1.
Non-standard analysis
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The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals, Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson and he wrote, the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the stages of the Differential and Integral Calculus. Robinson continued, However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort, as a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits. The key to our method is provided by the analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory. In 1973, intuitionist Arend Heyting praised non-standard analysis as a model of important mathematical research. A non-zero element of an ordered field F is infinitesimal if and only if its value is smaller than any element of F of the form 1 n, for n. Ordered fields that have infinitesimal elements are also called non-Archimedean, more generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. A field which satisfies the principle for real numbers is a hyperreal field. Robinsons original approach was based on these models of the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966 and is still in print, on page 88, Robinson writes, The existence of non-standard models of arithmetic was discovered by Thoralf Skolem. Skolems method foreshadows the ultrapower construction Several technical issues must be addressed to develop a calculus of infinitesimals, for example, it is not enough to construct an ordered field with infinitesimals. See the article on numbers for a discussion of some of the relevant ideas. In this section we outline one of the simplest approaches to defining a hyperreal field ∗ R, let R be the field of real numbers, and let N be the semiring of natural numbers. Denote by R N the set of sequences of real numbers, a field ∗ R is defined as a suitable quotient of R N, as follows. Take a nonprincipal ultrafilter F ⊂ P, in particular, F contains the Fréchet filter. There are at least three reasons to consider non-standard analysis, historical, pedagogical, and technical, much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity

2.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

3.
Bijection
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In mathematical terms, a bijective function f, X → Y is a one-to-one and onto mapping of a set X to a set Y. A bijection from the set X to the set Y has a function from Y to X. If X and Y are finite sets, then the existence of a means they have the same number of elements. For infinite sets the picture is complicated, leading to the concept of cardinal number. A bijective function from a set to itself is called a permutation. Bijective functions are essential to many areas of including the definitions of isomorphism, homeomorphism, diffeomorphism, permutation group. Satisfying properties and means that a bijection is a function with domain X and it is more common to see properties and written as a single statement, Every element of X is paired with exactly one element of Y. Functions which satisfy property are said to be onto Y and are called surjections, Functions which satisfy property are said to be one-to-one functions and are called injections. With this terminology, a bijection is a function which is both a surjection and an injection, or using words, a bijection is a function which is both one-to-one and onto. Consider the batting line-up of a baseball or cricket team, the set X will be the players on the team and the set Y will be the positions in the batting order The pairing is given by which player is in what position in this order. Property is satisfied since each player is somewhere in the list, property is satisfied since no player bats in two positions in the order. Property says that for each position in the order, there is some player batting in that position, in a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks them all to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. The instructor was able to conclude there were just as many seats as there were students. For any set X, the identity function 1X, X → X, the function f, R → R, f = 2x +1 is bijective, since for each y there is a unique x = /2 such that f = y. In more generality, any linear function over the reals, f, R → R, f = ax + b is a bijection, each real number y is obtained from the real number x = /a. The function f, R →, given by f = arctan is bijective since each real x is paired with exactly one angle y in the interval so that tan = x

4.
Integration (mathematics)
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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two operations of calculus, with its inverse, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total, roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative. In this case, it is called an integral and is written. The integrals discussed in this article are those termed definite integrals, a rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space and this method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was developed in China around the 3rd century AD by Liu Hui. This method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi. The next significant advances in integral calculus did not begin to appear until the 17th century, further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the theorem of calculus. Wallis generalized Cavalieris method, computing integrals of x to a power, including negative powers. The major advance in integration came in the 17th century with the independent discovery of the theorem of calculus by Newton. The theorem demonstrates a connection between integration and differentiation and this connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the mathematical framework that both Newton and Leibniz developed

