GetSpace - Maple Help

GetSpace

get space coordinates of a LAVF, Distribution, VFPDO object

 Calling Sequence GetSpace( obj)

Parameters

 obj - a LAVF, Distribution, or VFPDO object.

Description

 • The GetSpace method returns a list of space coordinate variables where the LAVF, Distribution, or VFPDO object lives.
 • To be more specific, the space where these object live is the space where their vector fields live. For example, let L be a LAVF object, then the call GetSpace(L) is equivalent to GetSpace(GetVectorField(L)). See GetSpace of a VectorField object for more detail.
 • This method is associated with the LAVF, Distribution, and VFPDO objects. For more detail, see Overview of the LAVF object, overview of the Distribution object, and overview of the VFPDO object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{ξ}\left(x,y\right),\mathrm{η}\left(x,y\right)\right]\right):$

First, construct an indeterminate vector field and a determining system, then construct an LAVF object from them...

 > $V≔\mathrm{VectorField}\left(\mathrm{ξ}\left(x,y\right){\mathrm{D}}_{x}+\mathrm{η}\left(x,y\right){\mathrm{D}}_{y},\mathrm{space}=\left[x,y\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (1)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ξ}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{η}\left(x,y\right)=-\left(\frac{\partial }{\partial y}\mathrm{ξ}\left(x,y\right)\right),\frac{\partial }{\partial y}\mathrm{η}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{ξ}\left(x,y\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{ξ},\mathrm{η}\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)
 > $L≔\mathrm{LAVF}\left(V,\mathrm{E2}\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (3)

Now we can apply the method to find the space of L (and  various objects associated with it).

 > $\mathrm{GetSpace}\left(L\right)$
 $\left[{x}{,}{y}\right]$ (4)
 > $\mathrm{OD}≔\mathrm{OrbitDistribution}\left(L\right)$
 ${\mathrm{OD}}{≔}\left\{\frac{{ⅆ}}{{ⅆ}{y}}{,}\frac{{ⅆ}}{{ⅆ}{x}}\right\}$ (5)
 > $\mathrm{GetSpace}\left(\mathrm{OD}\right)$
 $\left[{x}{,}{y}\right]$ (6)
 > $\mathrm{Δ}≔\mathrm{VFPDO}\left(L\right)$
 ${\mathrm{\Delta }}{≔}{X}{→}\left[\frac{{\partial }}{{\partial }{y}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{X}{}\left({x}\right)\right){,}\frac{{ⅆ}}{{ⅆ}{x}}{}{X}{}\left({x}\right){,}\frac{{\partial }}{{\partial }{x}}{}{X}{}\left({y}\right){+}\frac{{\partial }}{{\partial }{y}}{}{X}{}\left({x}\right){,}\frac{{ⅆ}}{{ⅆ}{y}}{}{X}{}\left({y}\right)\right]$ (7)
 > $\mathrm{GetSpace}\left(\mathrm{Δ}\right)$
 $\left[{x}{,}{y}\right]$ (8)

Compatibility

 • The GetSpace command was introduced in Maple 2020.