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...

 > $\mathrm{V1}≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y,z,w\right]\right)$
 ${\mathrm{V1}}{≔}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $\mathrm{V2}≔\mathrm{VectorField}\left(\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z,w\right]\right)$
 ${\mathrm{V2}}{≔}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (2)
 > $\mathrm{V3}≔\mathrm{VectorField}\left(\mathrm{D}\left[y\right]+z\mathrm{D}\left[w\right],\mathrm{space}=\left[x,y,z,w\right]\right)$
 ${\mathrm{V3}}{≔}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{z}{}\frac{{\partial }}{{\partial }{w}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (3)

Construct the associated distribution...

 > $\mathrm{\Sigma }≔\mathrm{Distribution}\left(\mathrm{V1},\mathrm{V2},\mathrm{V3}\right)$
 ${\mathrm{\Sigma }}{≔}\left\{\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{z}}{+}\frac{{\partial }}{{\partial }{w}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (4)

Construct Cauchy vectors...

 > $\mathrm{CauchyDistribution}\left(\mathrm{\Sigma }\right)$
 $\left\{\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (5)
 > $\mathrm{IsInvolutive}\left(\mathrm{\Sigma }\right)$
 ${\mathrm{false}}$ (6)

Compatibility

 • The CauchyDistribution command was introduced in Maple 2020.
 • For more information on Maple 2020 changes, see Updates in Maple 2020.