Calculus as dx = f(t)dt as x = integral f(t) dt

Watch and play with this code. containing the essence of Calculus in few lines.

The Calculus of Newton/Leibniz opened to the Industrial Revolution of the late 17th century forever changing human conditions in the form of mathematics of change as the initial value problem of finding a function x(t) of the (time) variable t which solves the initial value problem

  • \frac{dx}{dt} = f(t) for t>0
  • x(0)=x_0

where f(t) is a given function, x_0 a given initial value and dt a given time step.

Computing x(t) by time stepping in the form

  • x(t+dt) = x(t) + f(t)\, dt,

successively computes x(t+dt) at a next time t+dt from x(t) at former time t plus the change of f(t)dt,  which can be captured by summation over all time steps to a final time T=Ndt after N steps:

  • x(T) =\sum_{n=1}^{N-1}f(ndt)\, dt + x_0.

To capture this summation formula Leibniz introduced the notation

  •  x(T)=\int_0^Tf(t)\, dt + x_0

where the integral sign \int represents the summation with t as integration variable and dt as time step.

We can now express the Fundamental Theorem of Calculus as:

  1. Time stepping of \frac{dx}{dt} = f(t) constructs the integral x=\int f(t)\, dt of the integrand f(t).
  2. Differentiation of the integral x=\int f(t)\, dt produces the integrand f(t).

Here the constructive step is the time stepping 1 with f(t) given, captured as summation by repetition,  while 2 expresses the starting point as the initial value problem with dx =f(t)dt or \frac{dx}{dt}=f(t).

Extension to f(x,t) depending also on x direct.

A basic case is f(x,t)=x for which the integral is x =\exp(t) with the remarkable property of being equal to its derivative as shown here.

To do:

  • Study experimentally by computing the dependence of \int_0^T f(t)\, dt on the time step dt.