 Introduction
 What is differential Geometry
 The geometry of Euclidean spac (Tapp Chapter 1.8)
 Curves
 Curves in R^n (Tapp Chapter 1.3 to 1.5, do Carmo Chapter 12, 13, and 15)
 Curves in R^2
 Local theorey: signed curvature (Tapp Chapter 1.6)
 Rotation number, total curvature, and regular homotopy (Tapp Chapter 1.6, do Carmo Chapter 17)
 Convexity and curvature (Tapp Chapter 2.2 and 2.3, do Carmo Chapter 17)
 Length, width, and area (Tapp 2.5, do Carmo 17, do Carmo Chapter 17)
 Curves in R^3 (Tapp Chapter 1.7, do Carmo Chapter 15 and 16)
 Manifolds
 Recollections from Calculus (Tapp Chapter 3.1)
 Manifolds in R^n (partially Tapp Chapter 3.2 and 3.3, do Carmo Chapter 2.2)
 Surfaces
 Geometry of surfaces in R^3 (Tapp Chapter 4.1 and 4.2, do Carmo Chapter 32 and 33)
 Local coordinates, curvature, and area (Tapp Chapter 3.6, 3.7, and 4.5, do Carmo Chapters 25, 33)
 Some implications of curvature
 Intrinsic Geoemtry of Surfaces
 What is Intrinsic Geometry? (Tapp Chapter 5, do Carmo Chapters 41 and 43)
 Covariant derivatives and parallel transport (Tapp Chapter 5, do Carmo Chapter 44)
 Geodesics (Tapp Chapter 5, do Carmo 44)
 Curvature and Surfaces
 Classification of Surfaces (Tapp Chapter 6)
 GaussBonnet Theorem (Tapp Chapter 6.1 and 6.2, do Carmo Chapter 45)
 Geodesic coordinates
 Prescribing curvature
 Mean Curvature



