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...

 > $T\left[x\right]≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{T}}_{{x}}{≔}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $T\left[y\right]≔\mathrm{VectorField}\left(\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{T}}_{{y}}{≔}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (2)
 > $T\left[z\right]≔\mathrm{VectorField}\left(\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{T}}_{{z}}{≔}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (3)
 > $R≔\mathrm{VectorField}\left(-y\mathrm{D}\left[x\right]+x\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${R}{≔}{-}{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (4)

Construct two distributions...

 > $\mathrm{\Sigma }≔\mathrm{Distribution}\left(T\left[z\right],R\right)$
 ${\mathrm{\Sigma }}{≔}\left\{{-}\frac{{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (5)
 > $\mathrm{Gamma}≔\mathrm{Distribution}\left(T\left[x\right],T\left[y\right]\right)$
 ${\mathrm{Γ}}{≔}\left\{\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (6)

Intersection and sum...

 > $\mathrm{Intersection}\left(\mathrm{Gamma},\mathrm{\Sigma }\right)$
 $\left\{{-}\frac{{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (7)
 > $\mathrm{VectorSpaceSum}\left(\mathrm{\Sigma },\mathrm{Gamma}\right)$
 $\left\{\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (8)

Compatibility

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