mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a fundamental theorem is a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

which is considered to be central and conceptually important for some topic. For example, the

fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...

gives the relationship between

differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...

and

integral calculus In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...

. The names are mostly traditional, so that for example the

fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...

is basic to what would now be called

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

. Some of these are

classification theorems In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class. A few issues relat ...

of objects which are mainly dealt with in the field. For instance, the

fundamental theorem of curves In differential geometry, the fundamental theorem of space curves states that every regular curve in three-dimensional space, with non-zero curvature, has its shape (and size or scale) completely determined by its curvature and torsion.. Use A ...

describe classification of regular curves in space up to

translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...

rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

. Likewise, the mathematical literature sometimes refers to the fundamental lemma of a field. The term lemma is conventionally used to denote a proven proposition which is used as a stepping stone to a larger result, rather than as a useful statement in-and-of itself.

Fundamental theorems of mathematical topics

Fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomial ...

Fundamental theorem of algebraic K-theory In algebra, the fundamental theorem of algebraic K-theory, algebraic ''K''-theory describes the effects of change of rings, changing the ring of ''K''-groups from a ring ''R'' to R or R

, t^ The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...

The theorem was first proved by Hyman Bass for K_0 ...

Fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...

Fundamental theorem of Boolean algebra Boole's expansion theorem, often referred to as the Shannon expansion or decomposition, is the identity: F = x \cdot F_x + x' \cdot F_, where F is any Boolean function, x is a variable, x' is the complement of x, and F_xand F_ are F with the argum ...

Fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...

Fundamental theorem of calculus for line integrals The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...

Fundamental theorem of curves In differential geometry, the fundamental theorem of space curves states that every regular curve in three-dimensional space, with non-zero curvature, has its shape (and size or scale) completely determined by its curvature and torsion.. Use A ...

Fundamental theorem of cyclic groups In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order ''n'', every subgroup's order is a divisor of ''n'', and there is exactly one subgroup for each divisor. This result has been called the f ...

Fundamental theorem of equivalence relations In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relati ...

Fundamental theorem of exterior calculus In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integration of differential forms ...

Fundamental theorem of finitely generated abelian groups In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...

Fundamental theorem of finitely generated modules over a principal ideal domain In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finite ...

Fundamental theorem of finite distributive lattices :''This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).'' In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice ...

Fundamental theorem of Galois theory In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory. In its most basi ...

Fundamental theorem of geometric calculus In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including differential geometry and different ...

Fundamental theorem on homomorphisms In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and ...

Fundamental theorem of ideal theory in number fields In number theory, the fundamental theorem of ideal theory in number fields states that every nonzero proper ideal in the ring of integers of a number field admits unique factorization into a product of nonzero prime ideals. In other words, every ...

Fundamental theorem of Lebesgue integral calculus In calculus, absolute continuity is a smoothness (mathematics), smoothness property of function (mathematics), functions that is stronger than continuous function, continuity and uniform continuity. The notion of absolute continuity allows one to o ...

Fundamental theorem of linear algebra Fundamental may refer to: * Foundation of reality * Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental" * Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" idea ...

Fundamental theorem of linear programming In mathematical optimization, the fundamental theorem of linear programming states, in a weak formulation, that the maxima and minima of a linear function over a convex polygonal region occur at the region's corners. Further, if an extreme value oc ...

* Fundamental theorem of noncommutative algebra *

Fundamental theorem of projective geometry In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...

* Fundamental theorem of random fields *

Fundamental theorem of Riemannian geometry In the mathematical field of Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection that is torsion-free and metric-compatibl ...

Fundamental theorem of tessarine algebra In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as :(u,v)(w,z) = (u w - v z, u z ...

Fundamental theorem of symmetric polynomials In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...

Fundamental theorem of topos theory In mathematics, The fundamental theorem of topos theory states that the comma category, slice \mathbf / X of a topos \mathbf over any one of its object (category theory), objects X is itself a topos. Moreover, if there is a morphism f : A \rightar ...

* Fundamental theorem of ultraproducts *

Fundamental theorem of vector analysis In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

referred to the law of

quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...

as the "fundamental theorem" of

quadratic residue In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that: :x^2\equiv q \pmod. Otherwise, ''q'' is called a quadratic no ...

Applied or informally stated "fundamental theorems"

There are also a number of "fundamental theorems" that are not directly related to mathematics: *

Fundamental theorem of arbitrage-free pricing The fundamental theorems of asset pricing (also: of arbitrage, of finance), in both financial economics and mathematical finance, provide necessary and sufficient conditions for a market to be arbitrage-free, and for a market to be complete. An arb ...

Fisher's fundamental theorem of natural selection Fisher's fundamental theorem of natural selection is an idea about genetic variance in population genetics developed by the statistician and evolutionary biologist Ronald Fisher. The proper way of applying the abstract mathematics of the theorem to ...

Fundamental theorems of welfare economics There are two fundamental theorems of welfare economics. The first states that in economic equilibrium, a set of complete markets, with complete information, and in perfect competition, will be Pareto optimal (in the sense that no further exchang ...

Fundamental equations of thermodynamics A thermodynamic potential (or more accurately, a thermodynamic potential energy)ISO/IEC 80000-5, Quantities an units, Part 5 - Thermodynamics, item 5-20.4 Helmholtz energy, Helmholtz functionISO/IEC 80000-5, Quantities an units, Part 5 - Thermod ...

Fundamental theorem of poker The fundamental theorem of poker is a principle first articulated by David Sklansky that he believes expresses the essential nature of poker as a game of decision-making in the face of incomplete information. The fundamental theorem is stated ...

Holland's schema theorem Holland's schema theorem, also called the fundamental theorem of genetic algorithms, is an inequality that results from coarse-graining an equation for evolutionary dynamics. The Schema Theorem says that short, low-order schemata with above-average ...

, or the "fundamental theorem of

genetic algorithm In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to gene ...

s" *

Glivenko–Cantelli theorem In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the Fundamental Theorem of Statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determines the asymptotic behaviour of the empiric ...

, or the "fundamental theorem of statistics" *

Fundamental theorem of software engineering The fundamental theorem of software engineering (FTSE) is a term originated by Andrew Koenig to describe a remark by Butler Lampson attributed to David J. Wheeler: The theorem does not describe an actual theorem that can be proven; rather, it is ...

Fundamental lemmata

Fundamental lemma of calculus of variations In mathematics, specifically in the calculus of variations, a variation of a function can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (functional derivative equal zero ...

Fundamental lemma of Langlands and Shelstad In the mathematical theory of automorphic forms, the fundamental lemma relates orbital integrals on a reductive group over a local field to stable orbital integrals on its endoscopic groups. It was conjectured by in the course of developing the ...

* Fundamental lemma of sieve theory

References

External links

*{{Commons category-inline, Fundamental theorems
"Some Fundamental Theorems in Mathematics" (Knill, 2018)
- self-described "expository hitchhikers guide", or exploration, of around 130 fundamental/influential mathematical results and their significance, across a range of mathematical fields. Theorems,Fundamental

Fundamental Fundamental may refer to: * Foundation of reality * Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental" * Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" idea ...

es:Teorema fundamental ka:ფუნდამენტური თეორემა

Fundamental theorems of mathematical topics

Applied or informally stated "fundamental theorems"

Fundamental lemmata

See also

References

External links