 CauchyDistribution - Maple Help

CauchyDistribution

calculate Distribution spanned by Cauchy vector fields of a distribution Calling Sequence CauchyDistribution(dist) Parameters

 dist - a Distribution object Description

 • The CauchyDistribution method returns a Distribution object spanned by the Cauchy vector fields of input Distribution dist.
 • A vector field C is a Cauchy vector field of distribution dist if C within dist, then [C,X] likewise lies within dist.  That is, C lies in dist and is a symmetry of dist. Such vector fields span a distribution, which is always in involution.
 • This method is of little interest if the input Distribution dist is involutive, since in that case CauchyDistribution(dist) will simply return dist itself.
 • This use of the term 'Cauchy distribution' is unrelated to its use in statistics (see Statistics[Distributions][Cauchy]).
 • This method is associated with the Distribution object. For more detail see Overview of the Distribution object. Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$

Build vector fields...

 > $\mathrm{V1}≔\mathrm{VectorField}\left({\mathrm{D}}_{x},\mathrm{space}=\left[x,y,z,w\right]\right)$
 ${\mathrm{V1}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}$ (1)
 > $\mathrm{V2}≔\mathrm{VectorField}\left({\mathrm{D}}_{z},\mathrm{space}=\left[x,y,z,w\right]\right)$
 ${\mathrm{V2}}{≔}\frac{{ⅆ}}{{ⅆ}{z}}$ (2)
 > $\mathrm{V3}≔\mathrm{VectorField}\left({\mathrm{D}}_{y}+z{\mathrm{D}}_{w},\mathrm{space}=\left[x,y,z,w\right]\right)$
 ${\mathrm{V3}}{≔}\frac{{ⅆ}}{{ⅆ}{y}}{+}{z}{}\left(\frac{{ⅆ}}{{ⅆ}{w}}\right)$ (3)

Construct the associated distribution...

 > $\mathrm{Σ}≔\mathrm{Distribution}\left(\mathrm{V1},\mathrm{V2},\mathrm{V3}\right)$
 ${\mathrm{\Sigma }}{≔}\left\{\frac{{ⅆ}}{{ⅆ}{x}}{,}\frac{{ⅆ}}{{ⅆ}{z}}{,}\frac{\frac{{ⅆ}}{{ⅆ}{y}}}{{z}}{+}\frac{{ⅆ}}{{ⅆ}{w}}\right\}$ (4)

Construct Cauchy vectors...

 > $\mathrm{CauchyDistribution}\left(\mathrm{Σ}\right)$
 $\left\{\frac{{ⅆ}}{{ⅆ}{x}}\right\}$ (5)
 > $\mathrm{IsInvolutive}\left(\mathrm{Σ}\right)$
 ${\mathrm{false}}$ (6) Compatibility

 • The CauchyDistribution command was introduced in Maple 2020.