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# Integral

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. [1]

## Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

Adaptive quadrature is a numerical integration method in which the integral of a function f(x) is approximated using static quadrature rules on adaptively refined subintervals of the integration domain.

Addison-Wesley is a publisher of textbooks and computer literature.

## Alfréd Haar

Alfréd Haar (Haar Alfréd; 11 October 1885, Budapest &ndash; 16 March 1933, Szeged) was a Hungarian mathematician.

## American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

## Ancient Greek

The Ancient Greek language includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD.

## Antiderivative

In calculus, an antiderivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function.

## Arc length

Determining the length of an irregular arc segment is also called rectification of a curve.

## Archimedes

Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.

## Area

Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.

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## Area of a circle

In geometry, the area enclosed by a circle of radius is.

## Area under the curve (pharmacokinetics)

In the field of pharmacokinetics, the area under the curve (AUC) is the definite integral in a plot of drug concentration in blood plasma vs.

## Arnaud Denjoy

Arnaud Denjoy (1884–1974) was a French mathematician.

## Bernhard Riemann

Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.

## Bonaventura Cavalieri

Bonaventura Francesco Cavalieri (Cavalerius; 1598 &ndash; 30 November 1647) was an Italian mathematician and a Jesuate.

## Bounded function

In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded.

## Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size.

## Bounded variation

In mathematical analysis, a function of bounded variation, also known as function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense.

## Bulletin of the American Mathematical Society

The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.

## Calculus

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

## Cartesian product

In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.

## Cauchy–Schwarz inequality

In mathematics, the Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality, is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, vector algebra and other areas.

## Cavalieri's principle

In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows.

## Cavalieri's quadrature formula

In calculus, Cavalieri's quadrature formula, named for 17th-century Italian mathematician Bonaventura Cavalieri, is the integral and generalizations thereof.

## Chebyshev polynomials

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively.

## Classical physics

Classical physics refers to theories of physics that predate modern, more complete, or more widely applicable theories.

Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the integrand in terms of Chebyshev polynomials.

## Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

## Compass-and-straightedge construction

Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.

## Complete metric space

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).

## Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

## Complex-valued function

In mathematics, a complex-valued function (not to be confused with complex variable function) is a function whose values are complex numbers.

## Computer algebra system

A computer algebra system (CAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists.

## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

## Contour integration

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.

## Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers.

## Cross product

In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space \left(\mathbb^3\right) and is denoted by the symbol \times.

## Curl (mathematics)

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space.

## Curve

In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.

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## Curvilinear coordinates

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved.

## Daniell integral

In mathematics, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the Riemann integral to which students are typically first introduced.

## Darboux integral

In real analysis, a branch of mathematics, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function.

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

## Differential (infinitesimal)

The term differential is used in calculus to refer to an infinitesimal (infinitely small) change in some varying quantity.

## Differential calculus

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change.

## Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.

## Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.

## Disc integration

Disc integration, also known in integral calculus as the disc method, is a means of calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution.

## Displacement (fluid)

In fluid mechanics, displacement occurs when an object is immersed in a fluid, pushing it out of the way and taking its place.

## Displacement (vector)

A displacement is a vector whose length is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a straight line from the initial position to the final position of the point.

## Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.

## Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

## Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

## Electric field

An electric field is a vector field surrounding an electric charge that exerts force on other charges, attracting or repelling them.

## Electromagnetism

Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles.

## Element (mathematics)

In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.

## Elementary function

In mathematics, an elementary function is a function of one variable which is the composition of a finite number of arithmetic operations, exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of ''n''th roots).

## Eudoxus of Cnidus

Eudoxus of Cnidus (Εὔδοξος ὁ Κνίδιος, Eúdoxos ho Knídios) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato.

## Euler substitution

Euler Substitution is a method for evaluating integrals of the form: where R is a rational function of x and \sqrt.

## Evangelista Torricelli

Evangelista Torricelli; 15 October 1608 – 25 October 1647) was an Italian physicist and mathematician, best known for his invention of the barometer, but is also known for his advances in optics and work on the method of indivisibles.

## Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

## Exterior algebra

In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.

## Exterior derivative

On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.

## Flux

Flux describes the quantity which passes through a surface or substance.

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## Force

In physics, a force is any interaction that, when unopposed, will change the motion of an object.

