Differential Geometr 
  
Notes 
    
  1. Introduction
    1. What is differential Geometry
    2. The geometry of Euclidean spac (Tapp Chapter 1.8)
  1. Curves
    1. Curves in R^n (Tapp Chapter 1.3 to 1.5, do Carmo Chapter 1-2, 1-3, and 1-5)
    2. Curves in R^2
      1. Local theorey: signed curvature (Tapp Chapter 1.6)
      2. Rotation number, total curvature, and regular homotopy (Tapp Chapter 1.6, do Carmo Chapter 1-7)
      3. Convexity and curvature (Tapp Chapter 2.2 and 2.3, do Carmo Chapter 1-7)
      4. Length, width, and area (Tapp 2.5, do Carmo 1-7, do Carmo Chapter 1-7)
    3. Curves in R^3 (Tapp Chapter 1.7, do Carmo Chapter 1-5 and 1-6)
  1. Manifolds
    1. Recollections from Calculus (Tapp Chapter 3.1)
    2. Manifolds in R^n (partially Tapp Chapter 3.2 and 3.3, do Carmo Chapter 2.2)
  1. Surfaces
    1. Geometry of surfaces in R^3 (Tapp Chapter 4.1 and 4.2, do Carmo Chapter 3-2 and 3-3)
    2. Local coordinates, curvature, and area (Tapp Chapter 3.6, 3.7, and 4.5, do Carmo Chapters 2-5, 3-3)
    3. Some implications of curvature
  1. Intrinsic Geoemtry of Surfaces
    1. What is Intrinsic Geometry? (Tapp Chapter 5, do Carmo Chapters 4-1 and 4-3)
    2. Covariant derivatives and parallel transport (Tapp Chapter 5, do Carmo Chapter 4-4)
    3. Geodesics (Tapp Chapter 5, do Carmo 4-4)
  1. Curvature and Surfaces
    1. Classification of Surfaces (Tapp Chapter 6)
    2. Gauss-Bonnet Theorem (Tapp Chapter 6.1 and 6.2, do Carmo Chapter 4-5)
    3. Geodesic coordinates
    4. Prescribing curvature
  2. Mean Curvature