Calculus

History of calculus Calculus (Latin, calculus, a small stone used for counting) is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is thestudy of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering andcan solve many problems for which algebra alone is insufficient. Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) may refer to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus,lambda calculus, pi calculus, and join calculus. History Ancient The ancient period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus (c. 1820 BC), in which an Egyptian successfully calculated thevolume of a pyramidal frustum.[2][3] From the school of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus.[4] The method of exhaustion was later reinvented in China by Liu Hui in the 3rdcentury AD in order to find the area of a circle.[5] In the 5th century AD, Zu Chongzhi established a method which would later be called Cavalieri's principle to find the volume of a sphere.[6]

Medieval Around AD 1000, the Islamic mathematician Ibn al-Haytham (Alhacen) was the first to derive the formula for the sum of the fourth powers of an arithmetic progression, using a method that isreadily generalizable to finding the formula for the sum of any higher integral powers, which he used to perform an integration.[7] In the 11th century, the Chinese polymath Shen Kuo developed 'packing' equations that dealt with integration. In the 12th century, the Indian mathematician, Bhāskara II, developed an early derivative representing infinitesimal change, and he described an early form ofRolle's theorem.[8] Also in the 12th century, the Persian mathematician Sharaf al-Dīn al-Tūsī discovered the derivative of cubic polynomials, an important result in differential calculus.[9] In the 14th century, Indian mathematician Madhava of Sangamagrama, along with other mathematician-astronomers of the Kerala school of astronomy and mathematics, described special cases of Taylor series,[10] whichare treated in the text Yuktibhasa.[11][12][13]

Modern In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimal thin crosssections. The ideas were similar to Archimedes' in The Method, but this treatise was lost until the early part of the twentieth century.Cavalieri's work was not well respected since his methods can lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus combined Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two...