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99-486: The Yeloguy is 464 kilometres (288 mi) long, and the area of its basin is 25,100 square kilometres (9,700 sq mi). The lower reaches of the river are navigable downstream from Kellog . The Yeloguy was one of the places where Ket singer Alexander Kotusov found inspiration for his songs. The Yeloguy has its source in the West Siberian Plain . It forms at the confluence of two short rivers,

198-422: A ) = f ( b ) = 0 {\displaystyle f\left(a\right)=f\left(b\right)=0} , then f ′ ( x ) = 0 {\displaystyle f'\left(x\right)=0} for some x {\displaystyle x} with   a < x < b {\displaystyle \ a<x<b} . In his astronomical work, Bhāskara gives

297-400: A definite integral : The formula for the area enclosed by an ellipse is related to the formula of a circle; for an ellipse with semi-major and semi-minor axes x and y the formula is: Most basic formulas for surface area can be obtained by cutting surfaces and flattening them out (see: developable surfaces ). For example, if the side surface of a cylinder (or any prism )

396-602: A function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties: It can be proved that such an area function actually exists. Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m ), square centimetres (cm ), square millimetres (mm ), square kilometres (km ), square feet (ft ), square yards (yd ), square miles (mi ), and so forth. Algebraically, these units can be thought of as

495-437: A general framework of integral calculus . Archimedes was the first to find the tangent to a curve other than a circle, in a method akin to differential calculus. While studying the spiral, he separated a point's motion into two components, one radial motion component and one circular motion component, and then continued to add the two component motions together, thereby finding the tangent to the curve. The method of exhaustion

594-437: A more complicated shape usually requires multivariable calculus . Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra , and is a basic property of surfaces in differential geometry . In analysis , the area of a subset of the plane is defined using Lebesgue measure , though not every subset

693-450: A new formula where x = x + o (importantly, o is the letter, not the digit 0). He then recalculated the area with the aid of the binomial theorem, removed all quantities containing the letter o and re-formed an algebraic expression for the area. Significantly, Newton would then "blot out" the quantities containing o because terms "multiplied by it will be nothing in respect to the rest". At this point Newton had begun to realize

792-438: A new mathematical system. Importantly, Newton and Leibniz did not create the same calculus and they did not conceive of modern calculus. While they were both involved in the process of creating a mathematical system to deal with variable quantities their elementary base was different. For Newton, change was a variable quantity over time and for Leibniz it was the difference ranging over a sequence of infinitely close values. Notably,

891-428: A rectangle with length l and width w , the formula for the area is: That is, the area of the rectangle is the length multiplied by the width. As a special case, as l = w in the case of a square, the area of a square with side length s is given by the formula: The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom . On

990-569: A restricted version of the second fundamental theorem of calculus, that integrals can be computed using any of a function's antiderivatives. The first full proof of the fundamental theorem of calculus was given by Isaac Barrow . One prerequisite to the establishment of a calculus of functions of a real variable involved finding an antiderivative for the rational function f ( x )   =   1 x . {\displaystyle f(x)\ =\ {\frac {1}{x}}.} This problem can be phrased as quadrature of

1089-415: A result that looks like a precursor to infinitesimal methods: if x ≈ y {\displaystyle x\approx y} then sin ⁡ ( y ) − sin ⁡ ( x ) ≈ ( y − x ) cos ⁡ ( y ) {\displaystyle \sin(y)-\sin(x)\approx (y-x)\cos(y)} . This leads to the derivative of

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1188-596: A significant use of infinitesimals . Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they seemingly create. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat in his The Quadrature of

1287-653: A sphere was first obtained by Archimedes in his work On the Sphere and Cylinder . The formula is: where r is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus . History of calculus Calculus , 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

1386-554: A technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature. In the 17th century, European mathematicians Isaac Barrow , René Descartes , Pierre de Fermat , Blaise Pascal , John Wallis and others discussed the idea of a derivative . In particular, in Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum distributed in 1636, Fermat introduced

1485-413: A unit-radius circle) with his doubling method , in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular hexagon , then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with circumscribed polygons ). Heron of Alexandria found what is known as Heron's formula for

