Slides—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!)

These slides should also be helpful for those who have started on the homework. 🙂

Slides—Differential Forms in $R^n$

Following our lecture on exterior algebra, we will start building up differential forms, which is the next step on our journey toward doing computation on meshes with discrete exterior calculus. This material may be helpful for those of you working through the second part of the written homework:

Slides—Exterior Calculus in $R^n$

Later this week we’ll start talking about exterior calculus, which is a modern language used across differential geometry, mathematical physics, geometric computation… and the rest of our class! :-). Initially this language can look a bit daunting, but by making some connections with familiar ideas from vector calculus (like grad, div, and curl), we’ll see that it’s actually not so bad once you get down to concrete calculations. Slides here:

Slides — Curves

After our long journey to understand exterior calculus (and its discrete counterpart), we will start putting these tools to work to manipulate real curves and surfaces. This lecture studies smooth and discrete curves, which illustrate many of the important features of geometry embedded in $\mathbb{R}^n$.

Slides — Discrete Surfaces

Below are our slides about discrete surfaces. Here our hard work starts to pay off: since we’ve already discretized exterior calculus, and have described smooth surfaces in terms of (smooth) exterior calculus, the transition to the discrete setting takes very little additional work.