Author(s): Luther Eisenhart
Publisher: Princeton
Year: 1940
Title page
CHAPTER 1: CURVES IN SPACE
1. Curves and surfaces. The summation convention
2. Length of a curve. Linear element
3. Tangent to a curve. Order of contact. Osculating plane
4. Curvature. Principal normal. Circle of curvature
5. Binormal. Torsion
6. The Frenet formulas. The form of a curve in the neighborhood of a point
7. Intrinsic equations of a curve
8. Involutes and evolutes of a curve
!J. The tangent surface of a curve. The polar surface. Osculating sphere
10. Parametric equations of a surface. Coordinates and coordinate curves in a surface
11. Tangent plane to a surface
12. Developable surfaces. Envelope of a one-parameter family of surfaces
CHAPTER II: TRANSFORMATION OF COORDINATES. TENSOR CALCULUS
13. Transformation of coordinates. Curvilinear coordinates
14. The fundamental quadratic form of space
15. Contravariant vectors. Scalars
16. Length of a contravariant vector. Angle between two vectors
17. Covariant vectors. Contravariant and covariant components of a vector
18. Tensors. Symmetric and skew symmetric tensors
19. Addition, subtraction and multiplication of tensors. Contraction
20. The Christoffel symbols. The Riemann tensor
21. The Frenet formulas in general coordinates
22. Covariant differentiation
23. Systems of partial differential equations of the first order. Mixed systems
CHAPTER III: INTRINSIC GEOMETRY OF A SURFACE
24. Linear element of a surface. First fundamental quadratic form of a surface. Vectors in a surface
25. Angle of two intersecting curves in a surface. Element of area
26. Families of curves in a surface. Principal directions
27. The intrinsic geometry of a surface. Isometric surfaces
28. The Christoffel symbols for a surface. The Riemannian curvature tensor. The Gaussian curvature of a surface
29. Differentiai parameters
30. Isometric orthogonal nets. Isometric coordinates
31. Isometric surfaces
32. Geodesies
33. Geodesic polar coordinates. Geodesic triangles
34. Geodesic curvature
35. The vector asociate to a given vector with respect to a curve. Paralle!ism of vectors
36. Conformai correspondence of two surfaces
37. Geodesic correspondence of two surfaces
CHAPTER IV: SURFACES IN SPACE
38. The second fundamental form of a surface
39. The equation of Gauss and the equations of Codazzi
40. Normal curvature of a surface. Principal radii of normal curvature
41. Lines of curvature of a surface
42. Conjugate directions and conjugate nets. Isometric-conjugate nets
43. Asymptotic directions and asymptotic lines. Mean conjugate directions. The Dupin indicatrix
44. Geodesic curvaturc and geodesic torsion of a curve
45. Parallel vectors in a surface
46. Spherical representation of a surface. The Gaussian curvature of a surface
47. Tangential coordinates of a surface
48. Surfaces of center of a surface. Parallel surfaces
49. Spherical and pseudospherical surfaces
50. Minimal surfaces
Bib!iography
Index