5.
Unit circle
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In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1, the generalization to higher dimensions is the unit sphere, if is a point on the unit circles circumference, then | x | and | y | are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 =1. The interior of the circle is called the open unit disk. One may also use other notions of distance to define other unit circles, such as the Riemannian circle, see the article on mathematical norms for additional examples. The unit circle can be considered as the complex numbers. In quantum mechanics, this is referred to as phase factor, the equation x2 + y2 =1 gives the relation cos 2 + sin 2 =1. The unit circle also demonstrates that sine and cosine are periodic functions, triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P on the circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q and line segments PQ ⊥ OQ, the result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin = y1 and cos = x1. Having established these equivalences, take another radius OR from the origin to a point R on the circle such that the same angle t is formed with the arm of the x-axis. Now consider a point S and line segments RS ⊥ OS, the result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at in the way that P is at. The conclusion is that, since is the same as and is the same as, it is true that sin = sin and it may be inferred in a similar manner that tan = −tan, since tan = y1/x1 and tan = y1/−x1. A simple demonstration of the above can be seen in the equality sin = sin = 1/√2, when working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π

6.
Equivalence class
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In mathematics, when the elements of some set S have a notion of equivalence defined on them, then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the equivalence class if. Formally, given a set S and an equivalence relation ~ on S and it may be proven from the defining properties of equivalence relations that the equivalence classes form a partition of S. This partition – the set of equivalence classes – is sometimes called the quotient set or the quotient space of S by ~ and is denoted by S / ~. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories. If X is the set of all cars, and ~ is the relation has the same color as. X/~ could be identified with the set of all car colors. Let X be the set of all rectangles in a plane, for each positive real number A there will be an equivalence class of all the rectangles that have area A. Consider the modulo 2 equivalence relation on the set Z of integers, x ~ y if and this relation gives rise to exactly two equivalence classes, one class consisting of all even numbers, and the other consisting of all odd numbers. Under this relation, and all represent the element of Z/~. Let X be the set of ordered pairs of integers with b not zero, the same construction can be generalized to the field of fractions of any integral domain. In this situation, each equivalence class determines a point at infinity, the equivalence class of an element a is denoted and is defined as the set = of elements that are related to a by ~. An alternative notation R can be used to denote the class of the element a. This is said to be the R-equivalence class of a, the set of all equivalence classes in X with respect to an equivalence relation R is denoted as X/R and called X modulo R. The surjective map x ↦ from X onto X/R, which each element to its equivalence class, is called the canonical surjection or the canonical projection map. When an element is chosen in each class, this defines an injective map called a section. If this section is denoted by s, one has = c for every equivalence class c, the element s is called a representative of c. Any element of a class may be chosen as a representative of the class, sometimes, there is a section that is more natural than the other ones

7.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker

8.
Luigi Ambrosio
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Luigi Ambrosio is a professor at Scuola Normale Superiore in Pisa, Italy. His main fields of research are the calculus of variations and geometric measure theory, Ambrosio entered the Scuola Normale Superiore di Pisa in 1981. He obtained his degree under the guidance of Ennio de Giorgi in 1985 at University of Pisa, and he obtained his PhD in 1988. He is currently professor at the Scuola Normale, having taught previously at the University of Rome Tor Vergata, the University of Pisa, and the University of Pavia. Ambrosio also taught and conducted research at the Massachusetts Institute of Technology, the ETH in Zurich, and he is the Managing Editor of the scientific journal Calculus of Variations and Partial Differential Equations, and member of the editorial boards of scientific journals. In 1998 Ambrosio won the Caccioppoli Prize of the Italian Mathematical Union, in 2002 he was plenary speaker at the International Congress of Mathematicians in Beijing and in 2003 he has been awarded with the Fermat Prize. From 2005 he is a member of Accademia Nazionale dei Lincei. Ambrosio is listed as an ISI highly cited researcher, a compactness theorem for a new class of functions of bounded variation. New functionals in the calculus of variations, existence theory for a new class of variational problems. Ambrosio, Luigi, Fusco, Nicola, Pallara, Diego, functions of bounded variation and free discontinuity problems. The Clarendon Press, Oxford University Press, New York, currents in metric spaces, Acta Math. Ambrosio, Luigi, Gigli, Nicola, Savaré, Giuseppe, gradient flows in metric spaces and in the space of probability measures. Luigi Ambrosio, Edward Norman Dancer, Giuseppe Buttazzo, A. Marino, M. K. Venkatesha Murthy, Calculus of variations and partial differential equations. CS1 maint, Uses editors parameter Site of Caccioppoli Prize Luigi Ambrosio at the Mathematics Genealogy Project

