This is a continuation. We look more closely at the 2 dimensional case. The transition from 1 to 2 dimensions is crucial in the context of discretization. In the one dimensional case there is little choice on how to discretize calculus because there is only one possible discretization of the real line: see it is a graph with infinitesimal spacing (in the language of internal set theory of Nelson). 0-forms are the functions on the vertices of this graph, while 1-forms are functions on the directed edges of the graph. To the point of repeating myself again and again, there is a fundamental point to be made in single variable calculus, that there is a single variable calculus which is Fermionic and then there is a single variable calculus that is bosonic. The first one is the theory of differential forms, while the second is a theory of integral geometry. When we look at length for example, we do not care about the order, we just count edges. This is integral geometry. Most teachers do not realize that there are both versions appearing in introduction single variable calculus courses. Our course Math 1a is a Fermionic calculus course, its main point is the fundamental theorem of calculus. Our course Math 1b however is a almost pure Bosonic calculus course as it deals with Riemann sums, volumes, density problems, and summations where the order does not matter. The course Math 21a then is both, it deals with volumes, surface area and arc length (which are bosonic notions) as well as with integral theorems like the theorem of Stokes (which is a Fermionic notion). In more advanced courses like Math 22 one can point out such things and also look at the discrete case, where the distinction becomes crystal clear. Bosonic notions do not look at the orientations of simplices. For Fermionic notions, the orientation of simplices is important. Not that such things have to be pointed out to students in general. It is just important as a teacher that one is aware for example that surface integrals like (a Bosonic notion where the orientation of the surface does not matter) are a completely different beast than the flux integral where the orientation matters. When teaching such a course, it is therefore important to separate the two things and look at surface integrals of the first type as a generalization of surface area, while the second one produces an integral theorem, where the orientation matters. Now this is already interesting in one dimension and that’s where the Cartan formula for the Lie derivative (a Bosonic notion) plays a role. I wrote about Cartan’s magic formula some years ago. The point is that the Lie derivative is, like the Laplacian is a Bosonic notion. Now, in single variable calculus, we almost do not have a choice of X except to specify whether we go left or right. Assume we go right like in most calculus books, then on zero forms, applies first an exterior derivative , to get a then uses the inner derivative to get back a scalar function. When we look at Taylor series in Math 1b, then the derivative is the derivative . If we would take the Fermionic derivative (like in math 1a), then of course . Having taught calculus since 40 years now, I have witnessed a lot of confusion coming from the Bosonic-Fermionic mix-up. The brutal thing is that as a teacher one can not point it out because one can only appreciate the subtlety if one knows the exterior calculus in general, knows a bit of differential calculus on manifolds and (better even) understands how the calculus works in the discrete. In the discrete things are so simple that all the multi-variable calculus on discrete manifolds can be done in 2 one hour seminars. (see part I and part II, and see also the easter bunny episode from May 2022). An other way to see single variable calculus is to combine real 0 forms and real 1 forms as one complex form u + iv and see the Dirac operator in the form D=i d/dz. Unlike d, the powers of D are not zero and U=exp(i t D) generates translation on analytic functions. As discussed last time, the discretization is a discrete derivative.
Now lets look at the two dimensional case. Discretizing using a specific triangulation and in particular by a discrete grid breaks symmetries and is just ugly. It works as a numerical scheme. Closer to the classical theory is if we just deform the Dirac operator in such a way that it becomes bounded and keeps any rotational or translational symmetry, if present in the underlying manifold. For a 1-form and h positive, we can get a 2-form by taking the line integral along the boundary. If f is a 0 form, we can write it as f=Dg -dg where g is a 1-form. Now and as we know that depends only on the wave front, also only depends on the wave front.
