##### Why?

This page exists because I wanted to experiment with the WordPress content management system. The subject deals with calculus flavors related to quantum calculus or `calculus without limits'. Examples of non-traditional calculus flavors are non-standard analysis (i.e. internal set theory) or calculus on graphs (which is currently my focus). All these flavors of calculus are extensions of traditional calculus. The usual school calculus can be seen as a special or limiting case.##### How come?

The idea of calculus is to look at ``rates of changes" in order to ``predict the future". This idea is everywhere: look at the sequence 4, 15, 40, 85, 156, 259, 400, 585 for example. How does it continue? In order to figure that out, we take derivatives 11, 25, 45, 71, 103, 141, 185, then again derivatives 14, 20, 26, 32, 38, 44 and again derivatives 6,6,6,6,6,6. We see now a pattern and can integrate the three times starting with 6,6,6,6,6,6,6 always adjusting the constant. This gives us the next term 820. We can predict the future by analyzing the past. Thats what calculus is about in the simplest case.##### What for?

Even ergodic theory or operator theory are related to calculus. For example, as differentiation and integration are operators, one can not separate operator theory from calculus. Because a measure preserving transformation allows to define a derivative df = f(T)-f, there are calculus versions in ergodic theory for example (a lot of my thesis is about that) but the cohomologies are difficult to understand even in the simplest cases like the cohomology of the set of all measurable sets on the circle modulo all coboundaries Z +T(Z), where T is an irrational rotation. These cohomologies matter for example when studying chaos and in particular the Lyapunov exponents, which quantifies chaos.##### Why now?

The power, richness and applicability of calculus are all reasons why we teach it. Calculus is a wonderful and classical construct, rich of historical connections and related with many other fields. We hardly have to mention all the applications of calculus (the last 2 minutes of this 15 minute review for single variable calculus give a few). We have barely scratched the surface of what is possible when extending calculus and how it can be applied in the future. At the moment (2016), I work on a**generalized interaction calculus**.

##### Traditional calculus

The traditional exterior derivatives (like div,curl grad) or curvature notions based on differential forms define a**traditional calculus**. Integration and derivatives lead to theorems like Stokes, Gauss-Bonnet, Poincaré-Hopf or Brouwer-Lefschetz. In classical calculus, the basic building blocks of space are simplices. Differential forms are functions of these simplices. In the continuum, one can not see these infinitesimal simplices. To remedy this, sheaf theoretical constructs like tensor calculus have been developed, notably by Cartan. In the discrete, when looking at graphs, the structure is simple and transparent. The theorems become easy. Here, here and here are some write ups.

##### Calculus on graphs

Calculus on graphs is a quantum calculus with no limits: everything is finite and combinatorial. I prefer to work on graphs, despite the fact graphs are considered one-dimensional simplicial complexes for most of the mathematical community. It turns out that the category of graphs is much more powerful. This especially happens when a graph is equipped with the Whitney complex. Graphs can describe so an abstract finite simplicial complex: the barycentric refinement of an abstract finite simplicial complex is always the Whitney complex of a finite simple graph. Having ``trivialized calculus", one can look at more complex constructs.##### Interaction calculus

In**interaction calculus**, functions on pairs (or more generally k-tuples) of interacting (=intersecting) simplices in the graph are at the center of attention. Exterior derivatives again lead to cohomology. While in classical calculus, the derivative is df(x)=f(dx) (which is Stokes theorem as dx is the boundary chain), in second level interaction calculus, the derivative is df(x,y)=f(dx,y)+(-1)

^{dim(x)}f(x,dy). The chain complex is bigger. The cohomology groups more interesting.

##### Interaction cohomology

The analogue of Euler characteristic is the more general**Wu characteristic**. There are generalized versions of the just mentioned theorems. At the moment (2016), only a glimpse of the power of interaction calculus is visible. There are indications that it could be powerful: it allows to distinguish topological spaces, which traditional calculus does not: here is a case study.