9.
Leibniz's notation
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The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a tool of calculus. For example, the derivative of the position of an object with respect to time is the objects velocity. The derivative of a function of a variable at a chosen input value. The tangent line is the best linear approximation of the function near that input value, for this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives may be generalized to functions of real variables. In this generalization, the derivative is reinterpreted as a transformation whose graph is the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables and it can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation, the reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration, differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is the action of computing a derivative, the derivative of a function y = f of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x, If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. The simplest case, apart from the case of a constant function, is when y is a linear function of x. This formula is true because y + Δ y = f = m + b = m x + m Δ x + b = y + m Δ x. Thus, since y + Δ y = y + m Δ x and this gives an exact value for the slope of a line. If the function f is not linear, however, then the change in y divided by the change in x varies, differentiation is a method to find an exact value for this rate of change at any given value of x. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the value of the ratio of the differences Δy / Δx as Δx becomes infinitely small

10.
Integral symbol
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The integral symbol, ∫, ∫ is used to denote integrals and antiderivatives in mathematics. The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz towards the end of the 17th century, the symbol was based on the ſ character, and was chosen because Leibniz thought of the integral as an infinite sum of infinitesimal summands. The integral symbol is U+222B ∫ INTEGRAL in Unicode and \int in LaTeX, in HTML, it is written as ∫, ∫, and &int. The original IBM PC code page 437 character set included a couple of characters ⌠ and ⌡ to build the integral symbol and these were deprecated in subsequent MS-DOS code pages, but they still remain in Unicode for compatibility. The ∫ symbol is similar to, but not to be confused with. Related symbols include, In other languages, the shape of the integral symbol differs slightly from the commonly seen in English-language textbooks. While the English integral symbol leans to the right, the German symbol is upright, another difference is in the placement of limits for definite integrals. Generally, in English-language books, limits go to the right of the integral symbol, ∫0 T f d t. By contrast, in German and Russian texts, limits for definite integrals are placed above and below the symbol, and, as a result. Capital sigma notation Capital pi notation Stewart, James, zaitcev, V. Janishewsky, A. Berdnikov, A. Russian Typographical Traditions in Mathematical Literature, Russian Typographical Traditions in Mathematical Literature, EuroTeX99 Proceedings Fileformat. info

11.
The Method of Mechanical Theorems
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The Method of Mechanical Theorems, also referred to as The Method, is considered one of the major surviving works of the ancient Greek polymath Archimedes. The Method takes the form of a letter from Archimedes to Eratosthenes, the librarian at the Library of Alexandria. The work was thought to be lost, but in 1906 was rediscovered in the celebrated Archimedes Palimpsest. Archimedes did not admit the method of indivisibles as part of rigorous mathematics, in these treatises, he proves the same theorems by exhaustion, finding rigorous upper and lower bounds which both converge to the answer required. Nevertheless, the method was what he used to discover the relations for which he later gave rigorous proofs. To explain Archimedes method today, it is convenient to use of a little bit of Cartesian geometry. His idea is to use the law of the lever to determine the areas of figures from the center of mass of other figures. The simplest example in modern language is the area of the parabola. Archimedes uses an elegant method, but in Cartesian language, his method is calculating the integral ∫01 x 2 d x =13. The idea is to balance the parabola with a certain triangle that is made of the same material. The parabola is the region in the x-y plane between the x-axis and y = x2 as x varies from 0 to 1, the triangle is the region in the x-y plane between the x-axis and the line y = x, also as x varies from 0 to 1. Slice the parabola and triangle into vertical slices, one for each value of x, imagine that the x-axis is a lever, with a fulcrum at x =0. The law of the states that two objects on opposite sides of the fulcrum will balance if each has the same torque. Since each pair of balances, moving the whole parabola to x = −1 would balance the whole triangle. This means that if the original uncut parabola is hung by a hook from the point x = −1, the center of mass of a triangle can be easily found by the following method, also due to Archimedes. If a median line is drawn from any one of the vertices of a triangle to the opposite edge E, the triangle will balance on the median, considered as a fulcrum. The reason is if the triangle is divided into infinitesimal line segments parallel to E, each segment has equal length on opposite sides of the median. This argument can be made rigorous by exhaustion by using little rectangles instead of infinitesimal lines