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## Fourier analysis

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

## Fractional Brownian motion

In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion.

## Fubini's theorem

In mathematical analysis Fubini's theorem, introduced by, is a result that gives conditions under which it is possible to compute a double integral using iterated integrals.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Fundamental theorem of calculus

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function.

## Gamma function

In mathematics, the gamma function (represented by, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.

## Gauss–Kronrod quadrature formula

The Gauss–Kronrod quadrature formula is an adaptive method for numerical integration.

## Gaussian integral

The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.

In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration.

## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

## George Berkeley

George Berkeley (12 March 168514 January 1753) — known as Bishop Berkeley (Bishop of Cloyne) — was an Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism" (later referred to as "subjective idealism" by others).

## Gottfried Wilhelm Leibniz

Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.

In mathematics, the gradient is a multi-variable generalization of the derivative.

## Graduate Studies in Mathematics

Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS).

## Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

## Gravitational field

In physics, a gravitational field is a model used to explain the influence that a massive body extends into the space around itself, producing a force on another massive body.

## Green's theorem

In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is named after George Green, though its first proof is due to Bernhard Riemann and is the two-dimensional special case of the more general Kelvin–Stokes theorem.

## Haar measure

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

## HathiTrust

HathiTrust is a large-scale collaborative repository of digital content from research libraries including content digitized via the Google Books project and Internet Archive digitization initiatives, as well as content digitized locally by libraries.

## Hölder's inequality

In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of ''Lp'' spaces.

## Henri Lebesgue

Henri Léon Lebesgue (June 28, 1875 – July 26, 1941) was a French mathematician most famous for his theory of integration, which was a generalization of the 17th century concept of integration—summing the area between an axis and the curve of a function defined for that axis.

## Henstock–Kurzweil integral

In mathematics, the Henstock–Kurzweil integral or gauge integral (also known as the (narrow) Denjoy integral (pronounced), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral) is one of a number of definitions of the integral of a function.

## Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

## Holonomic function

In mathematics, and more specifically in analysis, a holonomic function is a smooth function in several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of D-modules theory.

## Hypergeometric function

In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases.

## Hyperreal number

The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities.

## Incomplete gamma function

In mathematics, the upper incomplete gamma function and lower incomplete gamma function are types of special functions, which arise as solutions to various mathematical problems such as certain integrals.

## Infinitesimal

In mathematics, infinitesimals are things so small that there is no way to measure them.

## Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

## Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign.

## Integral of inverse functions

In mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse f^ of a continuous and invertible function f, in terms of f^ and an antiderivative of f. This formula was published in 1905 by Charles-Ange Laisant.

## Integral symbol

The integral symbol: is used to denote integrals and antiderivatives in mathematics.

## Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative.

## Integration by reduction formulae

Integration by reduction formula in integral calculus is a technique of integration, in the form of a recurrence relation.

## Integration by substitution

In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals.

## Integration using Euler's formula

In integral calculus, complex numbers and Euler's formula may be used to evaluate integrals involving trigonometric functions.

## Integration using parametric derivatives

In mathematics, integration by parametric derivatives is a method of integrating certain functions.

## Interpolation

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.

## Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

## Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains).

## Isaac Barrow

Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for the discovery of the fundamental theorem of calculus.

## Isaac Newton

Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, astronomer, theologian, author and physicist (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution.

## Itô calculus

Itô calculus, named after Kiyoshi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process).

## Jaroslav Kurzweil

Jaroslav Kurzweil (born 1926) is a Czech mathematician.

Johann Karl August Radon (16 December 1887 – 25 May 1956) was an Austrian mathematician.

## John Wallis

John Wallis (3 December 1616 – 8 November 1703) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus.

## Joseph Fourier

Jean-Baptiste Joseph Fourier (21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations.

## Kelvin–Stokes theorem

The Kelvin–Stokes theoremThis proof is based on the Lecture Notes given by Prof.

## Kinematics

Kinematics is a branch of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the mass of each or the forces that caused the motion.

## Lagrange polynomial

In numerical analysis, Lagrange polynomials are used for polynomial interpolation.

## Lebesgue integration

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis.

## Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.

## Lebesgue–Stieltjes integration

In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework.

## Legendre function

In mathematics, the Legendre functions Pλ, Qλ and associated Legendre functions P, Q are generalizations of Legendre polynomials to non-integer degree.

