Fundamental Equation
   HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a fundamental theorem is a
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
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 derivative, differentiating a function (mathematics), function (calculating its slopes, or rate of change at every point on its domain) with the concept of integral, inte ...
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 is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Int ...
. 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 is prime or can be represented uniquely as a product of prime numbers, ...
is basic to what would now be called
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
. 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 rela ...
of objects which are mainly dealt with in the field. For instance, the fundamental theorem of curves describes classification of regular curves in space up to
translation Translation is the communication of the semantics, 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 English la ...
and
rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
. 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 called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one comp ...
* Fundamental theorem of algebraic K-theory *
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 is prime or can be represented uniquely as a product of prime numbers, ...
* Fundamental theorem of Boolean algebra *
Fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating its slopes, or rate of change at every point on its domain) with the concept of integral, inte ...
* Fundamental theorem of calculus for line integrals * Fundamental theorem of curves * Fundamental theorem of cyclic groups * Fundamental theorem of dynamical systems * Fundamental theorem of equivalence relations * Fundamental theorem of exterior calculus *
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, ...
* Fundamental theorem of finitely generated modules over a principal ideal domain * Fundamental theorem of finite distributive lattices *
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 bas ...
* Fundamental theorem of geometric calculus *
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 (alg ...
* Fundamental theorem of ideal theory in number fields * Fundamental theorem of Lebesgue integral calculus * Fundamental theorem of linear algebra * Fundamental theorem of linear programming * 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 * Fundamental theorem of tessarine algebra *
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 sym ...
* Fundamental theorem of topos theory * Fundamental theorem of ultraproducts * Fundamental theorem of vector analysis
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
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 a quadratic residue modulo operation, modulo ''n'' if it is Congruence relation, congruent to a Square number, perfect square modulo ''n''; that is, if there exists an integer ''x'' such that :x^2\equiv q \pm ...
s.


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 a ...
*
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 exchange ...
* Fundamental equations of thermodynamics *
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 g ...
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, describes the asymptotic behaviour of the empirica ...
, 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 ...


Fundamental lemmata

* Fundamental lemma of the calculus of variations * Fundamental lemma of Langlands and Shelstad *
Fundamental lemma of sieve theory In number theory, the fundamental lemma of sieve theory is any of several results that systematize the process of applying Sieve theory, sieve methods to particular problems. Heini Halberstam, Halberstam & Hans-Egon Richert, Richert write: Diamo ...


See also

* Main theorem of elimination theory *
List of theorems This is a list of notable theorems. Lists of theorems and similar statements include: * List of algebras *List of algorithms * List of axioms * List of conjectures *List of data structures * List of derivatives and integrals in alternative calcu ...
* Toy theorem


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 es:Teorema fundamental ka:ფუნდამენტური თეორემა