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.
- Study experimentally by computing the dependence of on the time step .