ⓘ Fundamental theorems ..

Fundamental theorem

In mathematics, the fundamental theorem of a field is the theorem which is considered to be the most central and the important one to that field. As such, its naming is not necessarily based on the difficulty of its proofs, or how often it is used. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus, which are two distinct branches of calculus that were not previously obviously related. On the other hand, being "fundamental" does not necessarily mean that it is the most basic result. For example, the proof of the funda ...

Collineation

In projective geometry, a collineation is a one-to-one and onto map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an isomorphism between projective spaces, or an automorphism from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism. The set of all collineations of a space to itself form a group, called the collineation group.

Finitely generated abelian group

In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x 1., x s in G such that every x in G can be written in the form x = n 1 x 1 + n 2 x 2 +. + n s x s with integers n 1., n s. In this case, we say that the set { x 1., x s } is a generating set of G or that x 1., x s generate G. Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.

Helmholtz decomposition

In physics and mathematics, in the area of vector calculus, Helmholtzs 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 the sum of an irrotational vector field and a solenoidal vector field, this is known as the Helmholtz decomposition or Helmholtz representation. It is named after Hermann von Helmholtz. As an irrotational vector field has a scalar potential and a solenoidal vector field has a vector potential, the Helmholtz decomposition states that a vector field ...

Thermodynamic potential

A thermodynamic potential is a scalar quantity used to represent the thermodynamic state of a system. The concept of thermodynamic potentials was introduced by Pierre Duhem in 1886. Josiah Willard Gibbs in his papers used the term fundamental functions. One main thermodynamic potential that has a physical interpretation is the internal energy U. It is the energy of configuration of a given system of conservative forces and only has meaning with respect to a defined set of references. Expressions for all other thermodynamic energy potentials are derivable via Legendre transforms from an exp ...

Fundamental theorem of algebra

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number can be considered a complex number with its imaginary part equal to zero. Equivalently by definition, the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients there, counted with multiplicity, exactly n complex roots. The equivalence ...

                                     

ⓘ Fundamental theorems

  • of fundamental theorems that are not directly related to mathematics: Fundamental theorem of arbitrage - free pricing Fisher s fundamental theorem of natural
  • kernel and image of the homomorphism. The homomorphism theorem is used to prove the isomorphism theorems Given two groups G and H and a group homomorphism
  • The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. The first
  • There are two fundamental theorems of welfare economics. The first theorem states that a market will tend toward a competitive equilibrium that is weakly
  • The fundamental theorems of asset pricing also: of arbitrage, of finance provide necessary and sufficient conditions for a market to be arbitrage free
  • 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
  • Fisher s fundamental theorem of natural selection is an idea about genetic variance in population genetics developed by the statistician and evolutionary
  • fundamentalism. Any of a number of fundamental theorems identified in mathematics, such as: The fundamental theorem of algebra, awe theorem regarding the factorization
  • number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique - prime - factorization theorem states that every
  • The fundamental theorem of algebra states that every non - constant single - variable polynomial with complex coefficients has at least one complex root.
  • In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces. Those statements may be given concretely in