Introduction to Differential Geometry of Space Curves and Surfaces

Cover Page; Title Page; Copyright Page; Preface; Contents; PART 1 THE THEORY OF CURVES AND SURFACES IN THREE-DIMENSIONAL EUCLIDEAN SPACE; I. THE THEORY OF SPACE CURVES; 1. Introductory remarks about space curves; 2. Definitions; 3. Arc length; 4. Tangent, normal, and binormal; 5. Curvature and torsion of a curve given as the intersection of two surfaces; 6. Contact between curves and surfaces; 7. Tangent surface, involutes, and evolutes; 8. Intrinsic equations, fundamental existence theorem for space curves; 9. Helices; Appendix I. 1. Existence theorem on linear differential equations.;A solid introduction to the methods of differential geometry and tensor calculus, this volume is suitable for advanced undergraduate and graduate students of mathematics, physics, and engineering. Rather than a comprehensive account, it offers an introduction to the essential ideas and methods of differential geometry. Part 1 begins by employing vector methods to explore the classical theory of curves and surfaces. An introduction to the differential geometry of surfaces in the large provides students with ideas and techniques involved in global research. Part 2 introduces the concept of a tensor, first in algebra, then in calculus. It covers the basic theory of the absolute calculus and the fundamentals of Riemannian geometry. Worked examples and exercises appear throughout the text.