The Transformation command creates an internal data structure for a mapping between two frames.  This internal data structure contains information regarding the transformation (domain, range, prolongation order, transformation type (projectable, point, contact, differential substitution etc.), and the Jacobian of the transformation).  Once a mapping between frames has been encoded using the Transformation command, the mapping can then be using to transform vectors, differential forms and tensors using the Pushforward, Pullback, and PushPullTensor commands in the DifferentialGeometry package and in the Tensor subpackage.

If M and N are the names of initialized Lie algebras, then the second calling sequence can be used to define a linear transformation from M to N with matrix representation A.

This command is part of the DifferentialGeometry package, and so can be used in the form Transformation(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-Transformation.

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

$\mathrm{DGsetup}\left(\left[t\right]\,R\right)\:$$\mathrm{DGsetup}\left(\left[x\,y\right]\,M\right)\:$$\mathrm{DGsetup}\left(\left[u\,v\right]\,N\right)\:$$\mathrm{DGsetup}\left(\left[r\,\mathrm{\theta }\right]\,P\right)\:$$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,Q\right)\:$

$\mathrm{T1}≔\mathrm{Transformation}\left(R\,Q\,\left[x=2t\sin \left(t\right)\,y=3t\cos \left(t\right)\,z=\frac{t}{t+1}\right]\right)$

$\mathrm{T1}≔\left[x=2t\sin \left(t\right)\,y=3t\cos \left(t\right)\,z=\frac{t}{t+1}\right]$

$w≔\mathrm{evalc}\left({\left(x+\mathrm{I}y\right)}^3\right)$

$w≔\mathrm{I}\left(3x^{2}y-y^3\right)+x^3-3x{y}^2$

$\mathrm{T2}≔\mathrm{Transformation}\left(M\,N\,\left[u=\mathrm{evalc}\left(\mathrm{\Re }\left(w\right)\right)\,v=\mathrm{evalc}\left(\mathrm{\Im }\left(w\right)\right)\right]\right)$

$\mathrm{T2}≔\left[u=x^3-3x{y}^2\,v=3x^{2}y-y^3\right]$

$\mathrm{T3}≔\mathrm{Transformation}\left(M\,M\,\left[x=\mathrm{evalc}\left(\mathrm{\Re }\left(w\right)\right)\,y=\mathrm{evalc}\left(\mathrm{\Im }\left(w\right)\right)\right]\right)$

$\mathrm{T3}≔\left[x=x^3-3x{y}^2\,y=3x^{2}y-y^3\right]$

$\mathrm{T4}≔\mathrm{Transformation}\left(P\,M\,\left[x=r\cos \left(\mathrm{\theta }\right)\,y=r\sin \left(\mathrm{\theta }\right)\right]\right)$

$\mathrm{T4}≔\left[x=r\cos \left(\mathrm{\theta }\right)\,y=r\sin \left(\mathrm{\theta }\right)\right]$

$\mathrm{T5}≔\mathrm{Transformation}\left(P\,Q\,\left[x=r\cos \left(\mathrm{\theta }\right)\,y=r\mathrm{\theta }\,z=\mathrm{sqrt}\left(1+x^2\right)\right]\right)$

$\mathrm{T5}≔\left[x=r\cos \left(\mathrm{\theta }\right)\,y=r\mathrm{\theta }\,z=\sqrt{x^2+1}\right]$

$\mathrm{T6}≔\mathrm{Transformation}\left(Q\,M\,\left[x=x\,y=y\right]\right)$

$\mathrm{T6}≔\left[x=x\,y=y\right]$

$\mathrm{T2}$

$\left[u=x^3-3x{y}^2\,v=3x^{2}y-y^3\right]$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{T2}\,''DomainFrame''\right)$

$M$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{T2}\,''RangeFrame''\right)$

$N$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{T2}\,''JacobianMatrix''\right)$

$\left[\begin{array}{cc}3x^2-3y^2& -6xy\\ 6xy& 3x^2-3y^2\end{array}\right]$

$\mathrm{ChangeFrame}\left(M\right)$

$Q$

$\mathrm{Fr}≔\left[\mathrm{D\_x}+y\mathrm{D\_y}\,\mathrm{D\_y}\right]$

$\mathrm{Fr}≔\left[\mathrm{D\_y}y+\mathrm{D\_x}\,\mathrm{D\_y}\right]$

$\mathrm{FrData}≔\mathrm{FrameData}\left(\mathrm{Fr}\,\mathrm{M1}\right)$

$\mathrm{FrData}≔\left[\left[\mathrm{E1}\,\mathrm{E2}\right]=-\mathrm{E2}\right]$