12.
Cavalieri's principle
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If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. 3-dimensional case, Suppose two regions in three-space are included two parallel planes. If every plane parallel to two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes. In the other direction, Cavalieris principle grew out of the ancient Greek method of exhaustion, Cavalieris principle was originally called the method of indivisibles, the name it was known by in Renaissance Europe. Archimedes was able to find the volume of a sphere given the volumes of a cone, in the 5th century AD, Zu Chongzhi and his son Zu Gengzhi established a similar method to find a spheres volume. The transition from Cavalieris indivisibles to Evangelista Torricellis and John Walliss infinitesimals was an advance in the history of the calculus. The indivisibles were entities of codimension 1, so that a figure was thought as made out of an infinity of 1-dimensional lines. Meanwhile, infinitesimals were entities of the dimension as the figure they make up, thus. Applying the formula for the sum of a progression, Wallis computed the area of a triangle by partitioning it into infinitesimal parallelograms of width 1/∞. If one knows that the volume of a cone is 13, then one can use Cavalieris principle to derive the fact that the volume of a sphere is 43 π r 3, where r is the radius. That is done as follows, Consider a sphere of radius r, within the cylinder is the cone whose apex is at the center of the sphere and whose base is the base of the cylinder. By the Pythagorean theorem, the plane located y units above the equator intersects the sphere in a circle of area π, the area of the planes intersection with the part of the cylinder that is outside of the cone is also π. The aforementioned volume of the cone is 13 of the volume of the cylinder, Therefore the volume of the upper half of the sphere is 23 of the volume of the cylinder. The volume of the cylinder is base × height = π r 2 ⋅ r = π r 3 Therefore the volume of the upper half-sphere is π r 3 and that of the whole sphere is π r 3. One may initially establish it in a case by partitioning the interior of a triangular prism into three pyramidal components of equal volumes. One may show the equality of three volumes by means of Cavalieris principle. The ancient Greeks used various techniques such as Archimedess mechanical arguments or method of exhaustion to compute these volumes. The cross-section of the ring is a plane annulus, whose area is the difference between the areas of two circles

13.
Infinitesimal strain theory
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With this assumption, the equations of continuum mechanics are considerably simplified. This approach may also be called small deformation theory, small displacement theory and it is contrasted with the finite strain theory where the opposite assumption is made. In such a linearization, the non-linear or second-order terms of the strain tensor are neglected. Therefore, the displacement gradient components and the spatial displacement gradient components are approximately equal. From the geometry of Figure 1 we have a b ¯ =2 +2 = d x 1 +2 ∂ u x ∂ x +2 +2 For very small displacement gradients, i. e. e. Therefore, the elements of the infinitesimal strain tensor are the normal strains in the coordinate directions. The results of operations are called strain invariants. Since there are no shear strain components in this coordinate system, an octahedral plane is one whose normal makes equal angles with the three principal directions. The engineering shear strain on a plane is called the octahedral shear strain and is given by γ o c t =232 +2 +2 where ε1, ε2, ε3 are the principal strains. Several definitions of equivalent strain can be found in the literature, thus, a solution does not generally exist for an arbitrary choice of strain components. Therefore, some restrictions, named compatibility equations, are imposed upon the strain components, with the addition of the three compatibility equations the number of independent equations are reduced to three, matching the number of unknown displacement components. These constraints on the strain tensor were discovered by Saint-Venant, and are called the Saint Venant compatibility equations, the compatibility functions serve to assure a single-valued continuous displacement function u i. The strains associated with length, i. e. the normal strain ε33, plane strain is then an acceptable approximation. The strain tensor for plane strain is written as, ε _ _ = in which the double underline indicates a second order tensor and this strain state is called plane strain. The corresponding stress tensor is, σ _ _ = in which the non-zero σ33 is needed to maintain the constraint ϵ33 =0. This stress term can be removed from the analysis to leave only the in-plane terms. Antiplane strain is another state of strain that can occur in a body. For infinitesimal deformations the scalar components of ω satisfy the condition | ω i j | ≪1, note that the displacement gradient is small only if both the strain tensor and the rotation tensor are infinitesimal

