---
tag: 30G1M35TK1
kind: concept
name: Time-scale calculus
msc: ["26E70", 39A12]
sources:
  - url: https://en.wikipedia.org/wiki/Time-scale_calculus
same_as:
  wikidata: Q7804838
  wikipedia: Time-scale calculus
---

## Summary

Time-scale calculus develops derivatives and integrals for functions defined on a time scale, that is, on any nonempty closed subset of $\R$.
On $\R$ its delta derivative is the ordinary [[derivative|0V1EJCT8G6]], and on $\Z$ it is the forward difference.
So it unifies differential equations and difference equations, and it covers mixed cases such as unions of intervals.

## Explanations

### L1 human

Some processes run in continuous time, some in steps, and some in both, like an insect population that grows in summer and is counted once each winter.
A time scale is the set of moments at which the process is observed.
At a moment followed by a gap, the derivative becomes a difference quotient across the gap; at a moment with no gap, it is the usual limit.
For example, on the time scale $\Z$ the derivative of $t^2$ is $(t + 1)^2 - t^2 = 2t + 1$, not $2t$.