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 of the (time) variable which solves the initial value problem
where is a given function, a given initial value and a given time step.
Computing by time stepping in the form
successively computes at a next time from at former time t plus the change of , which can be captured by summation over all time steps to a final time after N steps:
To capture this summation formula Leibniz introduced the notation
where the integral sign represents the summation with t as integration variable and dt as time step.
We can now express the Fundamental Theorem of Calculus as:
- Time stepping of constructs the integral of the integrand .
- Differentiation of the integral produces the integrand .
Here the constructive step is the time stepping 1 with given, captured as summation by repetition, while 2 expresses the starting point as the initial value problem with or .
Extension to depending also on x direct.
A basic case is for which the integral is 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.
- Study experimentally by computing the dependence of on the time step dt.