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Differential (mathematics), the Glossary

111 relations: Abraham Robinson, Affine variety, Algebra over a field, Algebraic curve, Algebraic geometry, Algebraic topology, Archimedes, Banach space, Basis (linear algebra), Calculus, Category (mathematics), Category of sets, Chain complex, Chain rule, Commutative ring, Complete metric space, Completeness of the real numbers, Connection (mathematics), Connection (vector bundle), Constructivism (philosophy of mathematics), Cotangent space, Covariance and contravariance of vectors, Covariant derivative, Delta (letter), Derivation (differential algebra), Derivative, Differentiable manifold, Differential (mathematics), Differential algebra, Differential equation, Differential form, Differential geometry, Differential of a function, Differential of the first kind, Differential topology, Dimensional analysis, Dual number, Exterior derivative, First-order logic, Fluent (mathematics), Function (mathematics), Gateaux derivative, George Berkeley, Gottfried Wilhelm Leibniz, Hilbert space, Hyperreal number, Ideal (ring theory), Identity function, Infinitesimal, Inner product space, ... Expand index (61 more) »
Set index articles on mathematics

Abraham Robinson

Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorporated into modern mathematics.

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Affine variety

In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field of some family of polynomials in the polynomial ring k. An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by the defining polynomials is prime.

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Algebra over a field

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.

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Algebraic curve

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables.

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Algebraic geometry

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.

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Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.

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Archimedes

Archimedes of Syracuse was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily.

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Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

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Basis (linear algebra)

In mathematics, a set of vectors in a vector space is called a basis (bases) if every element of may be written in a unique way as a finite linear combination of elements of.

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Calculus

Calculus 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.

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Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows".

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Category of sets

In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets.

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Chain complex

In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next.

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Chain rule

In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and.

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Commutative ring

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative.

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Complete metric space

In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in.

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Completeness of the real numbers

Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line.

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Connection (mathematics)

In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a parallel and consistent manner.

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Connection (vector bundle)

In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points.

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Constructivism (philosophy of mathematics)

In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists.

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Cotangent space

In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold.

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Covariance and contravariance of vectors

In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis.

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Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.

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Delta (letter)

Delta (uppercase Δ, lowercase δ; δέλτα, délta) is the fourth letter of the Greek alphabet.

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Derivation (differential algebra)

In mathematics, a derivation is a function on an algebra that generalizes certain features of the derivative operator.

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Derivative

The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input. Differential (mathematics) and derivative are differential calculus.

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Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus.

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Differential (mathematics)

In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions. Differential (mathematics) and differential (mathematics) are calculus, differential calculus, mathematical terminology and set index articles on mathematics.

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Differential algebra

In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras are used for the study of algebraic varieties, which are solution sets of systems of polynomial equations.

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Differential equation

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives.

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Differential form

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds.

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Differential geometry

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

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Differential of a function

In calculus, the differential represents the principal part of the change in a function y. Differential (mathematics) and differential of a function are differential calculus.

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Differential of the first kind

In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1-forms.

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Differential topology

In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds.

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Dimensional analysis

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measurement (such as metres and grams) and tracking these dimensions as calculations or comparisons are performed.

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Dual number

In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century.

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Exterior derivative

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

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First-order logic

First-order logic—also called predicate logic, predicate calculus, quantificational logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.

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Fluent (mathematics)

A fluent is a time-varying quantity or variable. Differential (mathematics) and fluent (mathematics) are differential calculus.

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Function (mathematics)

In mathematics, a function from a set to a set assigns to each element of exactly one element of.

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Gateaux derivative

In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus.

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George Berkeley

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

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Gottfried Wilhelm Leibniz

Gottfried Wilhelm Leibniz (– 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who invented calculus in addition to many other branches of mathematics, such as binary arithmetic, and statistics.

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Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional.

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Hyperreal number

In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers.

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Ideal (ring theory)

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements.

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Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged.

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Infinitesimal

In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. Differential (mathematics) and infinitesimal are calculus.

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Inner product space

In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product.

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Integral

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations.

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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 the product of their derivative and antiderivative.

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Integration by substitution

In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives.

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Invariant (mathematics)

In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. Differential (mathematics) and invariant (mathematics) are mathematical terminology.

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Isaac Newton

Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author who was described in his time as a natural philosopher.

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Jacobian matrix and determinant

In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. Differential (mathematics) and Jacobian matrix and determinant are differential calculus.

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Jet (mathematics)

In mathematics, the jet is an operation that takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain.

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John Lane Bell

John Lane Bell (born March 25, 1945) is an Anglo-Canadian philosopher, mathematician and logician.

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Joseph-Louis Lagrange

Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian mathematician, physicist and astronomer, later naturalized French.

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Kähler differential

In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.

