Dipping Your Toes in Differential Geometry

Course Description: 

Have you ever thought “parabolas are nice and all, but what happens if I roll a parabola on another parabola?” This course aims to show you how and why you should roll curves on each other. From the fundamentals of drawing a graph of a function to Wankel engines, and even understanding the curvature of space-time, we need a set of tools and problems which are collectively referred to as differential geometry. In fact, even the very letters you are reading right now are making use of differential geometry. 

The course will walk through the essential tools for understanding plane curves, with discussions on how these concepts extend to more advanced topics. The aim for this course is more exploratory than formal, and focused more on developing an intuitive understanding. Calculations and derivations will be important, but not central to the course. 

Prerequisites: No prerequisites, but some basic intuitive understanding of derivatives and integrals is recommended (minimal calculation). Likewise, some knowledge on vectors is also recommended. 3Blue1Brown’s Essence of Calculus and Essence of Linear Algebra series feature intuitive and easy-to-understand explanations of all necessary concepts (and more). Both series are free and available on YouTube.

Tentative schedule:

Week 1: What is differential geometry? Attempts at defining curves and manifolds

Week 2: How do we draw functions? Functions and coordinate systems, parametric curves

Week 3: Arc length, involutes and roulette curves

Week 4: Curvature, osculating circles and evolutes

Week 5: Applications and examples, centered trochoids, Bezier curves, catenary

Week 6: Parametric surfaces, curvature in higher dimensions, geodesics and minimal surfaces