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.

More generally, the elementary functions sin(t), cos(t), log(t) and exp(t), and special functions such as Bessel functions,  all arise as solutions to initial value problems of basic form, and their values are computed by time stepping.

To do:

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