IsSubspace - Maple Help

IsSubspace

check if solution space of a LAVF object is subspace of solution space of another LAVF object.

 Calling Sequence IsSubspace( L1, L2)

Parameters

 L1, L2 - LAVF objects

Description

 • Let L1, L2 be LAVF objects, the IsSubspace returns true if solution space of L1 is a subspace of solution space of L2. False otherwise.
 • More precisely, the method returns true if at each point x0, the local solution space of L1 at x0 is a subspace of the local solution space of L2 at x0.
 • This method eventually pass tasks down to IsSubspace of the LHPDE object
 • The determining systems of input arguments L1 and L2 need not have the same dependent variable names or dependencies.
 • This method is associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{ξ}\left(x,y\right),\mathrm{η}\left(x,y\right)\right]\right):$
 > $V≔\mathrm{VectorField}\left(\mathrm{ξ}\left(x,y\right){\mathrm{D}}_{x}+\mathrm{η}\left(x,y\right){\mathrm{D}}_{y},\mathrm{space}=\left[x,y\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (1)
 > $\mathrm{C2}≔\mathrm{LHPDE}\left(\left[\frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ξ}\left(x,y\right)=0,\frac{{\partial }^{2}}{\partial y\partial x}\mathrm{ξ}\left(x,y\right)=0,\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ξ}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{η}\left(x,y\right)=-\left(\frac{\partial }{\partial y}\mathrm{ξ}\left(x,y\right)\right),\frac{\partial }{\partial y}\mathrm{η}\left(x,y\right)=\frac{\partial }{\partial x}\mathrm{ξ}\left(x,y\right)\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{ξ},\mathrm{η}\right]\right)$
 ${\mathrm{C2}}{≔}\left[{{\mathrm{\xi }}}_{{x}{,}{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{{\mathrm{\xi }}}_{{x}}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ξ}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{η}\left(x,y\right)=-\left(\frac{\partial }{\partial y}\mathrm{ξ}\left(x,y\right)\right),\frac{\partial }{\partial y}\mathrm{η}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{ξ}\left(x,y\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{ξ},\mathrm{η}\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (3)

We first construct two LAVF objects for C2 and E2

 > $\mathrm{LC2}≔\mathrm{LAVF}\left(V,\mathrm{C2}\right)$
 ${\mathrm{LC2}}{≔}\left[{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{{\mathrm{\eta }}}_{{y}}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}\right]\right\}$ (4)
 > $\mathrm{LE2}≔\mathrm{LAVF}\left(V,\mathrm{E2}\right)$
 ${\mathrm{LE2}}{≔}\left[{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (5)
 > $\mathrm{IsSubspace}\left(\mathrm{LE2},\mathrm{LC2}\right)$
 ${\mathrm{true}}$ (6)

The determining system of this LAVF object's dependent variables have different names and dependencies.

 > $\mathrm{Va}≔\mathrm{VectorField}\left(\mathrm{α}\left(y\right){\mathrm{D}}_{x}+\mathrm{β}\left(x\right){\mathrm{D}}_{y},\mathrm{space}=\left[x,y\right]\right)$
 ${\mathrm{Va}}{≔}{\mathrm{\alpha }}{}\left({y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\beta }}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (7)
 > $\mathrm{E2p}≔\mathrm{LHPDE}\left(\left[\frac{{ⅆ}^{2}}{ⅆ{y}^{2}}\mathrm{α}\left(y\right)=0,\frac{ⅆ}{ⅆx}\mathrm{β}\left(x\right)=-\left(\frac{ⅆ}{ⅆy}\mathrm{α}\left(y\right)\right)\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{α},\mathrm{β}\right]\right)$
 ${\mathrm{E2p}}{≔}\left[\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({y}\right){=}{0}{,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\beta }}{}\left({x}\right){=}{-}\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({y}\right)\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\alpha }}{}\left({y}\right){,}{\mathrm{\beta }}{}\left({x}\right)\right]$ (8)
 > $\mathrm{LE2p}≔\mathrm{LAVF}\left(\mathrm{Va},\mathrm{E2p}\right)$
 ${\mathrm{LE2p}}{≔}\left[{\mathrm{\alpha }}{}\left({y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\beta }}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({y}\right){=}{0}{,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\beta }}{}\left({x}\right){=}{-}\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({y}\right)\right]\right\}$ (9)
 > $\mathrm{IsSubspace}\left(\mathrm{LE2p},\mathrm{LC2}\right)$
 ${\mathrm{true}}$ (10)

Compatibility

 • The IsSubspace command was introduced in Maple 2020.