Elements of Geometry： Geometry of Curves and Surfaces

Key Feature

The aim of this lecture is to understand curvature, which we use to quantitatively express how curves and surfaces are curved. First of all, calculating the curvature of a given curve or surface is the basis of this subject, so we took time for students to do calculations and to derive some important formulas. In addition to this, because we want students to understand the curvature conceptually, we tried to explain the process of formulating the curvature as carefully as possible.

Course Schedule

Session	Contents
1	Curves in R³ (1): definition, length and arc length parameter
2	Curves in R³ (2): curvature, torsion, and Frenet-Serret formulas
3	Curves in R³ (3): calculate curvature and torsion
4	Curves in R³ (4): meanings of curvature and torsion, fundamental theorem of curves
5	Surfaces in R³ (1): definition, tangential plane, unit normal vector, quick test
6	Surfaces in R³ (2): first and second fundamental forms, Gaussian curvature, mean curvature
7	Surfaces in R³ (3): meanings of Gaussian curvature and mean curvature
8	Surfaces in R³ (4): Theorema Egregium
9	Review & Mid-term exam
10	Surfaces in R³ (5): global surfaces
11	Geodesic curvature of curves, Gauss-Bonnet theorem
12	Differential forms and Stokes' theorem
13	Connection forms and structural equations
14	Proof of Gauss-Bonnet theorem
15	Review & Final exam

---