T-I-1: Difference between revisions
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(which is the composition of a reflection and an inversion in a sphere). However in two dimensions, <math> d=2 </math>, the space of conformal mappings is | (which is the composition of a reflection and an inversion in a sphere). However in two dimensions, <math> d=2 </math>, the space of conformal mappings is | ||
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that | much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that | ||
<math> f'(z) \neq 0</math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. | <math> f'(z) \neq 0</math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. The aim of this HW is to exploit | ||
this property to study some hydrodynamic flows in two spatial dimensions. | |||
= Joukovski's transformation = | |||
The Joukovski's transformation is defined by the following application | |||
<math> | |||
</math> | |||
Revision as of 15:26, 14 October 2011
Analytical functions: conformal map and applications to hydrodynamics
This homework deals with the application of conformal maps to the study of two-dimensional hydrodynamics. A conformal map is a geometrical transformation which preserves all (oriented) crossing angles between lines. In dimension a conformal map is necessarily composed from the following limited number of transformations: translations, rotations, homothetic transformation and special conformal transformation (which is the composition of a reflection and an inversion in a sphere). However in two dimensions, , the space of conformal mappings is much larger and one can show that, given an open set , any holomorphic function such that , defines a conformal map from to . The aim of this HW is to exploit this property to study some hydrodynamic flows in two spatial dimensions.
Joukovski's transformation
The Joukovski's transformation is defined by the following application