1584-451: A willingness to view infinite series not only as approximate devices, but also as alternative forms of expressing a term. Many of Newton's critical insights occurred during the plague years of 1665–1666, which he later described as, "the prime of my age for invention and minded mathematics and [natural] philosophy more than at any time since." It was during his plague-induced isolation that the first written conception of fluxionary calculus

1683-646: Is convex , which aesthetically justifies this analytic continuation of the factorial function over any other analytic continuation. To the subject Lejeune Dirichlet has contributed an important theorem (Liouville, 1839), which has been elaborated by Liouville , Catalan , Leslie Ellis , and others. Raabe (1843–44), Bauer (1859), and Gudermann (1845) have written about the evaluation of Γ ( x ) {\displaystyle \Gamma (x)} and log ⁡ Γ ( x ) {\displaystyle \log \Gamma (x)} . Legendre's great table appeared in 1816. The application of

1782-447: Is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is r , and the width is half the circumference of the circle, or π r . Thus, the total area of the circle is π r : Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of

1881-406: Is commonly used in describing the cross-sectional area of interaction in nuclear physics . In South Asia (mainly Indians), although the countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have same names, but with different values. There's no official consensus about the traditional units values. Thus,

1980-418: Is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone , the side surface can be flattened out into a sector of a circle, and the resulting area computed. The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature , it cannot be flattened out. The formula for the surface area of

2079-416: Is equivalent to 6 million square millimetres. Other useful conversions are: In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units. the relationship between square feet and square inches is where 144 = 12 = 12 × 12. Similarly: In addition, conversion factors include: There are several other common units for area. The are

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2178-443: Is measurable if one supposes the axiom of choice. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists. An approach to defining what is meant by "area" is through axioms . "Area" can be defined as

2277-500: Is reflected in the notation used today. Newton introduced the notation f ˙ {\displaystyle {\dot {f}}} for the derivative of a function f . Leibniz introduced the symbol ∫ {\displaystyle \int } for the integral and wrote the derivative of a function y of the variable x as d y d x {\displaystyle {\frac {dy}{dx}}} , both of which are still in use. Since

2376-415: Is used by modern mathematics, his logical base was different from our current one. Leibniz embraced infinitesimals and wrote extensively so as, "not to make of the infinitely small a mystery, as had Pascal." According to Gilles Deleuze , Leibniz's zeroes "are nothings, but they are not absolute nothings, they are nothings respectively" (quoting Leibniz' text "Justification of the calculus of infinitesimals by

2475-418: Is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus . For a solid shape such as a sphere , cone, or cylinder, the area of its boundary surface is called the surface area . Formulas for the surface areas of simple shapes were computed by the ancient Greeks , but computing the surface area of

2574-596: Is well established that Newton had started his work several years prior to Leibniz and had already developed a theory of tangents by the time Leibniz became interested in the question. It is not known how much this may have influenced Leibniz. The initial accusations were made by students and supporters of the two great scientists at the turn of the century, but after 1711 both of them became personally involved, accusing each other of plagiarism . The priority dispute had an effect of separating English-speaking mathematicians from those in continental Europe for many years. Only in

2673-460: The Cartesian coordinates ( x i , y i ) {\displaystyle (x_{i},y_{i})} ( i =0, 1, ..., n -1) of whose n vertices are known, the area is given by the surveyor's formula : where when i = n -1, then i +1 is expressed as modulus n and so refers to 0. The most basic area formula is the formula for the area of a rectangle . Given

2772-457: The calculus of finite differences developed in Europe at around the same time, and Fermat's adequality. The combination was achieved by John Wallis , Isaac Barrow , and James Gregory , the latter two proving predecessors to the second fundamental theorem of calculus around 1670. James Gregory , influenced by Fermat's contributions both to tangency and to quadrature, was then able to prove

2871-585: The felicific calculus in philosophy. The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volumes and areas, one goal of integral calculus, can be found in the Egyptian Moscow papyrus ( c.  1820 BC ), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning. Babylonians may have discovered

2970-406: The infinitesimal calculus to problems in physics and astronomy was contemporary with the origin of the science. All through the 18th 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

