Calculus on graphs

The article “If Archimedes would have known functions …” is an extended writeup about calculus on graphs. The paper contains also slides of a last years Pecha-Kucha event; but there is more material in it. We have added more exercises illustrating how texbook problems with calculus on graphs could look like. Every topic which appears in a multi-variable calculus course can be done also on a graph. Graphs are a natural frame work because one can avoid index notations and work with an intuitive geometric object. The graph concept allows naturally for a coordinate free calculus. The Dirac operator plays an important role as it rules what distances are. This is illustrated with the wave equation. The wave equation has historically played an important role in calculus and our understanding of functions. There had been a phase transition in particular at the time when Fourier series appeared: functions were no more considered equivalents to analytic expressions but rather general rules which assign to a number an other number. The wave equation can also not be more central to our understanding of nature as we use wave phenomena to measure virtually anything around us and use light to define the metric in a geometry. The wave equation is therefore also essential when doing calculus on graphs. Explicit formulas for the solution to a wave equation on a general graph can be seen in a quantum mechanical framework or as a generalization of the Taylor formula in calculus. Indeed, in one dimensions, the Taylor formula of Newton and Gregory describes the evolution of light on the one dimensional linear graph.