Parallel Vector Fields
8.4 Geometry of Space Curves
8.5 Serret-Frenet Formula
8.6 Equations of a Straight Line
8.7 Helix
8.8 Exercises
9 Intrinsic Geometry of Surfaces
9.1 Introduction
9.2 Curvilinear Coordinates on a Surface
9.3 Intrinsic Geometry: First Fundamental Quadratic Form
9.4 Angle Between Two Intersecting Curves on a Surface
9.5 Geodesic in Rn
9.6 Geodesic Coordinates
9.7 Parallel Vectors on a Surface
9.8 Isometric Surface
9.9 The Riemannian–Christoffel Tensor and Gaussian Curvature
9.10 The Geodesic Curvature
9.11 Exercises
10 Surfaces in Space
10.1 Introduction
10.2 The Tangent Vector
10.3 The Normal Line to the Surface
10.4 Tensor Derivatives
10.5 Second Fundamental Form of a Surface
10.6 The Integrability Condition
10.7 Formulas of Weingarten
10.8 Equations of Gauss and Codazzi
10.9 Mean and Total Curvatures of a Surface
10.10 Exercises
11 Curves on a Surface
11.1 Introduction
11.2 Curve on a Surface: Theorem of Meusnier
11.3 The Principal Curvatures of a Surface
11.4 Rodrigue’s Formula
11.5 Exercises
12 Curvature of Surface
12.1 Introduction
12.2 Surface of Positive and Negative Curvature
12.3 Parallel Surfaces
12.4 The Gauss-Bonnet Theorem
12.5 The n-Dimensional Manifolds
12.6 Hypersurfaces
12.7 Exercises
10 Part III: Analytical Mechanics 13 Classical Mechanics 13.1 Introduction 13.2 Newtonian Laws of Motion 13.3 Equations of Motion of Particles 13.4 Conservative Force Field 13.5 Lagrangean Equations of Motion 13.6 Applications of Lagrangean Equations 13.7 Himilton’s Principle 13.8 Principle of Least Action 13.9 Generalized Coordinates 13.10 Lagrangean Equations in Generalized Coordinates 13.11 Divergence Theorem, Green’s Theorem, Laplacian Operator, and Stoke’s Theorem in Tensor Notation 13.12 Hamilton’s Canonical Equations 13.13 Exercises 14 Newtonian Law of Gravitations 14.1 Introduction 14.2 Newtonian Laws of Gravitation 14.3 Theorem of Gauss 14.4 Poisson’s Equation 14.5 Solution of Poisson’s Equation 14.6 The Problem of Two Bodies 14.7 The Problem of Three Bodies 14.8 Exercises
11 Appendix A: Answers to Even-Numbered Exercises
12 References
13 Index
List of Illustrations
1 Chapter 7Figure 7.1Figure 7.2Figure 7.3Figure 7.4Figure 7.5Figure 7.6Figure 7.7
2 Chapter 8Figure 8.1Figure 8.2Figure 8.3Figure 8.4
3 Chapter 9Figure 9.1Figure 9.2Figure 9.3Figure 9.4Figure 9.5
4 Chapter 10Figure 10.1
5 Chapter 11Figure 11.1Figure 11.2
6 Chapter 12Figure 12.1Figure
