Why quantum calculus?

  • It leads to a natural calculus on graphs. Here is an attempt to summarize calculus. One can explain some main ideas in multivariable calculus, differential geometry and topology in discrete settings. Calculus on graphs by nature is coordinate free. Yes, one needs to introduce orientations of edges for example to define the gradient but essential notions like the gradient which determines how light or heat propagates are coordinate independent.
  • It allows experimentation with small worlds. One can look at the Maxwell equations, wave equations, heat flows etc on small graphs and experiment. Functional analysis reduces to linear algebra, there are no existence and uniqueness problems to solve. No unbounded operators, no regularity issues appear. Experimentation is esspecially interesting with discrete partial differential equations.
  • We do not need to change notation from traditional calculus. Discrete calculus as a reinterpretation of the calculus we know and hardly new. Newton did it, Archimedes did it. The point is that we do not have to change the language. Indeed, language for discrete calculus, including numerical methods is usually very difficult to read, with lots of indices etc. But the translation does not need any new language. We can talk about gradients, curl and divergence for example using the same notation. We just have to reinterpret things. By defining polynomials, exponentials in a convenient way, we have the same results as in the continuum.

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