$\mathrm{DGsetup}\left(\mathrm{FrData}\,\mathrm{verbose}\right)$

$\mathrm{The\; following\; coordinates\; have\; been\; protected:}$

$\left[x\,y\right]$

$\mathrm{The\; following\; vector\; fields\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{E1}\,\mathrm{E2}\right]$

$\mathrm{The\; following\; differential\; 1-forms\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{\Θ1}\,\mathrm{\Θ2}\right]$

$\mathrm{frame\; name:\; M1}$

$\mathrm{T7}≔\mathrm{Transformation}\left(M\,M\,\left[x=2x+y\,y=-x+3y\right]\right)$

$\mathrm{T7}≔\left[x=2x+y\,y=-x+3y\right]$

$\mathrm{J7}≔\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{T7}\,''JacobianMatrix''\right)$

$\mathrm{J7}≔\left[\begin{array}{cc}2& 1\\ \mathrm{-1}& 3\end{array}\right]$

$\mathrm{T8}≔\mathrm{Transformation}\left(M\,\mathrm{M1}\,\left[x=2x+y\,y=-x+3y\right]\right)$

$\mathrm{T8}≔\left[x=2x+y\,y=-x+3y\right]$

$\mathrm{J8}≔\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{T8}\,''JacobianMatrix''\right)$

$\mathrm{J8}≔\left[\begin{array}{cc}2& 1\\ 2x-6y-1& x-3y+3\end{array}\right]$

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

$\mathrm{L1}≔\mathrm{LieAlgebraData}\left(\left[\left[\mathrm{x1}\,\mathrm{x4}\right]=\mathrm{x1}\,\left[\mathrm{x2}\,\mathrm{x4}\right]=\mathrm{x1}+\mathrm{x2}\,\left[\mathrm{x3}\,\mathrm{x4}\right]=\mathrm{x2}+\mathrm{x3}\right]\,\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]\,\mathrm{Alg1}\right)$

$\mathrm{L1}≔\left[\left[\mathrm{e1}\,\mathrm{e4}\right]=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]=\mathrm{e1}+\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]=\mathrm{e2}+\mathrm{e3}\right]$

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

$\mathrm{Lie\; algebra:\; Alg1}$

$\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\left[\left[\mathrm{x1}\,\mathrm{x2}\right]=-\mathrm{x2}-\mathrm{x3}\,\left[\mathrm{x1}\,\mathrm{x3}\right]=-\mathrm{x3}\right]\,\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\right]\,\mathrm{Alg2}\right)$

$\mathrm{L2}≔\left[\left[\mathrm{e1}\,\mathrm{e2}\right]=-\mathrm{e2}-\mathrm{e3}\,\left[\mathrm{e1}\,\mathrm{e3}\right]=-\mathrm{e3}\right]$

$\mathrm{DGsetup}\left(\mathrm{L2}\,\left[f\right]\,\left[\mathrm{\theta }\right]\right)$

$\mathrm{Lie\; algebra:\; Alg2}$

$A≔\mathrm{Matrix}\left(\left[\left[0\,0\,0\,1\right]\,\left[0\,0\,1\,0\right]\,\left[0\,1\,0\,-1\right]\right]\right)$

$A≔\left[\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& \mathrm{-1}\end{array}\right]$

$\mathrm{\phi }≔\mathrm{Transformation}\left(\mathrm{Alg1}\,\mathrm{Alg2}\,A\right)$

$\mathrm{\phi }≔\left[\left[\mathrm{e1}\,0\mathrm{f1}\right]\,\left[\mathrm{e2}\,\mathrm{f3}\right]\,\left[\mathrm{e3}\,\mathrm{f2}\right]\,\left[\mathrm{e4}\,\mathrm{f1}-\mathrm{f3}\right]\right]$

$\mathrm{Query}\left(\mathrm{Alg1}\,\mathrm{Alg2}\,A\,''Homomorphism''\right)$

$\mathrm{true}$