3069-401: The lune of Hippocrates , but did not identify the constant of proportionality . Eudoxus of Cnidus , also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared. Subsequently, Book I of Euclid's Elements dealt with equality of areas between two-dimensional figures. The mathematician Archimedes used the tools of Euclidean geometry to show that

Yeloguy - Misplaced Pages Continue

3168-441: The squares of the corresponding length units. The SI unit of area is the square metre, which is considered an SI derived unit . Calculation of the area of a square whose length and width are 1 metre would be: 1 metre × 1 metre = 1 m and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as: 3 metres × 2 metres = 6 m . This

3267-417: The surveyor's formula for the area of any polygon with known vertex locations by Gauss in the 19th century. The development of integral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas of various curved three-dimensional objects. For a non-self-intersecting ( simple ) polygon,

3366-445: The trapezoidal rule while doing astronomical observations of Jupiter . From the age of Greek mathematics , Eudoxus (c. 408–355 BC) used the method of exhaustion , which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287–212 BC) developed this idea further , inventing heuristics which resemble the methods of integral calculus. Greek mathematicians are also credited with

3465-548: The 1820s, due to the efforts of the Analytical Society , did Leibnizian analytical calculus become accepted in England. Today, both Newton and Leibniz are given credit for independently developing the basics of calculus. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today: "calculus". Newton's name for it was "the science of fluents and fluxions ". While neither of

3564-592: The Levy Yeloguy and the Pravy Yeloguy, both roughly 20 kilometres (12 mi) long. It flows roughly northeastwards across the flatland and in its lower course it meanders in the mostly flat and swampy taiga . About 30 kilometres (19 mi) before the mouth, the Crooked Yeloguy (Krivoy Yeloguy) splits to the right and flows roughly parallel to the main river. The Yeloguy joins the left bank of

3663-586: 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. The development of calculus and its uses within the sciences have continued to

3762-500: The Parabola , The Method , and On the Sphere and Cylinder . It should not be thought that infinitesimals were put on a rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. It was not until the 17th century that the method was formalized by Cavalieri as the method of Indivisibles and eventually incorporated by Newton into

3861-640: The Yenisey forming a many-branched delta near Verkhne Imbatskoye (Verkhneimbatsk) village, located on the facing bank of the Yenisey. The confluence is located roughly halfway between the mouths of the rivers Sym and Turukhan . The river freezes in October and stays frozen until mid-May. Its main tributaries are the Kellog , Bolshaya Sigovaya and the Tyna . A 7,476 km (2,886 sq mi) taiga zone of

3960-403: The area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book Measurement of a Circle . (The circumference is 2 π r , and the area of a triangle is half the base times the height, yielding the area π r for the disk.) Archimedes approximated the value of π (and hence the area of

4059-470: The area of a cyclic quadrilateral (a quadrilateral inscribed in a circle) in terms of its sides. In 1842, the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt independently found a formula, known as Bretschneider's formula , for the area of any quadrilateral. The development of Cartesian coordinates by René Descartes in the 17th century allowed the development of

Yeloguy - Misplaced Pages Continue

4158-428: The area of a triangle in terms of its sides, and a proof can be found in his book, Metrica , written around 60 CE. It has been suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work. In 300 BCE Greek mathematician Euclid proved that

4257-507: The area of a triangle is half that of a parallelogram with the same base and height in his book Elements of Geometry . In 499 Aryabhata , a great mathematician - astronomer from the classical age of Indian mathematics and Indian astronomy , expressed the area of a triangle as one-half the base times the height in the Aryabhatiya . In the 7th century CE, Brahmagupta developed a formula, now known as Brahmagupta's formula , for

4356-467: The area of an open surface or the boundary of a three-dimensional object . Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). Two different regions may have

4455-417: The areas of the approximate parallelograms is exactly π r , which is the area of the circle. This argument is actually a simple application of the ideas of calculus . In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus . Using modern methods, the area of a circle can be computed using

4554-404: The calculus of ordinary algebra"). Alternatively, he defines them as, "less than any given quantity". For Leibniz, the world was an aggregate of infinitesimal points and the lack of scientific proof for their existence did not trouble him. Infinitesimals to Leibniz were ideal quantities of a different type from appreciable numbers. The truth of continuity was proven by existence itself. For Leibniz

