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Calculus

Calculus is the most important tool or respectfully branch of mathematics, which is fundamentally, used most of the major scientific and technological breakthrough inventions of modern world.

In essence, the calculus consists of a collection of methods to describe and handle patterns of infinity. The infinity large and the infinity small. Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches:

  1. Differential calculus (concerning rates of change and slopes of curves)
  2. Integral calculus (concerning accumulation of quantities and the areas under and between curves)

These two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.

The two mathematicians who invented Calculus

Modern calculus was developed in 17th century Europe at about the same time by two mathematicians working independently of each other. Sir Isaac Newton in England and Gottfried Wilhelm Leibniz in Germany, but elements of it have appeared in ancient Greece, China, medieval Europe, India, and the Middle East. It is believed that Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today.

Controversy

When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to be published in his Method of Fluxions), but Leibniz published his “Nova Methodus pro Maximis et Minimis” 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 Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society.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.

Application

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 modeled and an optimal solution is desired. It allows one to go from (non-constant) rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other.

Physics makes particular use of calculus; all concepts in classical mechanics and electromagnetism are related through calculus. The mass of an object of known density, the moment of inertia of objects, as well as the total energy of an object within a conservative field can be found by the use of calculus.

Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus. Chemistry also uses calculus in determining reaction rates and radioactive decay. In biology, population dynamics starts with reproduction and death rates to model population changes.