infinitesimal calculus

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infinitesimal calculus

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infinitesimal calculus (uncountable)

(calculus) Differential calculus and integral calculus considered together as a single subject.

Although calculus (in the sense of analysis) is usually synonymous with infinitesimal calculus, not all historical formulations have relied on infinitesimals (infinitely small numbers that are are nevertheless not zero). The original infinitesimal calculus of Newton and Leibniz did use them, but not in a demonstrably rigorous way, and many philosophers found the notion of an infinitesimal objectionable. Early attempts to prove the rigour of the approach were unsuccessful. A rigorous formulation (known also as standard calculus) was developed by Cauchy and Weierstrass, who avoided infinitesimals and made use of the concept of limit. In the 1960s, Robinson was able to develop a rigorous formulation (known as non-standard calculus) that makes use of infinitesimals. (See Wikipedia:infinitesimal calculus.) Another rigorous use of infinitesimals for calculus can be found in smooth infinitesimal analysis in which infinitesimals of a higher order than the first are 'neglected' (with their terms) as they arise in derivations.

Differential calculus and integral calculus considered together as a single subject