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Law of excluded middle

In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true.

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Leibniz's notation

In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols and to represent infinitely small (or infinitesimal) increments of and, respectively, just as and represent finite increments of and, respectively. Differential (mathematics) and Leibniz's notation are differential calculus.

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Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Differential (mathematics) and limit (mathematics) are differential calculus.

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Limit of a function

Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1.

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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).

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Linear map

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication.

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Linearization

In mathematics, linearization is finding the linear approximation to a function at a given point. Differential (mathematics) and linearization are differential calculus.

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Logic

Logic is the study of correct reasoning.

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Long s

The long s,, also known as the medial s or initial s, is an archaic form of the lowercase letter, found mostly in works from the late 8th to early 19th centuries.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Matrix (mathematics)

In mathematics, a matrix (matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.

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Method of Fluxions

Method of Fluxions (De Methodis Serierum et Fluxionum) is a mathematical treatise by Sir Isaac Newton which served as the earliest written formulation of modern calculus. Differential (mathematics) and Method of Fluxions are differential calculus.

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Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1.

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Necessity and sufficiency

In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. Differential (mathematics) and necessity and sufficiency are mathematical terminology.

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Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

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Nilpotent

In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n.

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Nonstandard analysis

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

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Norm (mathematics)

In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.

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Normed vector space

In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined.

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Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).

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Princeton University Press

Princeton University Press is an independent publisher with close connections to Princeton University.

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Product rule

In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions.

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Proof by contradiction

In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction.

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Pullback (differential geometry)

Let \phi:M\to N be a smooth map between smooth manifolds M and N. Then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is known as the pullback (by \phi), and is frequently denoted by \phi^*.

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Pushforward (differential)

In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces.

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Quadratic differential

In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle.

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Quotient space (linear algebra)

In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero.

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Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

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Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold.

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Riemann–Stieltjes integral

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

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Ringed space

In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions.

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Scheme (mathematics)

In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.

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Second-order logic

In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic.

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Slope

In mathematics, the slope or gradient of a line is a number that describes the direction and steepness of the line.

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Smooth infinitesimal analysis

Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals.

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Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number, called differentiability class, of continuous derivatives it has over its domain.

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Stochastic calculus

Stochastic calculus is a branch of mathematics that operates on stochastic processes.

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Stochastic differential equation

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process.

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Stochastic process

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a sequence of random variables in a probability space, where the index of the sequence often has the interpretation of time.

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Synthetic differential geometry

In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory.

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Tangent

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point.

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Tensor field

In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold).

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The Analyst

The Analyst (subtitled A Discourse Addressed to an Infidel Mathematician: Wherein It Is Examined Whether the Object, Principles, and Inferences of the Modern Analysis Are More Distinctly Conceived, or More Evidently Deduced, Than Religious Mysteries and Points of Faith) is a book by George Berkeley.

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Topological vector space

In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.

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Topos

In mathematics, a topos (plural topoi or, or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site).

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Total derivative

In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Differential (mathematics) and total derivative are differential calculus.

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Transfer principle

In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure.

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Variable (mathematics)

In mathematics, a variable (from Latin variabilis, "changeable") is a symbol that represents a mathematical object. Differential (mathematics) and variable (mathematics) are calculus.

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Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g.

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Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n.

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Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''.

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References

[1] https://en.wikipedia.org/wiki/Differential_(mathematics)

Also known as Differential (calculus), Differential (infinitesimal), Differential area, Differential element, Variable of integration, .

, Integral, Integration by parts, Integration by substitution, Invariant (mathematics), Isaac Newton, Jacobian matrix and determinant, Jet (mathematics), John Lane Bell, Joseph-Louis Lagrange, Kähler differential, Law of excluded middle, Leibniz's notation, Limit (mathematics), Limit of a function, Linear combination, Linear map, Linearization, Logic, Long s, Mathematics, Matrix (mathematics), Method of Fluxions, Multiplicative inverse, Necessity and sufficiency, Neighbourhood (mathematics), Nilpotent, Nonstandard analysis, Norm (mathematics), Normed vector space, Partial derivative, Princeton University Press, Product rule, Proof by contradiction, Pullback (differential geometry), Pushforward (differential), Quadratic differential, Quotient space (linear algebra), Real number, Riemann surface, Riemann–Stieltjes integral, Ringed space, Scheme (mathematics), Second-order logic, Slope, Smooth infinitesimal analysis, Smoothness, Stochastic calculus, Stochastic differential equation, Stochastic process, Synthetic differential geometry, Tangent, Tensor field, The Analyst, Topological vector space, Topos, Total derivative, Transfer principle, Variable (mathematics), Vector bundle, Vector field, Vector space.

Differential (mathematics), the Glossary

Table of Contents

See also

Set index articles on mathematics

References