Our first lecture on exterior calculus covered differentiation; our second lecture completes the picture by discussing integration of differential forms. The relationship between integration and differentiation is encapsulated by Stokes’ theorem, which generalizes the fundamental theorem of calculus, as well as many other important theorems from vector calculus and complex analysis (divergence theorem, Green’s theorem, Cauchy’s integral formula, etc.). Stokes’ theorem also plays a key role in numerical discretization of geometric problems, appearing for instance in finite volume methods and boundary element methods; for us it will be the essential tool for developing a discrete version of differential forms that we can actually compute with.
The next reading assignment will wrap up our discussion of exterior calculus, both smooth and discrete. In particular, it will explore how to differentiate and integrate \(k\)-forms, and how an important relationship between differentiation and integration (Stokes’ theorem) enables us to turn derivatives into discrete operations on meshes. In particular, the basic data we will work with in the computational setting are “integrals of derivatives,” which amount to simple scalar quantities we can associate with the vertices, edges, faces, etc. of a simplicial mesh. These tools will provide the basis for the algorithms we’ll explore throughout the rest of the semester.
The reading is the remainder of Chapter 4 from the course notes, “A Quick and Dirty Introduction to Exterior Calculus”, Sections 4.6 through 4.8 (pages 67–83). Note that you just have to read these sections; you do not have to do the written exercises; a different set of written problems will be posted later on. The reading is due Thursday, February 14 at 10am. See the assignments page for handin instructions.
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.
A differential \(k\)-form describes a \(k\)-dimensional measurement at each point of space. For instance, the area 2-form on a sphere tells us how much each little piece of the sphere should contribute to an integral over the sphere. In this lecture we give some basic facts and definitions about differential \(k\)-forms, and how to work with them in coordinates. Ultimately differential \(k\)-forms will pave the way to a general notion of integration, which in turn will be our basic mechanism for turning smooth equations into discrete ones (by integrating over elements of a mesh).