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Calculus

From Wikipedia, the free encyclopedia

This article is about the branch of mathematics. For other uses, see Calculus (disambiguation).

Topics in Calculus

Fundamental theorem

Limits of functions

Continuity

Mean value theorem

[show]Differential calculus

[show]Integral calculus

[show]Vector calculus

[show]Multivariable calculus

Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits,functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modernmathematics education. It has two major branches,differential calculus and integral calculus, which are related by the fundamental ...view middle of the document...

1 Notes

o 5.2 Books

• 6 Other resources

o 6.1 Further reading

o 6.2 Online books

• 7 External links

[edit]History

Main article: History of calculus

[edit]Ancient

Isaac Newton developed the use of calculus in his laws of motion andgravitation.

The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculations of volumes and areas, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are mere instructions, with no indication as to method, and some of them are wrong. Some, including Morris Kline inMathematical thought from ancient to modern times, Vol. I, suggest trial and error.[2] From the age 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 the methods of integral calculus.[3] The method of exhaustionwas later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle.[4] In the 5th century AD, Zu Chongzhiestablished a method which would later be called Cavalieri's principle to find the volume of a sphere.[5]

[edit]Modern

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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 cross-sections. 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 could 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 proving the second fundamental theorem of calculus around 1675.

The product rule and chain rule, the notion of higher derivatives, Taylor series, andanalytical functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve problems of mathematical physics. In his publications, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. 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...

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