Liouville's theorem (conformal mappings)
- Liouville's theorem (conformal mappings)
In mathematics, Liouville's theorem is a theorem about conformal mappings in Euclidean space. It states that any conformal mapping on a domain of R"n", where "n" > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are all Möbius transformations. This severely limits the variety of possible conformal mappings in R3 and higher-dimensional spaces.
By contrast, conformal mappings in R2 can be much more complicated - for example, all simply connected planar domains are conformally equivalent, by the Riemann mapping theorem.
Bibliography
*.
*citation|first=Gaspard|last=Monge|authorlink=Gaspard Monge|title=Application de l'analyse à la Géométrie|year=1850|publisher=Bachelier|pages=609–616|url=http://books.google.com/books?id=iCEOAAAAQAAJ&dq=%22Application+de+l%27analyse+%C3%A0+la+g%C3%A9om%C3%A9trie%22
*springer|id=L/l059680|title=Liouville theorems|first=E.D.|last=Solomentsev|year=2001
Wikimedia Foundation.
2010.
Look at other dictionaries:
Liouville's theorem — has various meanings, all mathematical results named after Joseph Liouville:*In complex analysis, see Liouville s theorem (complex analysis). *In conformal mappings, see Liouville s theorem (conformal mappings). *In Hamiltonian mechanics, see… … Wikipedia
Conformal map — For other uses, see Conformal (disambiguation). A rectangular grid (top) and its image under a conformal map f (bottom). It is seen that f maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°. In mathematics, a… … Wikipedia
List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… … Wikipedia
List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… … Wikipedia
Inversive geometry — Not to be confused with Inversive ring geometry. In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion. These… … Wikipedia
Möbius transformation — Not to be confused with Möbius transform or Möbius function. In geometry, a Möbius transformation of the plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − … Wikipedia
Complex analysis — Plot of the function f(x)=(x2 1)(x 2 i)2/(x2 + 2 + 2i). The hue represents the function argument, while the brightness represents the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch … Wikipedia
List of complex analysis topics — Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied … Wikipedia
Differential geometry of surfaces — Carl Friedrich Gauss in 1828 In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives:… … Wikipedia
List of important publications in mathematics — One of the oldest surviving fragments of Euclid s Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.[1] This is a list of important publications in mathematics, organized by field. Some… … Wikipedia
