 Dimension - Maple Help

Dimension

calculate the dimension of a Distribution object

Codimension

calculate the codimension of a Distribution object

IsTrivial

check if a Distribution object is trivial Calling Sequence Dimension( dist) Codimension( dist) IsTrivial( dist) Parameters

 dist - a Distribution object Description

 • The Dimension method returns the dimension of the subspace of tangent space spanned by a distribution.
 • The Codimension method returns the codimension of this subspace. If a distribution of dimension r lives on a space of dimension n, the codimension is n-r.
 • The IsTrivial method returns true if dist is of dimension 0 and false otherwise.
 • 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 associated with 3-d spatial rotations...

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

Construct the associated distribution....

 > $\mathrm{Σ}≔\mathrm{Distribution}\left({R}_{x},{R}_{y},{R}_{z}\right)$
 ${\mathrm{\Sigma }}{≔}\left\{{-}\frac{{y}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right)}{{x}}{+}\frac{{ⅆ}}{{ⅆ}{y}}{,}{-}\frac{{z}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right)}{{x}}{+}\frac{{ⅆ}}{{ⅆ}{z}}\right\}$ (4)
 > $\mathrm{Dimension}\left(\mathrm{Σ}\right)$
 ${2}$ (5)
 > $\mathrm{Codimension}\left(\mathrm{Σ}\right)$
 ${1}$ (6)
 > $\mathrm{IsTrivial}\left(\mathrm{Σ}\right)$
 ${\mathrm{false}}$ (7) Compatibility

 • The Dimension, Codimension and IsTrivial commands were introduced in Maple 2020.