Invented by : Isaac Newton / Leibniz Invented in year : 1693
Calculus is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient. The main idea behind the Calculus, developed over a very long period of time. Various cultures are linked with Calculus. These include Egyptians, Greeks, Chinese, Islamists, Indians, Persians, Europeans and Japanese. All these cultures contributed and helped in the development of Calculus. However these ideas were not systematic.
History of the Invention
In the modern period, independent discoveries relating to Calculus were being made in early 17th century Japan, by mathematicians such as Seki Kowa, who expanded upon the method of exhaustion. 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 Greek Scholar, 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 proving the second fundamental theorem of Calculus around 1675.
The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical 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 problems discussed in his Principia Mathematica. In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.
These ideas were systematized into a true Calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. When Newton published his first results in 1693 and when Leibniz published his first results in 1684, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first, but Leibniz published first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. Today, both Newton and Leibniz are given credit for developing Calculus independently. It is Leibniz, however, who gave the new discipline its name. Newton called his Calculus "The Science of Fluxions'. Leibniz, is now regarded as an independent inventor of and contributor to Calculus. His contribution was to provide a clear set of rules for manipulating infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz paid a lot of attention to the formalism – he often spent days determining appropriate symbols for concepts.
Leibniz and Newton are usually both credited with the invention of Calculus. Newton was the first to apply Calculus to general physics and Leibniz developed much of the notation used in Calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. Before Newton and Leibniz, the word “Calculus” was a general term used to refer to any body of mathematics, but in the following years, "Calculus" became a popular term for a field of mathematics based upon their insights. By Newton's time, the fundamental theorem of Calculus was known.
Development in the Invention of Calculus
Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of Calculus. In the 19th century, Calculus was put on a much more rigorous footing by mathematicians such as Cauchy, Riemann, and Weierstrass. It was also during this period that the ideas of Calculus were generalized to Euclidean space and the complex plane. Lebesgue generalized the notion of the integral so that virtually any function has an integral, while Laurent Schwartz extended differentiation in much the same way.
Role of Calculus in the Improvement Of Human Life
Calculus became a ubiquitous topic in most modern high schools and universities around the world.
Today, Calculus is used in every branch of the physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, demography and in other fields wherever a problem can be mathematically modelled and an optimal solution is desired.
Physics makes particular use of Calculus; all concepts in classical mechanics are interrelated through Calculus.
In the realm of medicine, Calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow.
In economics, Calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue.