226 relations: Absolute value, Adaptive quadrature, Addison-Wesley, Alfréd Haar, American Mathematical Society, Ancient Greek, Antiderivative, Arc length, Archimedes, Area, Area of a circle, Area under the curve (pharmacokinetics), Arnaud Denjoy, Bernhard Riemann, Bonaventura Cavalieri, Bounded function, Bounded set, Bounded variation, Bulletin of the American Mathematical Society, Calculus, Cartesian product, Cauchy–Schwarz inequality, Cavalieri's principle, Cavalieri's quadrature formula, Chebyshev polynomials, Classical physics, Clenshaw–Curtis quadrature, Closed set, Compass-and-straightedge construction, Complete metric space, Complex analysis, Complex-valued function, Computer algebra system, Continuous function, Contour integration, Convergent series, Cross product, Curl (mathematics), Curve, Curvilinear coordinates, Daniell integral, Darboux integral, Derivative, Differential (infinitesimal), Differential calculus, Differential form, Differential topology, Disc integration, Displacement (fluid), Displacement (vector), ..., Divergence theorem, Domain of a function, Dot product, Electric field, Electromagnetism, Element (mathematics), Elementary function, Eudoxus of Cnidus, Euler substitution, Evangelista Torricelli, Exponential function, Exterior algebra, Exterior derivative, Flux, Force, Fourier analysis, Fractional Brownian motion, Fubini's theorem, Function (mathematics), Fundamental theorem of calculus, Gamma function, Gauss–Kronrod quadrature formula, Gaussian integral, Gaussian quadrature, Geometry, George Berkeley, Gottfried Wilhelm Leibniz, Gradient, Graduate Studies in Mathematics, Graph of a function, Gravitational field, Green's theorem, Haar measure, HathiTrust, Hölder's inequality, Henri Lebesgue, Henstock–Kurzweil integral, Hilbert space, Holonomic function, Hypergeometric function, Hyperreal number, Incomplete gamma function, Infinitesimal, Inner product space, Integral equation, Integral of inverse functions, Integral symbol, Integration by parts, Integration by reduction formulae, Integration by substitution, Integration using Euler's formula, Integration using parametric derivatives, Interpolation, Interval (mathematics), Inverse trigonometric functions, Isaac Barrow, Isaac Newton, Itô calculus, Jaroslav Kurzweil, Johann Radon, John Wallis, Joseph Fourier, Kelvin–Stokes theorem, Kinematics, Lagrange polynomial, Lebesgue integration, Lebesgue measure, Lebesgue–Stieltjes integration, Legendre function, Leibniz integral rule, Limit (mathematics), Limits of integration, Line integral, Linear combination, Linear differential equation, Linear form, Lists of integrals, Liu Hui, Locally compact space, Logarithm, Long s, Lp space, Macsyma, Mathematics, Measurable function, Measure (mathematics), Meijer G-function, Method of exhaustion, Minkowski inequality, Modern Arabic mathematical notation, Monte Carlo integration, Multiple integral, Multivariable calculus, Newton–Cotes formulas, Nicolas Bourbaki, Non-standard analysis, Nonelementary integral, Normal (geometry), Nth root, Order of integration (calculus), Orthogonal polynomials, Oskar Perron, P-adic number, Parabola, Parseval's identity, Partial fraction decomposition, Paul Montel, Pierre de Fermat, Planimeter, Point (geometry), Pointwise, Pointwise product, Precision engineering, Probability density function, Probability theory, Ralph Henstock, Random variable, Rational function, Real analysis, Real line, Real number, Real-valued function, Riemann integral, Riemann sum, Riemann–Stieltjes integral, Risch algorithm, Romberg's method, Rough path, Runge's phenomenon, Scalar field, Semimartingale, Sequence, Shell integration, Simpson's rule, Singularity (mathematics), Society for Industrial and Applied Mathematics, Solid of revolution, Space, Special functions, Square, Square-integrable function, Standard part function, Stokes' theorem, Stratonovich integral, Summation, Surface (mathematics), Surface integral, Symbolic integration, Tangent half-angle substitution, Taylor series, Tensor, Thermodynamic integration, Thermodynamics, Three-dimensional space, Time, Time-scale calculus, Topological ring, Topological vector space, Trapezoidal rule, Trigonometric functions, Trigonometric substitution, Variable (mathematics), Vector field, Vector space, Velocity, Volume, Volume integral, Well-defined, Wiener process, Wolfram Alpha, Wolfram Mathematica, Work (physics), Zero of a function, Zu Chongzhi, Zu Gengzhi, 0. Expand index (176 more) » « Shrink index
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 (Haar Alfréd; 11 October 1885, Budapest – 16 March 1933, Szeged) was a Hungarian mathematician.
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.
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.
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.
Determining the length of an irregular arc segment is also called rectification of a curve.
Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.
In geometry, the area enclosed by a circle of radius is.
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 (1884–1974) was a French mathematician.
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 Francesco Cavalieri (Cavalerius; 1598 – 30 November 1647) was an Italian mathematician and a Jesuate.
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.
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size.
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.
The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.
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.
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.
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.
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows.
In calculus, Cavalieri's quadrature formula, named for 17th-century Italian mathematician Bonaventura Cavalieri, is the integral and generalizations thereof.
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 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.
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
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.
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, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.
In mathematics, a complex-valued function (not to be confused with complex variable function) is a function whose values are complex numbers.
