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Calculus : Differential calculus

This article is about a branch of mathematics. For the meaning in dentistry, see Calculus (dental).

Calculus is a branch of mathematics, developed from algebra and geometry. Calculus focuses on rates of change (within functions), such as accelerations, curves, and slopes. The development of calculus is credited to Leibniz and Newton; lesser credit is given to Barrow, Descartes, de Fermat, and Huygens. Fundamental to calculus are derivatives, integrals, and limits. One of the primary motives for the development of calculus was to solve the so-called "tangent line problem".

There are two main branches of calculus:

Differential calculus is concerned with finding the instantaneous rate of change (or derivative) of a function's value, with respect to changes within the function's arguments. Another application of differential calculus is Newton's method, an algorithm to find zeros of a function by approximating the function by its tangent. de Fermat is sometimes described as the "father" of differential calculus.

Integral calculus, studies methods for finding the integral of a function. An integral may be defined as the limit of a sum of terms which correspond to areas under the graph of a function. Considered as such, integration allows us to calculate the area under a curve and the surface area and volume of solids such as spheres and cones.

The fundamental theorem of calculus states that derivatives and integrals are inverse operations. It was this realization by Newton/Leibniz that was the key to the explosion of analytic results after their work became known. The understanding of the connection allows us to recover the total change in a function over some interval from its instantaneous rate of change, by integrating the latter. The fundamental theorem also provides a method to compute many integrals algebraically, without actually performing the limit process, by finding antiderivatives. It also allows us to solve some differential equations, equations that relate an unknown function to its derivative. Differential equations are ubiquitous in the sciences.

The conceptual foundations of calculus include the function, limit, infinite sequences, infinite series, and continuity. Its tools include the symbol manipulation techniques associated with elementary algebra, and mathematical induction.

Calculus has been extended to differential equations, vector calculus, calculus of variations, and differential topology. The modern, formally correct version of calculus is known as real analysis.

History

Although Archimedes and others have used integral methods throughout history, and a great many (Barrow, Fermat, Pascal, Wallis and others) had previously invented the idea of a derivative, Gottfried Wilhelm Leibniz and Sir Isaac Newton are usually credited with the invention, in the late 1600s, of differential and integral calculus as we know it today. Leibniz and Newton, apparently working independently, arrived at similar results. It is thought that Newton's discoveries were made earlier, but Leibniz' were the first to be published. Newton (who represented derivatives as <math>f'</math>, <math>f</math>, etc.) provided a host of applications in physics, but Leibniz' more flexible notation (<math>df/dx</math>, <math>d^2f/dx^2</math>, etc.) was eventually adopted. (The simpler <math>f'</math> notation is still used in some cases where it is sufficient.)

In 1704 an anonymous pamphlet, later determined to have been written by Leibniz, accused Newton of having plagiarised Leibniz' work. That claim is easily refuted as there is ample evidence to show that Newton commenced work on the calculus long before Leibniz can possibly have done, however the resulting controversy lead to suggestions that Leibniz may not have invented the calculus independently as he claimed, but may have been influenced by reading copies of Newtons early manuscripts. This claim isn't so easily dismissed and there is in fact considerable circumstantial evidence to support it. Leibnitz was not known at the time for his probity, and later admitted to falsifying the dates on certain of his manuscripts in an effort to bolster his claims. Furthermore a copy of one of Newtons very early manuscripts with annotations by Leibniz was found among Leibniz' papers after his death, although the exact date when Leibniz first aquired this is unknown. It is also interesting to note that a similar controversy exists in philosophy over whether or not Leibniz may have appropriated the ideas of Spinoza in his writings on that subject.

The truth of the matter will never be known, and in any case is unimportant to anyone alive today. Leibniz' great contribution to calculus was hisn'tation, and this is beyond doubt purely of Leibniz' invention. The controversy was unfortunate however in that it divided the mathematicians of Britain and Europe for many years. This set back British analysis (i.e. calculus based mathematics) for a very long time. Newton's terminology and notation was clearly less flexible than Leibniz', but was retained in British usage until the early 19th century, when the work of the Analytical Society successfully saw its introduction in Great Britain.

The limit definition of the derivative presented above was not evolved until much later, and neither Newton nor Leibniz, nor any of their followers until the mid-1800s, developed calculus with acceptable rigour. Nevertheless, the calculus was widely used, as it was a very powerful mathematical tool, but it was not until the nineteenth century that mathematicians like Augustin Louis Cauchy, Bernhard Bolzano[?], and Karl Weierstrass were able to provide a mathematically rigorous exposition. This eventually resulted in deep explorations of the concept of infinity by Georg Cantor and others.

See also: calculus with polynomials

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