Dimension - Maple Help

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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...

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

Construct the associated distribution....

 > $\mathrm{\Sigma }≔\mathrm{Distribution}\left(R\left[x\right],R\left[y\right],R\left[z\right]\right)$
 ${\mathrm{\Sigma }}{≔}\left\{{-}\frac{{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}{-}\frac{{z}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (4)
 > $\mathrm{Dimension}\left(\mathrm{\Sigma }\right)$
 ${2}$ (5)
 > $\mathrm{Codimension}\left(\mathrm{\Sigma }\right)$
 ${1}$ (6)
 > $\mathrm{IsTrivial}\left(\mathrm{\Sigma }\right)$
 ${\mathrm{false}}$ (7)

Compatibility

 • The Dimension, Codimension and IsTrivial commands were introduced in Maple 2020.
 • For more information on Maple 2020 changes, see Updates in Maple 2020.