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.
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
In mathematics, a series is the sum of the terms of an infinite sequence of numbers.
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.
In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space.
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.
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved.
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.
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.
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).
The term differential is used in calculus to refer to an infinitesimal (infinitely small) change in some varying quantity.
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change.
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.
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.
In fluid mechanics, displacement occurs when an object is immersed in a fluid, pushing it out of the way and taking its place.
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.
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.
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.
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.
An electric field is a vector field surrounding an electric charge that exerts force on other charges, attracting or repelling them.
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles.
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.
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 (Εὔδοξος ὁ Κνίδιος, Eúdoxos ho Knídios) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato.
Euler Substitution is a method for evaluating integrals of the form: where R is a rational function of x and \sqrt.
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.
In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.
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.
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.
Flux describes the quantity which passes through a surface or substance.
In physics, a force is any interaction that, when unopposed, will change the motion of an object.
In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.
In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion.
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.
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a 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.
The Gauss–Kronrod quadrature formula is an adaptive method for numerical integration.
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 (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 (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 (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 (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS).
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.
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.
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.
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 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.
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 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.
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.
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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.
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.
The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities.
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.
In mathematics, infinitesimals are things so small that there is no way to measure them.
In linear algebra, an inner product space is a vector space with an additional structure called an inner product.
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign.
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.
The integral symbol: is used to denote integrals and antiderivatives in mathematics.
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 formula in integral calculus is a technique of integration, in the form of a recurrence relation.
In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals.
In integral calculus, complex numbers and Euler's formula may be used to evaluate integrals involving trigonometric functions.
In mathematics, integration by parametric derivatives is a method of integrating certain functions.
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.
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.
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 (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.
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, named after Kiyoshi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process).
Jaroslav Kurzweil (born 1926) is a Czech mathematician.
Johann Karl August Radon (16 December 1887 – 25 May 1956) was an Austrian mathematician.
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.
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.
The Kelvin–Stokes theoremThis proof is based on the Lecture Notes given by Prof.
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.
In numerical analysis, Lagrange polynomials are used for polynomial interpolation.
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.
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.
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.
In mathematics, the Legendre functions Pλ, Qλ and associated Legendre functions P, Q are generalizations of Legendre polynomials to non-integer degree.
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.
In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.
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.
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.
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).
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.
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.
Integration is the basic operation in integral calculus.
Liu Hui was a Chinese mathematician who lived in the state of Cao Wei during the Three Kingdoms period (220–280) of China.
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.
In mathematics, the logarithm is the inverse function to exponentiation.
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").
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.
Macsyma (Project MAC’s SYmbolic MAnipulator) is one of the oldest general purpose computer algebra systems which is still widely used.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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.
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.
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.
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.
In mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are normed vector spaces.
Modern Arabic mathematical notation is a mathematical notation based on the Arabic script, used especially at pre-university levels of education.
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers.
The multiple integral is a definite integral of a function of more than one real variable, for example, or.
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.
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 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.
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers.
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).
In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.
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.
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.
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 (7 May 1880 – 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.
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped.
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.
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 Antoine Aristide Montel (29 April 1876 – 22 January 1975) was a French mathematician.
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.
A planimeter, also known as a platometer, is a measuring instrument used to determine the area of an arbitrary two-dimensional shape.
In modern mathematics, a point refers usually to an element of some set called a space.
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 — operations defined on functions by applying the operations to function values separately for each point in the domain of definition.
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 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.
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 is the branch of mathematics concerned with probability.
Ralph Henstock (2 June 1923 – 17 January 2007) was an English mathematician and author.
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.
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.
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.
In mathematics, the real line, or real number line is the line whose points are the real numbers.
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
In mathematics, a real-valued function is a function whose values are real numbers.
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.
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum.
In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes.
In symbolic computation (or computer algebra), at the intersection of mathematics and computer science, the Risch algorithm is an algorithm for indefinite integration.
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).
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.
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.
In mathematics and physics, a scalar field associates a scalar value to every point in a space – possibly physical space.
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.
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
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.
In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals.
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.
The Society for Industrial and Applied Mathematics (SIAM) is an academic association dedicated to the use of mathematics in industry.
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 is the boundless three-dimensional extent in which objects and events have relative position and direction.
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.
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.
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.
In non-standard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers.
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.
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.
In mathematics, summation (capital Greek sigma symbol: ∑) is the addition of a sequence of numbers; the result is their sum or total.
In mathematics, a surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero.
In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces.
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.
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.
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.
In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.
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 is the branch of physics concerned with heat and temperature and their relation to energy and work.
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 is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future.
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.
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 × R carries the product topology.
In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.
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.
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.
In mathematics, Trigonometric substitution is the substitution of trigonometric functions for other expressions.
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.
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of 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.
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 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.
In mathematics—in particular, in multivariable calculus—a volume integral refers to an integral over a 3-dimensional domain, that is, it is a special case of multiple integrals.
In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value.
In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener.
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 (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.
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.
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 (429–500 AD), courtesy name Wenyuan, was a Chinese mathematician, astronomer, writer and politician during the Liu Song and Southern Qi dynasties.
Zu Gengzhi (born ca. 450, died ca. 520) was a Chinese mathematician.
0 (zero) is both a number and the numerical digit used to represent that number in numerals.
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