Parameterized Differentiable Curve, Proof of Arc-Length Invariance under change of variables, Arc-Length Paramterization, Unit Tangent, Normal & Binormal Vectors, Frenet Frame, Bishop Frame, Elastic Rods Problem, Turning Angle Theorem, Curvature Estimate, 2D Gauss Map.

Tangent Spaces, Differential, Rank Theorem for Differential, Inverse Function Theorem, Implicit Function Theorem, Surface Representations.