4653-454: The central property of inversion. He had created an expression for the area under a curve by considering a momentary increase at a point. In effect, the fundamental theorem of calculus was built into his calculations. While his new formulation offered incredible potential, Newton was well aware of its logical limitations at the time. He admits that "errors are not to be disregarded in mathematics, no matter how small" and that what he had achieved

4752-512: The chosen heir of Isaac Barrow in Cambridge . His aptitude was recognized early and he quickly learned the current theories. By 1664 Newton had made his first important contribution by advancing the binomial theorem , which he had extended to include fractional and negative exponents . Newton succeeded in expanding the applicability of the binomial theorem by applying the algebra of finite quantities in an analysis of infinite series . He showed

4851-437: The concept of adequality , which represented equality up to an infinitesimal error term. This method could be used to determine the maxima, minima, and tangents to various curves and was closely related to differentiation. Isaac Newton would later write that his own early ideas about calculus came directly from "Fermat's way of drawing tangents." The formal study of calculus brought together Cavalieri's infinitesimals with

4950-480: The contributors. An important general work is that of Sarrus (1842) which was condensed and improved by Augustin Louis Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), Otto Hesse (1857), Alfred Clebsch (1858), and Carll (1885), but perhaps the most important work of the century is that of Karl Weierstrass . His course on the theory may be asserted to be

5049-457: The conversions between the SI units and the traditional units may have different results, depending on what reference that has been used. Some traditional South Asian units that have fixed value: In the 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his quadrature of

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5148-490: The derivative of the function to be known. Evidence suggests Bhāskara II was acquainted with some ideas of differential calculus. Bhāskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of ' infinitesimals '. There is evidence of an early form of Rolle's theorem in his work. The modern formulation of Rolle's theorem states that if f (

5247-627: The descriptive terms each system created to describe change was different. Historically, there was much debate over whether it was Newton or Leibniz who first "invented" calculus. This argument, the Leibniz and Newton calculus controversy , involving Leibniz, who was German, and the Englishman Newton, led to a rift in the European mathematical community lasting over a century. Leibniz was the first to publish his investigations; however, it

5346-405: The first and second species, as follows: although these were not the exact forms of Euler's study. If n is a positive integer : but the integral converges for all positive real n {\displaystyle n} and defines an analytic continuation of the factorial function to all of the complex plane except for poles at zero and the negative integers. To it Legendre assigned

5445-492: The first to conceive calculus as a system in which new rhetoric and descriptive terms were created. Newton completed no definitive publication formalizing his fluxional calculus; rather, many of his mathematical discoveries were transmitted through correspondence, smaller papers or as embedded aspects in his other definitive compilations, such as the Principia and Opticks . Newton would begin his mathematical training as

5544-486: The first to place calculus on a firm and rigorous foundation. Antoine Arbogast (1800) was the first to separate the symbol of operation from that of quantity in a differential equation. Francois-Joseph Servois (1814) seems to have been the first to give correct rules on the subject. Charles James Hargreave (1848) applied these methods in his memoir on differential equations, and George Boole freely employed them. Hermann Grassmann and Hermann Hankel made great use of

5643-458: The following years, "calculus" became a popular term for a field of mathematics based upon their insights. Newton and Leibniz, building on this work, independently developed the surrounding theory of infinitesimal calculus in the late 17th century. Also, Leibniz did a great deal of work with developing consistent and useful notation and concepts. Newton provided some of the most important applications to physics, especially of integral calculus . By

5742-673: The informal use of infinitesimals in his calculations. While Newton began development of his fluxional calculus in 1665–1666 his findings did not become widely circulated until later. In the intervening years Leibniz also strove to create his calculus. In comparison to Newton who came to math at an early age, Leibniz began his rigorous math studies with a mature intellect. He was a polymath , and his intellectual interests and achievements involved metaphysics , law , economics , politics , logic , and mathematics . In order to understand Leibniz's reasoning in calculus his background should be kept in mind. Particularly, his metaphysics which described

5841-463: The inverse relationship or differential became clear and Leibniz quickly realized the potential to form a whole new system of mathematics. Where Newton over the course of his career used several approaches in addition to an approach using infinitesimals , Leibniz made this the cornerstone of his notation and calculus. In the manuscripts of 25 October to 11 November 1675, Leibniz recorded his discoveries and experiments with various forms of notation. He

