Calculus

Calculus is the branch of mathematics that studies continuous change. It has two main parts. Differential calculus deals with instantaneous rates of change and the slopes of curves. Integral calculus deals with accumulation, such as areas, volumes and totals. The fundamental theorem of calculus links the two by showing that differentiation and integration are inverse operations. Both rest on the concept of a limit. Calculus took shape in the 17th century and became the basic mathematical language of physics, engineering and many other quantitative fields. Its rigorous foundations are studied in mathematical analysis.

Core concepts

The limit describes the value a function approaches as its input gets arbitrarily close to some point. Continuity, derivatives and integrals are all defined in terms of limits.

The derivative of a function measures how fast its output changes relative to its input at a single point. Geometrically, it is the slope of the tangent line to the function's graph. Physically, velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity. There are standard rules for computing derivatives, including the product, quotient and chain rules. Derivatives are used to find maxima and minima, so they are central to optimization problems.

The integral adds up infinitely many infinitesimally small pieces. A definite integral gives quantities such as the area under a curve, the work done by a variable force, or the total distance travelled at a changing speed. An indefinite integral, or antiderivative, is a function whose derivative is a given function.

The fundamental theorem of calculus says two things. First, if you integrate a function and then differentiate the result, you get the original function back. Second, you can evaluate a definite integral by taking any antiderivative and subtracting its values at the two endpoints. Before this result, finding areas required ingenious geometric arguments. Afterwards, it became a largely systematic procedure.

Infinite series let you write functions as sums of infinitely many terms. Taylor series, for example, express a function as an infinite polynomial. Series are a basic tool for approximation and computation.

Early history

Ideas that anticipate calculus go back to antiquity. In ancient Greece, Archimedes used the method of exhaustion to compute areas and volumes. He approximated curved figures with ever finer polygons, an approach close in spirit to integration. In China, Liu Hui used a similar polygon-based approach in the 3rd century to estimate π. In 14th- and 15th-century India, mathematicians of the Kerala school, beginning with Madhava of Sangamagrama, found infinite series for trigonometric functions.

During the Scientific Revolution, interest in motion and curves grew rapidly. Johannes Kepler, Bonaventura Cavalieri, Pierre de Fermat, Blaise Pascal, John Wallis and Isaac Barrow developed methods for tangents, maxima and areas. The coordinate geometry of René Descartes and Fermat made it possible to treat curves algebraically. That joining of geometry and algebra set the stage for a general calculus.

Newton, Leibniz and the priority dispute

Isaac Newton and Gottfried Wilhelm Leibniz are credited as the founders of calculus. Newton worked out his "method of fluxions" in the mid-1660s and used it in his work on gravity and classical mechanics. However, he published his main calculus writings only much later: in 1704 as an appendix to his Opticks, and in 1736, after his death. Leibniz developed his calculus independently in the 1670s and published it in the journal Acta Eruditorum in 1684 and 1686.

The different publication dates led to the Leibniz–Newton calculus controversy. Newton's supporters accused Leibniz of plagiarizing Newton's unpublished ideas. In 1712 the Royal Society set up a committee to investigate. Its members were mainly Newton's supporters, and its report sided with Newton. Most modern historians conclude that the two men developed calculus independently, using very different notations.

Leibniz's notation includes dy/dx for the derivative and the elongated S (∫) for the integral. It proved easier to use and is the notation used today. Continental mathematicians adopted it, while British mathematicians kept Newton's dot notation. This split slowed communication between the two mathematical communities for many years.

Rigor and foundations

Early calculus relied on "infinitesimals," quantities that were supposed to be smaller than any positive number but not zero. Their logical status was unclear. In 1734 the philosopher George Berkeley famously criticized them as "ghosts of departed quantities." Meanwhile, 18th-century mathematicians, especially the Bernoulli family, Leonhard Euler and Joseph-Louis Lagrange, greatly extended the subject's techniques and applications.

Calculus was given a rigorous footing in the 19th century. Augustin-Louis Cauchy built the subject around limits. Karl Weierstrass introduced the precise ε–δ definition of a limit. Bernhard Riemann gave a rigorous definition of the integral. Richard Dedekind and others built rigorous constructions of the real numbers, which supplied the completeness that calculus needs. Later, Henri Lebesgue developed measure-based integration. In the 1960s, Abraham Robinson's nonstandard analysis showed that infinitesimals can be given a rigorous logical foundation.

Extensions

Multivariable calculus extends derivatives and integrals to functions of several variables. It introduces partial derivatives, multiple integrals, and the theorems of vector calculus, such as Green's, Stokes' and the divergence theorems. Differential equations relate functions to their derivatives and are the main way to model change over time and space. Complex analysis applies calculus to functions of a complex variable. The calculus of variations finds functions that optimize quantities such as path length or energy.

Applications

Calculus is essential throughout science and technology. In physics, Newton's laws of motion, the equations of electromagnetism and the theory of relativity are all written in the language of calculus. Engineers use it to analyse structures, circuits, fluid flow and control systems. In economics, marginal cost and marginal utility are derivatives. In probability and statistics, continuous distributions are described by density functions and integrals. Biology and medicine use differential equations to model population growth, epidemics and how drugs are processed by the body. In machine learning, training algorithms such as gradient descent depend on computing derivatives of loss functions.