Calculus

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (mathematical analysis).

It has two major branches, differential calculus and integral calculus.

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change.

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity.

Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. The product rule and chain rule, the notions of higher derivatives and Taylor series, and of analytic functions were used by Isaac Newton in an idiosyncratic notation which he applied to solve problems of mathematical physics.

Newton also made seminal contributions to optics, and shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits.

Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).

In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits.

Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis.

Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.

Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.

It has two major branches, differential calculus and integral calculus. However, they were not able to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today".

Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other.

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from the 16th or 17th century.

408–355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c.

Generally considered the greatest mathematician of antiquity and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola.

Today, calculus has widespread uses in science, engineering, and economics.

The inventor and mathematician Archimedes of Syracuse made major contributions to the beginnings of calculus and has sometimes been credited as its inventor, although his proto-calculus lacked several defining features.

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

This was a necessary precursor to the development of calculus and a precise quantitative science of physics.

Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.

The study of series is a major part of calculus and its generalization, mathematical analysis.

Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (13th dynasty, c.

For shapes with curved boundary, calculus is usually required to compute the area.

In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits.

In most of the U.S., algebra, geometry and analysis (pre-calculus and calculus) are taught as separate courses in different years of high school.

The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving the second fundamental theorem of calculus around 1670.

John Wallis (3 December 1616 – 8 November 1703) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus.

The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving the second fundamental theorem of calculus around 1670.

Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for the discovery of the fundamental theorem of calculus.

The product rule and chain rule, the notions of higher derivatives and Taylor series, and of analytic functions were used by Isaac Newton in an idiosyncratic notation which he applied to solve problems of mathematical physics.

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions.

The product rule and chain rule, the notions of higher derivatives and Taylor series, and of analytic functions were used by Isaac Newton in an idiosyncratic notation which he applied to solve problems of mathematical physics.

In calculus, the product rule is a formula used to find the derivatives of products of two or more functions.

Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term.

Pierre de Fermat (between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.

The product rule and chain rule, the notions of higher derivatives and Taylor series, and of analytic functions were used by Isaac Newton in an idiosyncratic notation which he applied to solve problems of mathematical physics. However, they were not able to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today".

Derivatives are a fundamental tool of calculus.

Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics thereby stated components of calculus.

One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra.

Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz.

As a representative of the seventeenth-century tradition of rationalism, Leibniz's most prominent accomplishment was conceiving the ideas of differential and integral calculus, independently of Isaac Newton's contemporaneous developments.

In Europe, the foundational work was a treatise written by Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections.

He is known for his work on the problems of optics and motion, work on indivisibles, the precursors of infinitesimal calculus, and the introduction of logarithms to Italy.

He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his Principia Mathematica (1687).

In formulating his physical theories, Newton developed and used mathematical methods now included in the field of calculus.