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.

$\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right)\:$

$V≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x\,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x\,y\right)\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$V≔\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{C2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\,x\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{C2}≔\left[{\mathrm{\xi }}_{x,x}=0\,{\mathrm{\xi }}_{x,y}=0\,{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y={\mathrm{\xi }}_x\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{E2}≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{LC2}≔\mathrm{LAVF}\left(V\,\mathrm{C2}\right)$

$\mathrm{LC2}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_{y,y}=0\,{\mathrm{\xi }}_x={\mathrm{\eta }}_y\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\right]\right\}$

$\mathrm{LE2}≔\mathrm{LAVF}\left(V\,\mathrm{E2}\right)$

$\mathrm{LE2}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{IsSubspace}\left(\mathrm{LE2}\,\mathrm{LC2}\right)$

$\mathrm{true}$

$\mathrm{Va}≔\mathrm{VectorField}\left(\mathrm{\alpha }\left(y\right)\mathrm{D}\left[x\right]+\mathrm{\beta }\left(x\right)\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$\mathrm{Va}≔\mathrm{\alpha }\left(y\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\beta }\left(x\right)\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{E2p}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\alpha }\left(y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\beta }\left(x\right)\,x\right)=-\mathrm{diff}\left(\mathrm{\alpha }\left(y\right)\,y\right)\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\alpha }\,\mathrm{\beta }\right]\right)$

$\mathrm{E2p}≔\left[\frac{{\ⅆ}^2}{\ⅆy^2}\mathrm{\alpha }\left(y\right)=0\,\frac{\ⅆ}{\ⅆx}\mathrm{\beta }\left(x\right)=-\frac{\ⅆ}{\ⅆy}\mathrm{\alpha }\left(y\right)\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\alpha }\left(y\right)\,\mathrm{\beta }\left(x\right)\right]$

$\mathrm{LE2p}≔\mathrm{LAVF}\left(\mathrm{Va}\,\mathrm{E2p}\right)$

$\mathrm{LE2p}≔\left[\mathrm{\alpha }\left(y\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\beta }\left(x\right)\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\frac{{\ⅆ}^2}{\ⅆy^2}\mathrm{\alpha }\left(y\right)=0\,\frac{\ⅆ}{\ⅆx}\mathrm{\beta }\left(x\right)=-\frac{\ⅆ}{\ⅆy}\mathrm{\alpha }\left(y\right)\right]\right\}$

$\mathrm{IsSubspace}\left(\mathrm{LE2p}\,\mathrm{LC2}\right)$

$\mathrm{true}$

The IsSubspace command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.