History of calculusCalculus, originally called infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Many elements of calculus appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India. Infinitesimal calculus was developed in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz independently of each other. An argument over priority led to the Leibniz–Newton calculus controversy which continued until the death of Leibniz in 1716.
Notation for differentiationIn differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation (and its opposite operation, the antidifferentiation or indefinite integration) are listed below.
Differential equationIn mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
The AnalystThe Analyst (subtitled A Discourse Addressed to an Infidel Mathematician: Wherein It Is Examined Whether the Object, Principles, and Inferences of the Modern Analysis Are More Distinctly Conceived, or More Evidently Deduced, Than Religious Mysteries and Points of Faith) is a book by George Berkeley. It was first published in 1734, first by J. Tonson (London), then by S. Fuller (Dublin). The "infidel mathematician" is believed to have been Edmond Halley, though others have speculated Sir Isaac Newton was intended.
Limit of a functionAlthough the function \tfrac{\sin x}{x} is not defined at zero, as x becomes closer and closer to zero, \tfrac{\sin x}{x} becomes arbitrarily close to 1. In other words, the limit of \tfrac{\sin x}{x}, as x approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x.
Differential (mathematics)In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions. The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology. The term differential is used nonrigorously in calculus to refer to an infinitesimal ("infinitely small") change in some varying quantity.
