tensor
10.1 Multiplicative Tensors
10.2 Multiplicative Derivatives of Multiplicative Tensor Fields
10.3 Properties of the Multiplicative Curvature Tensor
10.4 The Multiplicative Sectional Curvature
10.5 The Multiplicative Ricci Tensor
10.6 The Multiplicative Einstein Tensor
Appendix A. The Multiplicative Lipschitz Condition
Appendix B. The Multiplicative Implicit Function Theorem
Bibliography
Index