请输入您要查询的字词:

 

单词 ConformalMapping
释义

conformal mapping


A mapping f:mn which preserves the magnitude and orientation of the angles between any two curves which intersect in a given point z0 is said to be conformal at z0. A mapping that is conformal at any point in a domain D is said to be conformal in D.

An important special case is when m=n=2. In this case, we may identify 2 with and speak of conformal mappingsMathworldPlanetmathPlanetmath of the complex planeMathworldPlanetmath. It can be shown that a mapping f: is conformal if and only if f is a complex analytic function.  The complex conjugateMathworldPlanetmath f¯ of a conformal mapping f is inversely conformal, i.e. it preserves the magnitude but reverses the orientation of angles.

If m=n, then we can study invertible conformal mappings. It is clear from the definition that the composition of two such maps and the inverse of any such map is again an invertible conformal mapping, so the set of such mappings forms a group. In the case m=n>2, the group of conformal mappings is finite dimensional and is generated by rotations, translationsPlanetmathPlanetmath, and spherical inversionsMathworldPlanetmath.

This notion of conformal mappings can be generalized to any setting in which it makes sense to speak of angles between curves or angles between tangent vectorsMathworldPlanetmath. In particular, one can consider conformal mappings of Riemannian manifolds. It can be shown that, if (M,g) and (N,h) are Riemannian manifolds, then a map f:MN is conformal if and only if f*h=sg for some scalar field s (on M).

Titleconformal mapping
Canonical nameConformalMapping
Date of creation2013-03-22 13:35:42
Last modified on2013-03-22 13:35:42
Ownerrspuzio (6075)
Last modified byrspuzio (6075)
Numerical id19
Authorrspuzio (6075)
Entry typeDefinition
Classificationmsc 30E20
Classificationmsc 53A30
Related topicQuasiconformalMapping
Related topicConformalMappingTheorem
Related topicAngleBetweenTwoLines
Related topicSchwarzChristoffelTransformation
Related topicCategoryOfRiemannianManifolds
Definesconformal
Definesinversely conformal
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 11:44:26