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Discrete Calculus -- Preface

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Discrete Calculus by Peter Saveliev

In calculus, one is concerned with a long list of issues some of which are listed below:

  • change and rate of change,
  • velocity and acceleration,
  • displacement and distance,
  • length and volume,
  • population growth,
  • fluid flow,
  • heat transfer,
  • wave propagation,
  • and so on.

So, we study functions. Now, we still would like to study functions but don't want to deal with limits and, therefore, concentrate only on functions that are defined at isolated points. We call them discrete functions. In calculus, we study the rates of change of functions. For a discrete function, this is a simple ratio. That's how we define the derivatives. Their values are constant within the interval between the points; the derivatives are discrete functions of a new type. All discrete functions have derivatives. There are also integrals and the main theorems of calculus hold... In the mean time, discrete ODEs and PDEs provide computer simulations.

Most of the material comes from Topology Illustrated by the author where it was used to illustrate some of the tools of algebraic topology. Here, the build-up is more gradual and purposeful. Some of the elementary topics receive detailed exposition while some of the more advanced material is omitted.

Linear algebra is required beyond Chapter 0. This chapter is meant to demonstrate the feasibility of teaching discrete calculus before calculus.

What is the point of discrete calculus?

  • 1. There are no limits which makes the theory simpler.
  • 2. And the theory is complete: derivatives and their applications to tangents lines and monotonicity, integrals and their applications to lengths, areas, and volumes, the Fundamental Theorem of Calculus, etc.
  • 3. The analytical tools are also available, even though some of the formulas are more complex as they depend of the fixed increment, $h$. For example, the derivative of $\sin$ is still $\cos$ but with a coefficient depending on $h$.
  • 4. With all that done, a transition to the conventional calculus is straight-forward: $h\to 0$.
  • 5. Discrete calculus provides instant tools for computer simulations: motion, population growth, predator-prey models, heat transfer, wave propagation, etc. without discretization.