## Leibniz integral rule

In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form where -\infty, the derivative of this integral is expressible as where the partial derivative indicates that inside the integral, only the variation of f(x, t) with x is considered in taking the derivative.

## Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.

## Limits of integration

In calculus and mathematical analysis the limits of integration of the integral of a Riemann integrable function f defined on a closed and bounded are the real numbers a and b.

## Line integral

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.

## Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

## Linear differential equation

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where,..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable.

## Linear form

In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars.

## Lists of integrals

Integration is the basic operation in integral calculus.

## Liu Hui

Liu Hui was a Chinese mathematician who lived in the state of Cao Wei during the Three Kingdoms period (220–280) of China.

## Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

## Logarithm

In mathematics, the logarithm is the inverse function to exponentiation.

## Long s

The long, medial, or descending s (ſ) is an archaic form of the lower case letter s. It replaced a single s, or the first in a double s, at the beginning or in the middle of a word (e.g. "ſinfulneſs" for "sinfulness" and "ſucceſsful" for "successful").

## Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.

## Macsyma

Macsyma (Project MAC’s SYmbolic MAnipulator) is one of the oldest general purpose computer algebra systems which is still widely used.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Measurable function

In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open.

## Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.

## Meijer G-function

In mathematics, the G-function was introduced by as a very general function intended to include most of the known special functions as particular cases.

## Method of exhaustion

The method of exhaustion (methodus exhaustionibus, or méthode des anciens) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape.

## Minkowski inequality

In mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are normed vector spaces.

## Modern Arabic mathematical notation

Modern Arabic mathematical notation is a mathematical notation based on the Arabic script, used especially at pre-university levels of education.

## Monte Carlo integration

In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers.

## Multiple integral

The multiple integral is a definite integral of a function of more than one real variable, for example, or.

## Multivariable calculus

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables, rather than just one.

## Newton–Cotes formulas

In numerical analysis, the Newton–Cotes formulae, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulae for numerical integration (also called quadrature) based on evaluating the integrand at equally spaced points.

## Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935.

## Non-standard analysis

The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers.

## Nonelementary integral

In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative that is, itself, not an elementary function (i.e. a function constructed from a finite number of quotients of constant, algebraic, exponential, and logarithmic functions using field operations).

## Normal (geometry)

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.

## Nth root

In mathematics, an nth root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power n yields x: where n is the degree of the root.

## Order of integration (calculus)

In calculus, interchange of the order of integration is a methodology that transforms iterated integrals (or multiple integrals through the use of Fubini's theorem) of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed.

## Orthogonal polynomials

In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.

## Oskar Perron

Oskar Perron (7 May 1880 &ndash; 22 February 1975) was a German mathematician.

In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

## Parabola

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped.

## Parseval's identity

In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function.

## Partial fraction decomposition

In algebra, the partial fraction decomposition or partial fraction expansion of a rational function (that is, a fraction such that the numerator and the denominator are both polynomials) is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

## Paul Montel

Paul Antoine Aristide Montel (29 April 1876 – 22 January 1975) was a French mathematician.

## Pierre de Fermat

Pierre de Fermat (Between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.

## Planimeter

A planimeter, also known as a platometer, is a measuring instrument used to determine the area of an arbitrary two-dimensional shape.

## Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space.

## Pointwise

In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the pointwise operations &mdash; operations defined on functions by applying the operations to function values separately for each point in the domain of definition.

## Pointwise product

The pointwise product of two functions is another function, obtained by multiplying the image of the two functions at each value in the domain.

## Precision engineering

Precision engineering is a subdiscipline of electrical engineering, software engineering, electronics engineering, mechanical engineering, and optical engineering concerned with designing machines, fixtures, and other structures that have exceptionally high tolerances, are repeatable, and are stable over time.

## Probability density function

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function, whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.

## Probability theory

Probability theory is the branch of mathematics concerned with probability.

## Ralph Henstock

Ralph Henstock (2 June 1923 &ndash; 17 January 2007) was an English mathematician and author.

## Random variable

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon.

## Rational function

In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.

## Real analysis

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions.

## Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Real-valued function

In mathematics, a real-valued function is a function whose values are real numbers.

## Riemann integral

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.

## Riemann sum

In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum.