14.
Surreal number
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The surreals share many properties with the reals, including the usual arithmetic operations, as such, they form an ordered field. The surreals also contain all transfinite ordinal numbers, the arithmetic on them is given by the natural operations, research on the go endgame by John Horton Conway led to another definition and construction of the surreal numbers. Conways construction was introduced in Donald Knuths 1974 book Surreal Numbers, How Two Ex-Students Turned on to Pure Mathematics, in his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers. Conway later adopted Knuths term, and used surreals for analyzing games in his 1976 book On Numbers and Games. In the Conway construction, the numbers are constructed in stages. Different subsets may end up defining the same number, and may define the number even if L ≠ L′. So strictly speaking, the numbers are equivalence classes of representations of form that designate the same number. In the first stage of construction, there are no previously existing numbers so the representation must use the empty set. This representation, where L and R are both empty, is called 0, subsequent stages yield forms like, =1 =2 =3 and = −1 = −2 = −3 The integers are thus contained within the surreal numbers. Similarly, representations arise like, = 1/2 = 1/4 = 3/4 so that the rationals are contained within the surreal numbers. Thus the real numbers are also embedded within the surreals, but there are also representations like = ω = ε where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. The construction consists of three interdependent parts, the rule, the comparison rule and the equivalence rule. A form is a pair of sets of numbers, called its left set. A form with left set L and right set R is written, when L and R are given as lists of elements, the braces around them are omitted. Either or both of the left and right set of a form may be the empty set, the form with both left and right set empty is also written. The numeric forms are placed in classes, each such equivalence class is a surreal number. The elements of the left and right set of a form are drawn from the universe of the surreal numbers, equivalence Rule Two numeric forms x and y are forms of the same number if and only if both x ≤ y and y ≤ x. An ordering relationship must be antisymmetric, i. e. it must have the property that x = y only when x and y are the same object and this is not the case for surreal number forms, but is true by construction for surreal numbers

15.
Standard part function
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In non-standard analysis, the standard part function is a function from the limited hyperreal numbers to the real numbers. Briefly, the standard part function rounds off a finite hyperreal to the nearest real and it associates to every such hyperreal x, the unique real x 0 infinitely close to it, i. e. x − x 0 is infinitesimal. As such, it is an implementation of the historical concept of adequality introduced by Pierre de Fermat. The standard part function was first defined by Abraham Robinson who used the notation ∘ x for the part of a hyperreal x. This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivative, the latter theory is a rigorous formalisation of calculations with infinitesimals. The standard part of x is sometimes referred to as its shadow, nonstandard analysis deals primarily with the pair R ⊂ ∗ R, where the hyperreals ∗ R are an ordered field extension of the reals R, and contain infinitesimals, in addition to the reals. In the hyperreal line every real number has a collection of numbers of hyperreals infinitely close to it, the standard part function associates to a finite hyperreal x, the unique standard real number x0 which is infinitely close to it. The relationship is expressed symbolically by writing s t = x 0, the standard part of any infinitesimal is 0. Thus if N is an infinite hypernatural, then 1/N is infinitesimal, if a hyperreal u is represented by a Cauchy sequence ⟨ u n, n ∈ N ⟩ in the ultrapower construction, then st = lim n → ∞ u n. More generally, each finite u ∈ ∗ R defines a Dedekind cut on the subset R ⊂ ∗ R, the standard part function st is not defined by an internal set. There are several ways of explaining this, perhaps the simplest is that its domain L, which is the collection of limited hyperreals, is not an internal set. Namely, since L is bounded, L would have to have a least upper bound if L were internal, but L doesnt have a least upper bound. Alternatively, the range of st is R ⊂ ∗ R which is not internal, in every internal set in ∗ R which is a subset of R is necessarily finite. All the traditional notions of calculus are expressed in terms of the standard part function, the standard part function is used to define the derivative of a function f. If f is a function, and h is infinitesimal. Alternatively, if y = f, one takes an infinitesimal increment Δ x, one forms the ratio Δ y Δ x. The derivative is defined as the standard part of the ratio. Given a sequence, its limit is defined by lim n → ∞ u n = st where H ∈ ∗ N ∖ N is an infinite index, here the limit is said to exist if the standard part is the same regardless of the infinite index chosen

