Intersection - Maple Help

Intersection

calculate intersection of distributions

VectorSpaceSum

calculate vector space sum of distributions

 Calling Sequence Intersection(dist1, ...) VectorSpaceSum(dist1, ...)

Parameters

 dist1, ... - sequence of Distribution objects.

Description

 • The Intersection method returns a Distribution object representing the intersection of the input Distribution objects.  More precisely, the result is a distribution such that at each point, the subspace of tangent space is the intersection of the subspaces spanned by the input distributions.
 • Similarly, the VectorSpaceSum command returns a Distribution object representing the vector space sum of the input Distribution objects.  More precisely, the result is a distribution such that at each point, the subspace of tangent space is the vector space sum of the subspaces spanned by the input distributions.
 • These methods are associated with the Distribution object. For more detail see Overview of the Distribution object.

Examples

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

Build vector fields...

 > ${T}_{x}≔\mathrm{VectorField}\left({\mathrm{D}}_{x},\mathrm{space}=\left[x,y,z\right]\right)$
 ${{T}}_{{x}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}$ (1)
 > ${T}_{y}≔\mathrm{VectorField}\left({\mathrm{D}}_{y},\mathrm{space}=\left[x,y,z\right]\right)$
 ${{T}}_{{y}}{≔}\frac{{ⅆ}}{{ⅆ}{y}}$ (2)
 > ${T}_{z}≔\mathrm{VectorField}\left({\mathrm{D}}_{z},\mathrm{space}=\left[x,y,z\right]\right)$
 ${{T}}_{{z}}{≔}\frac{{ⅆ}}{{ⅆ}{z}}$ (3)
 > $R≔\mathrm{VectorField}\left(-y{\mathrm{D}}_{x}+x{\mathrm{D}}_{y},\mathrm{space}=\left[x,y,z\right]\right)$
 ${R}{≔}{-}{y}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (4)

Construct two distributions...

 > $\mathrm{Σ}≔\mathrm{Distribution}\left({T}_{z},R\right)$
 ${\mathrm{\Sigma }}{≔}\left\{{-}\frac{{y}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right)}{{x}}{+}\frac{{ⅆ}}{{ⅆ}{y}}{,}\frac{{ⅆ}}{{ⅆ}{z}}\right\}$ (5)
 > $\mathrm{Gamma}≔\mathrm{Distribution}\left({T}_{x},{T}_{y}\right)$
 ${\mathrm{Γ}}{≔}\left\{\frac{{ⅆ}}{{ⅆ}{x}}{,}\frac{{ⅆ}}{{ⅆ}{y}}\right\}$ (6)

Intersection and sum...

 > $\mathrm{Intersection}\left(\mathrm{Gamma},\mathrm{Σ}\right)$
 $\left\{{-}\frac{{y}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right)}{{x}}{+}\frac{{ⅆ}}{{ⅆ}{y}}\right\}$ (7)
 > $\mathrm{VectorSpaceSum}\left(\mathrm{Σ},\mathrm{Gamma}\right)$
 $\left\{\frac{{ⅆ}}{{ⅆ}{x}}{,}\frac{{ⅆ}}{{ⅆ}{y}}{,}\frac{{ⅆ}}{{ⅆ}{z}}\right\}$ (8)

Compatibility

 • The Intersection and VectorSpaceSum commands were introduced in Maple 2020.