## Riemann–Stieltjes integral

In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes.

## Risch algorithm

In symbolic computation (or computer algebra), at the intersection of mathematics and computer science, the Risch algorithm is an algorithm for indefinite integration.

## Romberg's method

In numerical analysis, Romberg's method is used to estimate the definite integral by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule).

## Rough path

In stochastic analysis, a rough path is a generalization of the notion of smooth path allowing to construct a robust solution theory for controlled differential equations driven by classically irregular signals, for example a Wiener process.

## Runge's phenomenon

In the mathematical field of numerical analysis, Runge's phenomenon is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points.

## Scalar field

In mathematics and physics, a scalar field associates a scalar value to every point in a space – possibly physical space.

## Semimartingale

In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process.

## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

## Shell integration

Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution.

## Simpson's rule

In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals.

## Singularity (mathematics)

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.

## Society for Industrial and Applied Mathematics

The Society for Industrial and Applied Mathematics (SIAM) is an academic association dedicated to the use of mathematics in industry.

## Solid of revolution

In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line (the axis of revolution) that lies on the same plane.

## Space

Space is the boundless three-dimensional extent in which objects and events have relative position and direction.

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## Special functions

Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.

## Square

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted.

## Square-integrable function

In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.

## Standard part function

In non-standard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers.

## Stokes' theorem

In vector calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes theorem or the Stokes–Cartan theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.

## Stratonovich integral

In stochastic processes, the Stratonovich integral (developed simultaneously by Ruslan Stratonovich and Donald Fisk) is a stochastic integral, the most common alternative to the Itô integral.

## Summation

In mathematics, summation (capital Greek sigma symbol: ∑) is the addition of a sequence of numbers; the result is their sum or total.

## Surface (mathematics)

In mathematics, a surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero.

## Surface integral

In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces.

## Symbolic integration

In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f(x), i.e. to find a differentiable function F(x) such that This is also denoted.

## Tangent half-angle substitution

In integral calculus, the tangent half-angle substitution is a substitution used for finding antiderivatives, and hence definite integrals, of rational functions of trigonometric functions.

## Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

## Tensor

In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.

## Thermodynamic integration

Thermodynamic integration is a method used to compare the difference in free energy between two given states (e.g., A and B) whose potential energies U_A and U_B have different dependences on the spatial coordinates.

## Thermodynamics

Thermodynamics is the branch of physics concerned with heat and temperature and their relation to energy and work.

## Three-dimensional space

Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point).

## Time

Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future.

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## Time-scale calculus

In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid discrete–continuous dynamical systems.

## Topological ring

In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps where R &times; R carries the product topology.

## Topological vector space

In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.

## Trapezoidal rule

In mathematics, and more specifically in numerical analysis, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area.

## Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

## Trigonometric substitution

In mathematics, Trigonometric substitution is the substitution of trigonometric functions for other expressions.

## Variable (mathematics)

In elementary mathematics, a variable is a symbol, commonly an alphabetic character, that represents a number, called the value of the variable, which is either arbitrary, not fully specified, or unknown.

## Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## Velocity

The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time.

## Volume

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.

## Volume integral

In mathematics&mdash;in particular, in multivariable calculus&mdash;a volume integral refers to an integral over a 3-dimensional domain, that is, it is a special case of multiple integrals.

## Well-defined

In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value.

## Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener.

## Wolfram Alpha

Wolfram Alpha (also styled WolframAlpha, and Wolfram|Alpha) is a computational knowledge engine or answer engine developed by Wolfram Alpha LLC, a subsidiary of Wolfram Research.

## Wolfram Mathematica

Wolfram Mathematica (usually termed Mathematica) is a modern technical computing system spanning most areas of technical computing — including neural networks, machine learning, image processing, geometry, data science, visualizations, and others.

## Work (physics)

In physics, a force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force.

## Zero of a function

In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).

## Zu Chongzhi

Zu Chongzhi (429–500 AD), courtesy name Wenyuan, was a Chinese mathematician, astronomer, writer and politician during the Liu Song and Southern Qi dynasties.

## Zu Gengzhi

Zu Gengzhi (born ca. 450, died ca. 520) was a Chinese mathematician.

## 0

0 (zero) is both a number and the numerical digit used to represent that number in numerals.

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## References

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