5940-580: The leading physicists of the century. It is impossible in this article 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

6039-406: The left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle: However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of

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6138-644: The lower course of the river, including its confluence with the Tyna, was established as the Yeloguy Nature Reserve (Елогуйский Заказник) on 10 March 1987. The protected area is under the Central Siberia Nature Reserve . Area Area is the measure of a region 's size on a surface . The area of a plane region or plane area refers to the area of a shape or planar lamina , while surface area refers to

6237-426: The middle of the 17th century, European mathematics had changed its primary repository of knowledge. In comparison to the last century which maintained Hellenistic mathematics as the starting point for research, Newton, Leibniz and their contemporaries increasingly looked towards the works of more modern thinkers. Newton came to calculus as part of his investigations in physics and geometry . He viewed calculus as

6336-424: 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 Lejeune Dirichlet , Riemann , von Neumann , Heine , Kronecker , Lipschitz , Christoffel , Kirchhoff , Beltrami , and many of

6435-432: The other hand, if geometry is developed before arithmetic , this formula can be used to define multiplication of real numbers . Most other simple formulas for area follow from the method of dissection . This involves cutting a shape into pieces, whose areas must sum to the area of the original shape. For an example, any parallelogram can be subdivided into a trapezoid and a right triangle , as shown in figure to

6534-454: The other hand, the calculus of Leibniz was from a "logical point of view, distinctly inferior to that of Newton, for it never transcended the view of d y d x {\displaystyle {\frac {dy}{dx}}} as a quotient of infinitely small changes or differences in y and x ." However, heuristically, it was a success, despite being a "failure" from a logical point of view. The work of both Newton and Leibniz

6633-421: The parallelogram: Similar arguments can be used to find area formulas for the trapezoid as well as more complicated polygons . The formula for the area of a circle (more properly called the area enclosed by a circle or the area of a disk ) is based on a similar method. Given a circle of radius r , it is possible to partition the circle into sectors , as shown in the figure to the right. Each sector

6732-420: The physical world. The base of Newton's revised calculus became continuity; as such he redefined his calculations in terms of continual flowing motion. For Newton, variable magnitudes are not aggregates of infinitesimal elements, but are generated by the indisputable fact of motion. As with many of his works, Newton delayed publication. Methodus Fluxionum was not published until 1736. Newton attempted to avoid

6831-425: The present. In mathematics education , calculus denotes courses of elementary mathematical analysis , which are mainly devoted to the study of functions and limits. The word calculus is Latin for "small pebble" (the diminutive of calx , meaning "stone"), a meaning which still persists in medicine . Because such pebbles were used for counting out distances, tallying votes, and doing abacus arithmetic,

6930-401: The principle of continuity and thus the validity of his calculus was assured. Three hundred years after Leibniz's work, Abraham Robinson showed that using infinitesimal quantities in calculus could be given a solid foundation. The rise of calculus stands out as a unique moment in mathematics. Calculus is the mathematics of motion and change, and as such, its invention required the creation of

7029-499: The rectangular hyperbola xy = 1. In 1647 Gregoire de Saint-Vincent noted that the required function F satisfied F ( s t ) = F ( s ) + F ( t ) , {\displaystyle F(st)=F(s)+F(t),} so that a geometric sequence became, under F , an arithmetic sequence . A. A. de Sarasa associated this feature with contemporary algorithms called logarithms that economized arithmetic by rendering multiplications into additions. So F

7128-509: The results to carry out what would now be called an integration , where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid . Roshdi Rashed has argued that the 12th century mathematician Sharaf al-Dīn al-Tūsī must have used the derivative of cubic polynomials in his Treatise on Equations . Rashed's conclusion has been contested by other scholars, who argue that he could have obtained his results by other methods which do not require

7227-532: The same area (as in squaring the circle ); by synecdoche , "area" sometimes is used to refer to the region, as in a " polygonal area ". The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m ), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have

7326-453: The same area as three such squares. In mathematics , the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number . There are several well-known formulas for the areas of simple shapes such as triangles , rectangles , and circles . Using these formulas, the area of any polygon can be found by dividing the polygon into triangles . For shapes with curved boundary, calculus

