The command Retrieve() returns the calling sequences for all the tables in the DifferentialGeometry Library.

Each table in the DifferentialGeometry Library is uniquely specified by its call sequence author, n.  The command Retrieve(author, n) returns all the indices for the table specified by the given call sequence.

The command Retrieve(author, n, index, options) returns the data in the table for the given call sequence and table index.  The index refers to the exact reference scheme used by the author in the original source article or book.  The command returns the data from the source article or book in a format which can be immediately used by the appropriate DifferentialGeometry commands.

When an entry from a table of Lie algebras of vector fields is to be retrieved, the argument manifold = M is required.  Here M is the name of the manifold, initialized via DGsetup, upon which the vector fields are to be defined.

When an entry from a table of tensor fields is to be retrieved, the argument manifoldname = M is required.  Here M is the name of the manifold, initialized via DGsetup, upon which the tensor fields are to be defined.

When an entry from a table of Lie algebras is to be retrieved, an optional 4th argument can be used to specify the name that will be used to refer to the algebra.

When an entry from a table of differential equations is to be retrieved, the argument variables = variablelist is required.  The independent variables should be listed first and the dependent variables second.

The list of parameters appearing in the table entry can be obtained with the optional argument parameters = "yes".

The command Retrieve is part of the DifferentialGeometry:-Library package.  It can be used in the form Retrieve(...) only after executing the commands with(DifferentialGeometry) and with(Library), but can always be used by executing DifferentialGeometry:-Library:-Retrieve(...).

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Library}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$\mathrm{Retrieve}\left(\right)$

$\left[\left[''Doubrov''\,1\right]\,\left[''Gong''\,1\right]\,\left[''Gonzalez-Lopez''\,1\right]\,\left[''HawkingEllis''\,1\right]\,\left[''Kamke''\,1\right]\,\left[''Morozov''\,1\right]\,\left[''Mubarakyzanov''\,1\right]\,\left[''Mubarakyzanov''\,2\right]\,\left[''Mubarakyzanov''\,3\right]\,\left[''Olver''\,1\right]\,\left[''Petrov''\,1\right]\,\left[''Stephani''\,1\right]\,\left[''Turkowski''\,1\right]\,\left[''Turkowski''\,2\right]\,\left[''USU''\,2\right]\,\left[''Winternitz''\,1\right]\right]$

$\mathrm{Retrieve}\left(''Winternitz''\,1\right)$

$\left[\left[3\,0\right]\,\left[3\,1\right]\,\left[3\,2\right]\,\left[3\,3\right]\,\left[3\,4\right]\,\left[3\,5\right]\,\left[3\,6\right]\,\left[3\,7\right]\,\left[3\,8\right]\,\left[3\,9\right]\,\left[4\,0\right]\,\left[4\,1\right]\,\left[4\,2\right]\,\left[4\,3\right]\,\left[4\,4\right]\,\left[4\,5\right]\,\left[4\,6\right]\,\left[4\,7\right]\,\left[4\,8\right]\,\left[4\,9\right]\,\left[4\,10\right]\,\left[4\,11\right]\,\left[4\,12\right]\,\left[5\,0\right]\,\left[5\,1\right]\,\left[5\,2\right]\,\left[5\,3\right]\,\left[5\,4\right]\,\left[5\,5\right]\,\left[5\,6\right]\,\left[5\,7\right]\,\left[5\,8\right]\,\left[5\,9\right]\,\left[5\,10\right]\,\left[5\,11\right]\,\left[5\,12\right]\,\left[5\,13\right]\,\left[5\,14\right]\,\left[5\,15\right]\,\left[5\,16\right]\,\left[5\,17\right]\,\left[5\,18\right]\,\left[5\,19\right]\,\left[5\,20\right]\,\left[5\,21\right]\,\left[5\,22\right]\,\left[5\,23\right]\,\left[5\,24\right]\,\left[5\,25\right]\,\left[5\,26\right]\,\left[5\,27\right]\,\left[5\,28\right]\,\left[5\,29\right]\,\left[5\,30\right]\,\left[5\,31\right]\,\left[5\,32\right]\,\left[5\,33\right]\,\left[5\,34\right]\,\left[5\,35\right]\,\left[5\,36\right]\,\left[5\,37\right]\,\left[5\,38\right]\,\left[5\,39\right]\,\left[5\,40\right]\,\left[6\,1\right]\,\left[6\,2\right]\,\left[6\,3\right]\,\left[6\,4\right]\,\left[6\,5\right]\,\left[6\,6\right]\,\left[6\,7\right]\,\left[6\,8\right]\,\left[6\,9\right]\,\left[6\,10\right]\,\left[6\,11\right]\,\left[6\,12\right]\,\left[6\,13\right]\,\left[6\,14\right]\,\left[6\,15\right]\,\left[6\,16\right]\,\left[6\,17\right]\,\left[6\,18\right]\,\left[6\,19\right]\,\left[6\,20\right]\,\left[6\,21\right]\,\left[6\,22\right]\right]$

