calculus

Branch of mathematics which uses the concept of a derivative (see differentiation) to analyse the way in which the values of a function vary. Calculus is probably the most widely used part of mathematics. Many real-life problems are analysed by expressing one quantity as a function of another – position of a moving object as a function of time, temperature of an object as a function of distance from a heat source, force on an object as a function of distance from the source of the force, and so on – and calculus is concerned with such functions.

There are several branches of calculus. Differential and integral calculus, both dealing with small quantities which during manipulation are made smaller and smaller, compose the infinitesimal calculus. Differential equations relate to the derivatives of a set of variables and may include the variables. Many give the mathematical models for physical phenomena such as simple harmonic motion. Differential equations are solved generally by integration, depending on their degree. If no analytical processes are available, integration can be performed numerically. Other branches of calculus include calculus of variations and calculus of errors.

Calculus methods have been developed slowly since the ancient Greek mathematicians. In the 17th century Isaac Newton and Gottfried Leibniz were the first to give (independently) general rules for calculus but it was very difficult to put the subject on a secure logical basis, mainly because of the difficult concepts of limit and continuity involved. Instead of using the idea of limit, 18th- and 19th-century mathematicians sought to base calculus on the ideas of ‘infinitesimals’ (roughly, very small quantities) and ‘differentials’ and the subject has in the past been known as ‘infinitesimal calculus’ or ‘differential calculus’. The first complete presentation of calculus using limits was given by Augustin Cauchy in 1821, but his ideas were not generally adopted (particularly in Britain) for many years.

calculus

Another name for a stone formed in the body, notably in the gall bladder (see gallstone) or urinary tract. It may need to be removed surgically or by ultrasound (lithotripsy).