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Calculus
Topics in calculus
Function
Limits of functions
Continuity
Calculus with polynomials
Differentiation
Product rule
Quotient rule
Chain rule
Implicit differentiation
Taylor's theorem
Integration
Integration by substitution
Integration by parts
Integration by trigonometric substitution
Solids of revolution
Integration by disks
Integration by cylindrical shells
List of integrals
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 Archimedes, Leibniz and Newton; lesser credit is given to Barrow, Descartes, de Fermat, Huygens, and Wallis. Fundamental to calculus are derivatives, integrals, and limits. One of the primary motives for the development of modern 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, time scale calculus 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
,
,
etc.) provided a host of applications in physics, but Leibniz' more flexible differential notation (df / dx, d2f / dx2, etc.) was eventually adopted. (The simpler f' 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 Newton's early manuscripts. This claim is not so easily dismissed and there is in fact considerable circumstantial evidence to support it. Leibniz 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 Newton's very early manuscripts with annotations by Leibniz was found among Leibniz' papers after his death, although the exact date when Leibniz first acquired 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 his notation, and this is beyond doubt purely of Leibniz's 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 that of Leibniz, yet it was retained in British usage until the early 19th century, when the work of the Analytical Society successfully saw the introduction of Leibniz's notation in Great Britain.
The strict 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, Bernard 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.
Abel seems to have been the first to consider in a general way the question as to what differential expressions can be integrated in a finite form by the aid of ordinary functions, an investigation extended by Liouville. Cauchy early undertook the general theory of determining definite integrals, and the subject has been prominent during the 19th century. Frullani's theorem (1821), Bierens de Haan's
work on the theory (1862) and his elaborate tables (1867),
Dirichlet's lectures (1858) embodied in Meyer's treatise (1871), and
numerous memoirs of Legendre, Poisson, Plana, Raabe, Sohncke,
Schlömilch, Elliott, Leudesdorf, and Kronecker are among the
noteworthy contributions.
Eulerian integrals were first studied by Euler and afterwards investigated by Legendre, by whom they were classed as Eulerian integrals of the first and second species, as follows:
,
,
although these were not the exact forms of Euler's study. If n is integral, it follows that
,
but if n is fractional it is a transcendent function. To it Legendre assigned the symbol Γ, and it is now called the
gamma function. To the subject Dirichlet has contributed an
important theorem (Liouville, 1839), which has been elaborated by
Liouville, Catalan, Leslie Ellis, and others. On the evaluation of
Γx and logΓx Raabe (1843-44), Bauer (1859), and
Gudermann (1845) have written. Legendre's great table appeared in
1816.
Symbolic methods may be traced back to Taylor, and the analogy
between successive differentiation and ordinary exponentials had
been observed by numerous writers before the nineteenth
century. Arbogast (1800) was the first, however, to separate the
symbol of operation from that of quantity in a differential
equation. François (1812) and Servois (1814) seem to have been
the first to give correct rules on the subject. Hargreave (1848)
applied these methods in his memoir on differential equations, and
Boole freely employed them. Grassmann and Hankel made great use of
the theory, the former in studying equations, the latter in his
theory of complex numbers.
The calculus of variations may be said to begin with a problem of Johann Bernoulli's (1696). It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but Euler first elaborated
the subject. His contributions began in 1733, and his Elementa Calculi Variationum gave to the science its name. Lagrange
contributed extensively to the theory, and Legendre (1786) laid down
a method, not entirely satisfactory, for the discrimination of
maxima and minima. To this discrimination Brunacci (1810), Gauss
(1829), Poisson (1831), Ostrogradsky (1834), and Jacobi (1837) have
been among the contributors. An important general work is that of
Sarrus (1842) which was condensed and improved by Cauchy
(1844). Other valuable treatises and memoirs have been written by
Strauch (1849), Jellett (1850), Hesse (1857), Clebsch (1858), and
Carll (1885), but perhaps the most important work of the century is
that of Weierstrass. His celebrated course on the theory is
epoch-making, and it may be asserted that he was the first to place
it on a firm and unquestionable foundation.
The application of the infinitesimal calculus to problems in physics and astronomy was contemporary with the origin of the science. All through the eighteenth century these applications were multiplied, until at its close Laplace and Lagrange had brought the whole range of the study of forces into the realm of analysis. To Lagrange (1773) we owe the introduction of the theory of the
potential into dynamics, although the name "potential function" and the
fundamental memoir of the subject are due to Green (1827, printed in 1828). The name "potential" is due to Gauss (1840), and the
distinction between potential and potential function to
Clausius. With its development are connected the names of Dirichlet,
Riemann, Neumann, Heine, Kronecker, Lipschitz, Christoffel,
Kirchhoff, Beltrami, and many of the leading physicists of the
century.
It is impossible in this place to enter into the great variety of
other applications of analysis to physical problems. Among them are
the investigations of Euler on vibrating chords; Sophie Germain on
elastic membranes; Poisson, Lamé, Saint-Venant, and Clebsch on
the elasticity of three-dimensional bodies; Fourier on heat
diffusion; Fresnel on light; Maxwell, Helmholtz, and Hertz on
electricity; Hansen, Hill, and Gyldén on astronomy; Maxwell on
spherical harmonics; Lord Rayleigh on acoustics; and the
contributions of Dirichlet, Weber, Kirchhoff, F. Neumann, Lord
Kelvin, Clausius, Bjerknes, MacCullagh, and Fuhrmann to physics in
general. The labors of Helmholtz should be especially mentioned,
since he contributed to the theories of dynamics, electricity, etc.,
and brought his great analytical powers to bear on the fundamental
axioms of mechanics as well as on those of pure mathematics.
See also: calculus with polynomials
Further reading
In mathematics and related fields, the term calculus more generally refers to a system of formal rules of inference and axioms that are used for computation.
This usage is particularly common in mathematical logic, where a calculus is applied to compute universally true statements of a certain formal logic. Examples include the calculus of natural deduction, the sequent calculus, as well as many other calculi that are deviced in proof theory.
Derived from the Latin word for "pebble", calculus in its most general sense can mean any method or system of calculation. Other topics where the term calculus is used in this sense include:
Topics in mathematics related to change
Arithmetic | Calculus | Vector calculus | Analysis | Differential equations | Dynamical systems and chaos theory | Fractional calculus | List of functions
This article is adapted from from Wikipedia All Wikipedia article text is available under the terms of the GNU Free Documentation License
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