7425-557: The same distance as a body with uniform speed whose speed is half the final velocity of the accelerated body. Johannes Kepler 's work Stereometrica Doliorum published in 1615 formed the basis of integral calculus. Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse. A significant work was a treatise inspired by Kepler's methods published in 1635 by Bonaventura Cavalieri on his method of indivisibles . He argued that volumes and areas should be computed as

7524-425: The scientific description of the generation of motion and magnitudes . In comparison, Leibniz focused on the tangent problem and came to believe that calculus was a metaphysical explanation of change. Importantly, the core of their insight was the formalization of the inverse properties between the integral and the differential of a function . This insight had been anticipated by their predecessors, but they were

7623-650: The sine function, although he did not develop the notion of a derivative. Some ideas on calculus later appeared in Indian mathematics, at the Kerala school of astronomy and mathematics . Madhava of Sangamagrama in the 14th century, and later mathematicians of the Kerala school, stated components of calculus such as the Taylor series and infinite series approximations. However, they did not combine many differing ideas under

7722-577: The subject has been prominent during the 19th century. Frullani integrals , David Bierens de Haan 's work on the theory and his elaborate tables, Lejeune Dirichlet 's lectures embodied in Meyer 's treatise, 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

7821-503: The subject. His contributions began in 1733, and his Elementa Calculi Variationum gave to the science its name. Joseph Louis Lagrange contributed extensively to the theory, and Adrien-Marie Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. To this discrimination Brunacci (1810), Carl Friedrich Gauss (1829), Siméon Denis Poisson (1831), Mikhail Vasilievich Ostrogradsky (1834), and Carl Gustav Jakob Jacobi (1837) have been among

7920-522: The sums of the volumes and areas of infinitesimally thin cross-sections. He discovered Cavalieri's quadrature formula which gave the area under the curves x of higher degree. This had previously been computed in a similar way for the parabola by Archimedes in The Method , but this treatise is believed to have been lost in the 13th century, and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri's work

8019-422: The symbol Γ {\displaystyle \Gamma } , and it is now called the gamma function . Besides being analytic over positive reals R + {\displaystyle \mathbb {R} ^{+}} , Γ {\displaystyle \Gamma } also enjoys the uniquely defining property that log ⁡ Γ {\displaystyle \log \Gamma }

8118-419: The theory, the former in studying equations , the latter in his theory of complex numbers . Niels Henrik Abel seems to have been the first to consider in a general way the question as to what differential equations 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

8217-441: The time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi . The calculus of variations may be said to begin with a problem of Johann Bernoulli (1696). It immediately occupied the attention of Jakob Bernoulli but Leonhard Euler first elaborated

8316-481: The two offered convincing logical foundations for their calculus according to mathematician Carl B. Boyer , Newton came the closest, with his best attempt coming in Principia , where he described his idea of "prime and ultimate ratios" and came extraordinarily close to the limit , and his ratio of velocities corresponded to a single real number , which would not be fully defined until the late nineteenth century. On

8415-479: The two unifying themes of the derivative and the integral , show the connection between the two, and turn calculus into the powerful problem-solving tool we have today. The mathematical study of continuity was revived in the 14th century by the Oxford Calculators and French collaborators such as Nicole Oresme . They proved the "Merton mean speed theorem ": that a uniformly accelerated body travels

8514-454: The ultimate ratio by appealing to motion: For by the ultimate velocity is meant that, with which the body is moved, neither before it arrives at its last place, when the motion ceases nor after but at the very instant when it arrives... the ultimate ratio of evanescent quantities is to be understood, the ratio of quantities not before they vanish, not after, but with which they vanish Newton developed his fluxional calculus in an attempt to evade

8613-410: The universe as a Monadology , and his plans of creating a precise formal logic whereby, "a general method in which all truths of the reason would be reduced to a kind of calculation". In 1672, Leibniz met the mathematician Huygens who convinced Leibniz to dedicate significant time to the study of mathematics. By 1673 he had progressed to reading Pascal 's Traité des Sinus du Quarte Cercle and it

