In the last part of the Proof seminar, we have done a bit of discrete calculus.
These were Unit 33 and
Unit 37. Calculus in the discrete
started with Gustav Kirchhoff.
The fact that the sum of the divergences values are zero ∑x d* F(x) = 0 is called the
Kirchhoff's current law.
It is a conservation law which we also know in the continuum. If we have a n-dimensional closed surface
which is also bounded, then the integral of the divergence is zero. The reason is the
divergence theorem which tells that this integral is the flux through the boundary. But
there is no boundary for a closed manifold. In the discrete, this applies more and less with traffic
of goods or money and be seen in the context of input-output analysis. The matrix L= d*d = div grad is the analogue of the
the classical Laplacian L f (x,y) = fxx + fyy. If written as a matrix it is called
the Kirchhoff matrix. You can get this matrix in Mathematica with
Normal[KirchhoffMatrix[WheelGraph[5]]]
Kirchhoff can be seen as the pioneer and early champion of discrete calculus.
It must be said that discrete calculus has been reemerged again and again, in particular by computer
scientists. It is useful in computer graphics for example, where surfaces are just graphs (discrete meshes
of triangles or tetrahedra). The discrete setup is implicitly used by any algebraic topologist
since the beginnings where algebra has started to be used for describing geometric objects. In order to compute
things, we need finite matrices and the matrices can be obtained by discretizing space.
You find in this document of 2012 the discrete Stokes theorem,
the discrete Poincaré-Hopf theorem (a generalization of the Island theorem we have seen in this course)
as well as Gauss-Bonnet all stated and proven in 2 pages. In classical mathematics, this takes maybe
100 times more, if everything is done in arbitrary dimensions (Gauss-Bonnet-Chern in higher dimensions).
Poincare-Hopf and Gauss-Bonnet are always done in differential geometry. Gauss-Bonnet-Chern are rarely proven
even in graduate level differential geometry courses. On this talk of 2013 you see all of single variable calculus (with Fundamental theorem of calculus, Taylor series, differential equations up to Stokes theorem) done and proven in 20 x 20 =400 seconds. The format of presenting in 20x20 seconds (20 slides which run for exactly 20 seconds) is called Pecha Kucha. See here for the annotated slides. But here are the slides (was recorded with the
Math 22 Tatoo, you might want to see in 4K resolution to see that).
