Category: Lectures

Supplemental: Vector-Valued Differential Forms

This short-but-important supplemental lecture introduces some language we’ll need for describing geometry (curves, surfaces, etc.) in terms of differential forms. So far, we’ve said that a differential \(k\)-form produces a scalar measurement. But when talking about geometry, we often care about quantities that are vector-valued rather than scalar-valued. For instance, positions in \(\mathbb{R}^n\), tangents, and normals are all vector-valued quantities. For the most part, all of our operations look pretty much the same as before. The one exception is the wedge product, which in \(\mathbb{R}^3\) we now define in terms of the cross product.

Lecture 8: Discrete Differential Forms

In this lecture, we turn smooth differential \(k\)-forms into discrete objects that we can actually compute with. The basic idea is actually quite simple: to capture some information about a differential \(k\)-form, we integrate it over each oriented \(k\)-simplex of a mesh. The resulting values are just ordinary numbers that give us some sense of what the original \(k\)-form must have looked like.

Lecture 6: Exterior Derivative

Ordinary calculus provides tools for understanding rates of change (via derivatives), total quantities (via integration), and the total change (via the fundamental theorem of calculus). Exterior calculus generalizes these ideas to \(n\)-dimensional quantities that arise throughout geometry and physics. Our first lecture on exterior calculus studies the exterior derivative, which describes the rate of change of a differential form, and (together with the Hodge star) generalizes the gradient, divergence, and curl operators from standard vector calculus.

Lecture 4: \(k\)-Forms

Today we continue our journey toward building up (discrete) exterior calculus by talking about how to measure little k-dimensional volumes. Just like rulers measure length, and cups measure volume, k-forms will be used to take measurements of the little k-dimensional volumes or k-vectors that we built up using exterior algebra in our previous lecture. Such measurements will ultimately allow us to talk about integration over curved spaces; in the discrete setting, these measurements will be the basic data we associate with the elements of mesh.

Lecture 3: Exterior Algebra

Our next lecture will cover one of the basic tools we’ll use throughout the rest of the course: exterior algebra. The basic idea is to add a couple new operations to our usual list of vector operations (dot product, cross product, etc.) that make it easy to talk about volumes rather than just vectors. If you felt ok working with things like the cross product and the determinant in your linear algebra/vector calculus courses, this shouldn’t be too big of a leap. (If not, could be a good moment for a review!)

Lecture 2B: Introduction to Manifolds

In tomorrow’s lecture we will catch a first glimpse of the idea of manifolds, which are pretty central to differential geometry. Rather than giving a formal definition in the smooth case, we’ll introduce a notion of discrete manifolds that capture the most important ideas. These discrete manifolds build on the idea of a simplicial complex, introduced in the previous lecture.

Lecture 1: Overview

Our first lecture provides motivation for the topics we’ll cover in the course, and takes a deep dive into one specific example (curvature of curves in the plane) to highlight some of the basic principles of discrete differential geometry. This example moves pretty fast and uses some ideas that we’ll study at a slower, more careful pace later on. For now, don’t worry too much about the details—the goal here is to just get a sense of what the course is all about!