8712-727: The use of the infinitesimal by forming calculations based on ratios of changes. In the Methodus Fluxionum he defined the rate of generated change as a fluxion , which he represented by a dotted letter, and the quantity generated he defined as a fluent . For example, if x {\displaystyle {x}} and y {\displaystyle {y}} are fluents, then x ˙ {\displaystyle {\dot {x}}} and y ˙ {\displaystyle {\dot {y}}} are their respective fluxions. This revised calculus of ratios continued to be developed and

8811-511: The word came to mean a method of computation. In this sense, it was used in English at least as early as 1672, several years prior to the publications of Leibniz and Newton. In addition to the differential calculus and integral calculus, the term is also used widely for naming specific methods of calculation. Examples of this include propositional calculus in logic, the calculus of variations in mathematics, process calculus in computing, and

8910-484: Was "shortly explained rather than accurately demonstrated". In an effort to give calculus a more rigorous explication and framework, Newton compiled in 1671 the Methodus Fluxionum et Serierum Infinitarum . In this book, Newton's strict empiricism shaped and defined his fluxional calculus. He exploited instantaneous motion and infinitesimals informally. He used math as a methodological tool to explain

9009-459: Was acutely aware of the notational terms used and his earlier plans to form a precise logical symbolism became evident. Eventually, Leibniz denoted the infinitesimal increments of abscissas and ordinates dx and dy , and the summation of infinitely many infinitesimally thin rectangles as a long s (∫ ), which became the present integral symbol ∫ {\displaystyle \scriptstyle \int } . While Leibniz's notation

9108-424: Was during his largely autodidactic research that Leibniz said "a light turned on". Like Newton, Leibniz saw the tangent as a ratio but declared it as simply the ratio between ordinates and abscissas . He continued this reasoning to argue that the integral was in fact the sum of the ordinates for infinitesimal intervals in the abscissa; in effect, the sum of an infinite number of rectangles. From these definitions

9207-410: Was first known as the hyperbolic logarithm . After Euler exploited e = 2.71828..., and F was identified as the inverse function of the exponential function , it became the natural logarithm , satisfying d F d x   =   1 x . {\displaystyle {\frac {dF}{dx}}\ =\ {\frac {1}{x}}.} The first proof of Rolle's theorem

9306-502: Was given by Michel Rolle in 1691 using methods developed by the Dutch mathematician Johann van Waveren Hudde . The mean value theorem in its modern form was stated by Bernard Bolzano and Augustin-Louis Cauchy (1789–1857) also after the founding of modern calculus. Important contributions were also made by Barrow , Huygens , and many others. Before Newton and Leibniz , the word "calculus" referred to any body of mathematics, but in

9405-479: Was independently invented in China by Liu Hui in the 4th century AD in order to find the area of a circle. In the 5th century, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere . In the Middle East, Hasan Ibn al-Haytham , Latinized as Alhazen ( c.  965  – c.  1040  AD) derived a formula for the sum of fourth powers . He used

9504-402: Was maturely stated in the 1676 text De Quadratura Curvarum where Newton came to define the present day derivative as the ultimate ratio of change, which he defined as the ratio between evanescent increments (the ratio of fluxions) purely at the moment in question. Essentially, the ultimate ratio is the ratio as the increments vanish into nothingness. Importantly, Newton explained the existence of

9603-465: Was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. Torricelli extended Cavalieri's work to other curves such as the cycloid , and then the formula was generalized to fractional and negative powers by Wallis in 1656. In a 1659 treatise, Fermat is credited with an ingenious trick for evaluating the integral of any power function directly. Fermat also obtained

9702-415: Was recorded in the unpublished De Analysi per Aequationes Numero Terminorum Infinitas . In this paper, Newton determined the area under a curve by first calculating a momentary rate of change and then extrapolating the total area. He began by reasoning about an indefinitely small triangle whose area is a function of x and y . He then reasoned that the infinitesimal increase in the abscissa will create

9801-439: Was the original unit of area in the metric system , with: Though the are has fallen out of use, the hectare is still commonly used to measure land: Other uncommon metric units of area include the tetrad , the hectad , and the myriad . The acre is also commonly used to measure land areas, where An acre is approximately 40% of a hectare. On the atomic scale, area is measured in units of barns , such that: The barn

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