16.
Gottfried Wilhelm Leibniz
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Leibnizs notation has been widely used ever since it was published. It was only in the 20th century that his Law of Continuity and he became one of the most prolific inventors in the field of mechanical calculators. He also refined the number system, which is the foundation of virtually all digital computers. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th-century advocates of rationalism and he wrote works on philosophy, politics, law, ethics, theology, history, and philology. Leibnizs contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters and he wrote in several languages, but primarily in Latin, French, and German. There is no complete gathering of the writings of Leibniz in English, Gottfried Leibniz was born on July 1,1646, toward the end of the Thirty Years War, in Leipzig, Saxony, to Friedrich Leibniz and Catharina Schmuck. Friedrich noted in his journal,21. Juny am Sontag 1646 Ist mein Sohn Gottfried Wilhelm, post sextam vespertinam 1/4 uff 7 uhr abents zur welt gebohren, in English, On Sunday 21 June 1646, my son Gottfried Wilhelm is born into the world a quarter after six in the evening, in Aquarius. Leibniz was baptized on July 3 of that year at St. Nicholas Church, Leipzig and his father died when he was six and a half years old, and from that point on he was raised by his mother. Her teachings influenced Leibnizs philosophical thoughts in his later life, Leibnizs father had been a Professor of Moral Philosophy at the University of Leipzig, and the boy later inherited his fathers personal library. He was given access to it from the age of seven. Access to his fathers library, largely written in Latin, also led to his proficiency in the Latin language and he also composed 300 hexameters of Latin verse, in a single morning, for a special event at school at the age of 13. In April 1661 he enrolled in his fathers former university at age 15 and he defended his Disputatio Metaphysica de Principio Individui, which addressed the principle of individuation, on June 9,1663. Leibniz earned his masters degree in Philosophy on February 7,1664, after one year of legal studies, he was awarded his bachelors degree in Law on September 28,1665. His dissertation was titled De conditionibus, in early 1666, at age 19, Leibniz wrote his first book, De Arte Combinatoria, the first part of which was also his habilitation thesis in Philosophy, which he defended in March 1666. His next goal was to earn his license and Doctorate in Law, in 1666, the University of Leipzig turned down Leibnizs doctoral application and refused to grant him a Doctorate in Law, most likely due to his relative youth. Leibniz then enrolled in the University of Altdorf and quickly submitted a thesis, the title of his thesis was Disputatio Inauguralis de Casibus Perplexis in Jure. Leibniz earned his license to practice law and his Doctorate in Law in November 1666 and he next declined the offer of an academic appointment at Altdorf, saying that my thoughts were turned in an entirely different direction

17.
Abraham Robinson
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Nearly half of Robinsons papers were in applied mathematics rather than in pure mathematics. He was born to a Jewish family with strong Zionist beliefs, in Waldenburg, Germany, in 1933, he emigrated to British Mandate of Palestine, where he earned a first degree from the Hebrew University. While in London, he joined the Free French Air Force and contributed to the war effort by teaching himself aerodynamics, after the war, Robinson worked in London, Toronto, and Jerusalem, but ended up at University of California, Los Angeles in 1962. He became known for his approach of using the methods of logic to attack problems in analysis. He introduced many of the notions of model theory. Using these methods, he found a way of using formal logic to show there are self-consistent nonstandard models of the real number system that include infinite. Robinsons book Non-standard Analysis was published in 1966, while at UCLA his colleagues remember him as working hard to accommodate PhD students of all levels of ability by finding them projects of the appropriate difficulty. He was courted by Yale, and after initial reluctance. He died of cancer in 1974. Robinson, Abraham, Introduction to model theory and to the metamathematics of algebra, Amsterdam, North-Holland, ISBN 978-0-7204-2222-1, MR0153570 Robinson, Abraham, Keisler, H. Jerome, ed. Complete theories, Studies in Logic and the Foundations of Mathematics, Amsterdam, North-Holland, ISBN 978-0-7204-0690-0, MR0472504 Robinson, Abraham, Keisler, H. Jerome, ed. I Model theory and algebra, Yale University Press, ISBN 978-0-300-02071-7, MR533887 Robinson, Abraham, Luxemburg, W. A. J. Körner, S. eds. Vol. II Nonstandard analysis and philosophy, Yale University Press, ISBN 978-0-300-02072-4, MR533888 Robinson, Abraham, Young, A. D. ed. Vol. W. Dauben, Abraham Robinson, The Creation of Nonstandard Analysis, A Personal and Mathematical Odyssey, Princeton, NJ, Princeton University Press,1998 OConnor, John J. Robertson, Abraham Robinson, MacTutor History of Mathematics archive, University of St Andrews. Abraham Robinson at the Mathematics Genealogy Project Abraham Robinson — Biographical Memoirs of the National Academy of Sciences

