A series of mathematical techniques developed independently by Isaac Newton and Gottfried Leibniz (1646–1716). Differential calculus treats a continuously varying quantity as if it consisted of an infinitely large number of infinitely small changes. For example, the velocity v of a body at a particular instant can be regarded as the infinitesimal distance, written ds, that it travels in the vanishingly small time interval dt; the instantaneous velocity v is then ds/dt, which is called the derivative of s with respect to t. If s is a known function of t, v at any instant can be calculated by the process of differentiation. The differential calculus is a powerful technique for solving many problems concerned with rate processes, maxima and minima, and similar problems.
Integral calculus is the opposite technique. For example, if the velocity of a body is a known function of time, the infinitesimal distance ds travelled in the brief instant dt is given by ds=vdt. The measurable distance s travelled between two instants t1 and t2 can then be found by a process of summation, called integration, i.e.
The technique is used for finding areas under curves and volumes and other problems involving the summation of infinitesimals.
For problems such as Brownian motion and fractals, in which functions that are not smoothly varying appear, a rigorous mathematical treatment requires a more general concept of calculus. Such a concept was developed by Norbert Wiener (1894–1964) and others in the first half of the 20th century.