$\mathrm{L1}≔\mathrm{Retrieve}\left(''Winternitz''\,1\,\left[5\,40\right]\,\mathrm{Alg}\right)$

$\mathrm{L1}:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=2\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=-\mathrm{e2}\,\left[\mathrm{e1}\,\mathrm{e4}\right]\=\mathrm{e5}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=2\mathrm{e3}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e4}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=-\mathrm{e5}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=\mathrm{e4}\right]$

$\mathrm{DGsetup}\left(\mathrm{L1}\right)$

$\mathrm{Lie\; algebra:\; Alg}$

$\mathrm{LeviDecomposition}\left(\right)$

$\left[\left[\mathrm{e4}\,\mathrm{e5}\right]\,\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\right]\right]$

$\mathrm{with}\left(\mathrm{JetCalculus}\right)\:$$\mathrm{with}\left(\mathrm{GroupActions}\right)\:$

$\mathrm{DGsetup}\left(\left[x\right]\,\left[y\right]\,M\,1\right)$

$\mathrm{frame\; name:\; M}$

$V≔\mathrm{Retrieve}\left(''Gonzalez-Lopez''\,1\,\left[5\right]\,\mathrm{manifold}=M\right)$

$V:=\left[\mathrm{D\_x}\,{\mathrm{D\_y}}_{\[\]}\,x\mathrm{D\_x}-y_{\[\]}{\mathrm{D\_y}}_{\[\]}\,y_{\[\]}\mathrm{D\_x}\,x{\mathrm{D\_y}}_{\[\]}\right]$

$\mathrm{V2}≔\mathrm{map}\left(\mathrm{Prolong}\,V\,1\right)$

$\mathrm{V2}:=\left[\mathrm{D\_x}\,{\mathrm{D\_y}}_{\[\]}\,x\mathrm{D\_x}-y_{\[\]}{\mathrm{D\_y}}_{\[\]}-2y_1{\mathrm{D\_y}}_1\,y_{\[\]}\mathrm{D\_x}-y_1^2{\mathrm{D\_y}}_1\,x{\mathrm{D\_y}}_{\[\]}\+{\mathrm{D\_y}}_1\right]$

$\mathrm{Alg}≔\mathrm{IsotropySubalgebra}\left(\mathrm{V2}\,\left[x=a\,y\left[\right]=b\,y\left[1\right]=c\right]\right)$

$\mathrm{Alg}:=\left[\frac{\left(-y_{\[\]}b\+2y_{\[\]}ac-a{c}^{2}x-2bca\+a^2{c}^2\+b^2\right)\mathrm{D\_x}}{-2bca\+a^2{c}^2\+b^2}\+\frac{c^2\left(-bx\+a{y}_{\[\]}\right){\mathrm{D\_y}}_{\[\]}}{-2bca\+a^2{c}^2\+b^2}\+\frac{\left(-b{c}^2\+y_1^{2}b-2y_1^{2}ac\+2a{c}^2{y}_1\right){\mathrm{D\_y}}_1}{-2bca\+a^2{c}^2\+b^2}\,-\frac{\left(-bx\+a{y}_{\[\]}\right)\mathrm{D\_x}}{-2bca\+a^2{c}^2\+b^2}\+\frac{\left(-a{c}^{2}x\+2cxb-y_{\[\]}b-2bca\+a^2{c}^2\+b^2\right){\mathrm{D\_y}}_{\[\]}}{-2bca\+a^2{c}^2\+b^2}-\frac{\left(a{c}^2-2bc-a{y}_1^2\+2y_{1}b\right){\mathrm{D\_y}}_1}{-2bca\+a^2{c}^2\+b^2}\right]$

$\mathrm{nops}\left(\mathrm{Alg}\right)$

$2$

$\mathrm{DE}≔\mathrm{Retrieve}\left(''Kamke''\,1\,\left[7\,8\right]\,\mathrm{variables}=\left[x\,y\right]\right)$

$\mathrm{DE}:=4{y\left(x\right)}^2\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)-18y\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)\+15{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^3$

$S≔\mathrm{PDETools}:-\mathrm{Infinitesimals}\left(\mathrm{DE}\,\left[y\right]\right)$

$S:=\left[{\mathrm{\_\ξ}}_1\left(x\,y\right)\=x\,{\mathrm{\_\η}}_1\left(x\,y\right)\=0\right]\,\left[{\mathrm{\_\ξ}}_1\left(x\,y\right)\=1\,{\mathrm{\_\η}}_1\left(x\,y\right)\=0\right]\,\left[{\mathrm{\_\ξ}}_1\left(x\,y\right)\=0\,{\mathrm{\_\η}}_1\left(x\,y\right)\=y\right]\,\left[{\mathrm{\_\ξ}}_1\left(x\,y\right)\=0\,{\mathrm{\_\η}}_1\left(x\,y\right)\=y^{3\/2}\right]\,\left[{\mathrm{\_\ξ}}_1\left(x\,y\right)\=0\,{\mathrm{\_\η}}_1\left(x\,y\right)\=x{y}^{3\/2}\right]\,\left[{\mathrm{\_\ξ}}_1\left(x\,y\right)\=0\,{\mathrm{\_\η}}_1\left(x\,y\right)\=\frac{1}{2}{x}^2{y}^{3\/2}\right]\,\left[{\mathrm{\_\ξ}}_1\left(x\,y\right)\=\frac{1}{2}{x}^2\,{\mathrm{\_\η}}_1\left(x\,y\right)\=-2xy\right]$

$\mathrm{DGsetup}\left(\left[x\right]\,\left[y\right]\,M\right)\:$

$V≔\mathrm{convert}\left(\left[S\right]\,\mathrm{DGvector}\right)$

$V:=\left[x\mathrm{D\_x}\,\mathrm{D\_x}\,y\mathrm{D\_y}\,y^{3\/2}\mathrm{D\_y}\,x{y}^{3\/2}\mathrm{D\_y}\,\frac{1}{2}{x}^2{y}^{3\/2}\mathrm{D\_y}\,\frac{1}{2}{x}^2\mathrm{D\_x}-2xy\mathrm{D\_y}\right]$

$L≔\mathrm{LieAlgebraData}\left(V\,\mathrm{K7\_8}\,\mathrm{initialpointlist}=\left[\left[x=1\,y=1\right]\,\left[x=0\,y=1\right]\right]\right)$

$L:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=-\mathrm{e2}\,\left[\mathrm{e1}\,\mathrm{e5}\right]\=\mathrm{e5}\,\left[\mathrm{e1}\,\mathrm{e6}\right]\=2\mathrm{e6}\,\left[\mathrm{e1}\,\mathrm{e7}\right]\=\mathrm{e7}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=\mathrm{e4}\,\left[\mathrm{e2}\,\mathrm{e6}\right]\=\mathrm{e5}\,\left[\mathrm{e2}\,\mathrm{e7}\right]\=\mathrm{e1}-2\mathrm{e3}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\frac{1}{2}\mathrm{e4}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=\frac{1}{2}\mathrm{e5}\,\left[\mathrm{e3}\,\mathrm{e6}\right]\=\frac{1}{2}\mathrm{e6}\,\left[\mathrm{e4}\,\mathrm{e7}\right]\=\mathrm{e5}\,\left[\mathrm{e5}\,\mathrm{e7}\right]\=\mathrm{e6}\right]$

$\mathrm{DGsetup}\left(L\right)$

$\mathrm{Lie\; algebra:\; K7\_8}$

$\mathrm{LeviDecomposition}\left(\right)$

$\left[\left[\mathrm{e3}\,\mathrm{e4}\,\mathrm{e5}\,\mathrm{e6}\right]\,\left[\mathrm{e1}-2\mathrm{e3}\,\mathrm{e2}\,\mathrm{e7}\right]\right]$

$\mathrm{DGsetup}\left(\left[t\,x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{Retrieve}\left(''Stephani''\,1\,\left[12\,14\,1\right]\,\mathrm{manifoldname}=M\,\mathrm{output}=\left[''Fields''\right]\right)$

$\left[-\frac{{\ⅇ}^x\cos \left(\sqrt{3}x\right)\mathrm{dt}\mathrm{dt}}{{\mathrm{\_k}}^2}-\frac{{\ⅇ}^x\sin \left(\sqrt{3}x\right)\mathrm{dt}\mathrm{dz}}{{\mathrm{\_k}}^2}\+\frac{\mathrm{dx}\mathrm{dx}}{{\mathrm{\_k}}^2}\+\frac{{\ⅇ}^{-2x}\mathrm{dy}\mathrm{dy}}{{\mathrm{\_k}}^2}-\frac{{\ⅇ}^x\sin \left(\sqrt{3}x\right)\mathrm{dz}\mathrm{dt}}{{\mathrm{\_k}}^2}\+\frac{{\ⅇ}^x\cos \left(\sqrt{3}x\right)\mathrm{dz}\mathrm{dz}}{{\mathrm{\_k}}^2}\right]$

$\mathrm{Retrieve}\left(''Stephani''\,1\,\left[12\,14\,1\right]\,\mathrm{manifoldname}=M\,\mathrm{output}=\left[''NullTetrad''\right]\right)$

$\left[\left[-\frac{1}{4}\sqrt{2}\mathrm{\_k}{\ⅇ}^{-\frac{1}{2}x}\left(\cos \left(\frac{1}{2}\sqrt{3}x\right)-\sin \left(\frac{1}{2}\sqrt{3}x\right)\right)\mathrm{D\_t}-\frac{1}{4}\sqrt{2}\mathrm{\_k}{\ⅇ}^{-\frac{1}{2}x}\left(\sin \left(\frac{1}{2}\sqrt{3}x\right)\+\cos \left(\frac{1}{2}\sqrt{3}x\right)\right)\mathrm{D\_z}\,-\sqrt{2}\mathrm{\_k}{\ⅇ}^{-\frac{1}{2}x}\left(\sin \left(\frac{1}{2}\sqrt{3}x\right)\+\cos \left(\frac{1}{2}\sqrt{3}x\right)\right)\mathrm{D\_t}\+\sqrt{2}\mathrm{\_k}{\ⅇ}^{-\frac{1}{2}x}\left(\cos \left(\frac{1}{2}\sqrt{3}x\right)-\sin \left(\frac{1}{2}\sqrt{3}x\right)\right)\mathrm{D\_z}\,\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{\_k}\mathrm{D\_x}\+\frac{1}{2}\sqrt{2}\mathrm{\_k}{\ⅇ}^x\mathrm{D\_y}\,-\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{\_k}\mathrm{D\_x}\+\frac{1}{2}\sqrt{2}\mathrm{\_k}{\ⅇ}^x\mathrm{D\_y}\right]\right]$

$\mathrm{Retrieve}\left(''Stephani''\,1\,\left[12\,14\,1\right]\,\mathrm{manifoldname}=M\,\mathrm{output}=\left[''PetrovType''\right]\right)$

$\left[''I''\right]$

$\mathrm{Retrieve}\left(''Stephani''\,1\,\left[12\,14\,1\right]\,\mathrm{manifoldname}=M\,\mathrm{output}=\left[''KillingVectors''\right]\right)$

$\left[\left[\mathrm{D\_t}\,\mathrm{D\_y}\,\mathrm{D\_z}\,\left(-\frac{1}{2}\sqrt{3}z-\frac{1}{2}t\right)\mathrm{D\_t}\+\mathrm{D\_x}\+y\mathrm{D\_y}\+\left(\frac{1}{2}\sqrt{3}t-\frac{1}{2}z\right)\mathrm{D\_z}\right]\right]$

$L,P≔\mathrm{Retrieve}\left(''Winternitz''\,1\,\left[5\,17\right]\,\mathrm{Alg1}\,\mathrm{parameters}=''yes''\right)$

$L\,P:=\left[\left[\mathrm{e1}\,\mathrm{e5}\right]\=p\mathrm{e1}-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=\mathrm{e1}\+p\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=q\mathrm{e3}-s\mathrm{e4}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=s\mathrm{e3}\+q\mathrm{e4}\right]\,\left[p\,q\,s\right]$

$\mathrm{L1}≔\mathrm{eval}\left(L\,\left[P\left[1\right]=3\,P\left[2\right]=7\,P\left[3\right]=K\right]\right)$

$\mathrm{L1}:=\left[\left[\mathrm{e1}\,\mathrm{e5}\right]\=3\mathrm{e1}-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=\mathrm{e1}\+3\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=7\mathrm{e3}-K\mathrm{e4}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=K\mathrm{e3}\+7\mathrm{e4}\right]$