18.
Pierre de Fermat
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He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermats principle for light propagation and his Fermats Last Theorem in number theory, Fermat was born in the first decade of the 17th century in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Dominique Fermat, was a leather merchant. Pierre had one brother and two sisters and was almost certainly brought up in the town of his birth, there is little evidence concerning his school education, but it was probably at the Collège de Navarre in Montauban. He attended the University of Orléans from 1623 and received a bachelor in law in 1626. In Bordeaux he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apolloniuss De Locis Planis to one of the mathematicians there, there he became much influenced by the work of François Viète. In 1630, he bought the office of a councillor at the Parlement de Toulouse, one of the High Courts of Judicature in France and he held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat, fluent in six languages, Fermat was praised for his written verse in several languages and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends, often little or no proof of his theorems. In some of these letters to his friends he explored many of the ideas of calculus before Newton or Leibniz. Fermat was a trained lawyer making mathematics more of a hobby than a profession, nevertheless, he made important contributions to analytical geometry, probability, number theory and calculus. Secrecy was common in European mathematical circles at the time and this naturally led to priority disputes with contemporaries such as Descartes and Wallis. Anders Hald writes that, The basis of Fermats mathematics was the classical Greek treatises combined with Vietas new algebraic methods, Fermats pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes famous La géométrie. This manuscript was published posthumously in 1679 in Varia opera mathematica, in these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature. Fermat was the first person known to have evaluated the integral of power functions. With his method, he was able to reduce this evaluation to the sum of geometric series, the resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus. In number theory, Fermat studied Pells equation, perfect numbers, amicable numbers and it was while researching perfect numbers that he discovered Fermats little theorem. Fermat developed the two-square theorem, and the polygonal number theorem, although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived

19.
Augustin-Louis Cauchy
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Baron Augustin-Louis Cauchy FRS FRSE was a French mathematician who made pioneering contributions to analysis. He was one of the first to state and prove theorems of calculus rigorously and he almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had an influence over his contemporaries. His writings range widely in mathematics and mathematical physics, more concepts and theorems have been named for Cauchy than for any other mathematician. Cauchy was a writer, he wrote approximately eight hundred research articles. Cauchy was the son of Louis François Cauchy and Marie-Madeleine Desestre, Cauchy married Aloise de Bure in 1818. She was a relative of the publisher who published most of Cauchys works. By her he had two daughters, Marie Françoise Alicia and Marie Mathilde, Cauchys father was a high official in the Parisian Police of the New Régime. He lost his position because of the French Revolution that broke out one month before Augustin-Louis was born, the Cauchy family survived the revolution and the following Reign of Terror by escaping to Arcueil, where Cauchy received his first education, from his father. After the execution of Robespierre, it was safe for the family to return to Paris, there Louis-François Cauchy found himself a new bureaucratic job, and quickly moved up the ranks. When Napoleon Bonaparte came to power, Louis-François Cauchy was further promoted, the famous mathematician Lagrange was also a friend of the Cauchy family. On Lagranges advice, Augustin-Louis was enrolled in the École Centrale du Panthéon, most of the curriculum consisted of classical languages, the young and ambitious Cauchy, being a brilliant student, won many prizes in Latin and Humanities. In spite of successes, Augustin-Louis chose an engineering career. In 1805 he placed second out of 293 applicants on this exam, one of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young, nevertheless, he finished the Polytechnique in 1807, at the age of 18, and went on to the École des Ponts et Chaussées. He graduated in engineering, with the highest honors. After finishing school in 1810, Cauchy accepted a job as an engineer in Cherbourg. Cauchys first two manuscripts were accepted, the one was rejected

20.
Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler