This procedures finds all vector fields X in the span of S such that Phi_*(X) = Y, where Phi_* is the Jacobian of Phi. If Phi is a local immersion, then Phi_* is injective and the vector X, if it exists, is unique.  If  Phi is not a local immersion, then the optional argument freevariable = k can be used to specify the name of the indexed variable that will be used to parameterize the possibilities for X.

The kernel of  Phi_* can be computed by taking Y to be the zero vector.

If no vector field X exists such that Phi_*(X) = Y, then NULL is returned.

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

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

$\mathrm{DGsetup}\left(\left[x\,y\,z\,w\right]\,\mathrm{E4}\right)\:$$\mathrm{DGsetup}\left(\left[r\,s\,t\right]\,\mathrm{E3}\right)\:$

$\mathrm{\Φ1}≔\mathrm{Transformation}\left(\mathrm{E3}\,\mathrm{E4}\,\left[x=\frac{2r}{1+r^2+s^2+t^2}\,y=\frac{2s}{1+r^2+s^2+t^2}\,z=\frac{2t}{1+r^2+s^2+t^2}\,w=\frac{1-\left(r^2+s^2+t^2\right)}{1+r^2+s^2+t^2}\right]\right)$

$\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E3}\,0\right]\,\left[\mathrm{E4}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{ccc}\frac{2}{r^2\+s^2\+t^2\+1}-\frac{4r^2}{{\left(r^2\+s^2\+t^2\+1\right)}^2}& -\frac{4rs}{{\left(r^2\+s^2\+t^2\+1\right)}^2}& -\frac{4rt}{{\left(r^2\+s^2\+t^2\+1\right)}^2}\\ -\frac{4rs}{{\left(r^2\+s^2\+t^2\+1\right)}^2}& \frac{2}{r^2\+s^2\+t^2\+1}-\frac{4s^2}{{\left(r^2\+s^2\+t^2\+1\right)}^2}& -\frac{4st}{{\left(r^2\+s^2\+t^2\+1\right)}^2}\\ -\frac{4rt}{{\left(r^2\+s^2\+t^2\+1\right)}^2}& -\frac{4st}{{\left(r^2\+s^2\+t^2\+1\right)}^2}& \frac{2}{r^2\+s^2\+t^2\+1}-\frac{4t^2}{{\left(r^2\+s^2\+t^2\+1\right)}^2}\\ -\frac{2r}{r^2\+s^2\+t^2\+1}-\frac{2\left(-r^2-s^2-t^2\+1\right)r}{{\left(r^2\+s^2\+t^2\+1\right)}^2}& -\frac{2s}{r^2\+s^2\+t^2\+1}-\frac{2\left(-r^2-s^2-t^2\+1\right)s}{{\left(r^2\+s^2\+t^2\+1\right)}^2}& -\frac{2t}{r^2\+s^2\+t^2\+1}-\frac{2\left(-r^2-s^2-t^2\+1\right)t}{{\left(r^2\+s^2\+t^2\+1\right)}^2}\end{array}\right]\right]\right]\,\left[\left[\frac{2r}{r^2\+s^2\+t^2\+1}\,x\right]\,\left[\frac{2s}{r^2\+s^2\+t^2\+1}\,y\right]\,\left[\frac{2t}{r^2\+s^2\+t^2\+1}\,z\right]\,\left[\frac{-r^2-s^2-t^2\+1}{r^2\+s^2\+t^2\+1}\,w\right]\right]\right]\right)$

$\mathrm{Y1}≔\mathrm{evalDG}\left(-y\mathrm{D\_x}+x\mathrm{D\_y}+w\mathrm{D\_z}-z\mathrm{D\_w}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E4}\,\left[\right]\right]\,\left[\left[\left[1\right]\,-y\right]\,\left[\left[2\right]\,x\right]\,\left[\left[3\right]\,w\right]\,\left[\left[4\right]\,-z\right]\right]\right]\right)$

$\mathrm{Y2}≔\mathrm{evalDG}\left(-z\mathrm{D\_x}-w\mathrm{D\_y}+x\mathrm{D\_z}+y\mathrm{D\_w}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E4}\,\left[\right]\right]\,\left[\left[\left[1\right]\,-z\right]\,\left[\left[2\right]\,-w\right]\,\left[\left[3\right]\,x\right]\,\left[\left[4\right]\,y\right]\right]\right]\right)$

$\mathrm{X1}≔\mathrm{PullbackVector}\left(\mathrm{\Φ1}\,\mathrm{Y1}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E3}\,\left[\right]\right]\,\left[\left[\left[1\right]\,rt-s\right]\,\left[\left[2\right]\,st\+r\right]\,\left[\left[3\right]\,-\frac{1}{2}{r}^2-\frac{1}{2}{s}^2\+\frac{1}{2}{t}^2\+\frac{1}{2}\right]\right]\right]\right)$

$\mathrm{X2}≔\mathrm{PullbackVector}\left(\mathrm{\Φ1}\,\mathrm{Y2}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E3}\,\left[\right]\right]\,\left[\left[\left[1\right]\,-rs-t\right]\,\left[\left[2\right]\,\frac{1}{2}{r}^2-\frac{1}{2}{s}^2\+\frac{1}{2}{t}^2-\frac{1}{2}\right]\,\left[\left[3\right]\,-st\+r\right]\right]\right]\right)$

$\mathrm{Y3}≔\mathrm{LieBracket}\left(\mathrm{Y1}\,\mathrm{Y2}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E4}\,\left[\right]\right]\,\left[\left[\left[1\right]\,-2w\right]\,\left[\left[2\right]\,2z\right]\,\left[\left[3\right]\,-2y\right]\,\left[\left[4\right]\,2x\right]\right]\right]\right)$

$\mathrm{X3}≔\mathrm{PullbackVector}\left(\mathrm{\Φ1}\,\mathrm{Y3}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E3}\,\left[\right]\right]\,\left[\left[\left[1\right]\,-r^2\+s^2\+t^2-1\right]\,\left[\left[2\right]\,-2rs\+2t\right]\,\left[\left[3\right]\,-2rt-2s\right]\right]\right]\right)$

$\mathrm{X3}\&minus\mathrm{LieBracket}\left(\mathrm{X1}\,\mathrm{X2}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E3}\,\left[\right]\right]\,\left[\left[\left[1\right]\,0\right]\right]\right]\right)$

$\mathrm{DGsetup}\left(\left[x\,y\,z\,w\right]\,\mathrm{E4}\right)\:$$\mathrm{DGsetup}\left(\left[t\,u\,v\right]\,\mathrm{E2}\right)\:$

$\mathrm{\Φ2}≔\mathrm{Transformation}\left(\mathrm{E4}\,\mathrm{E2}\,\left[u=x\,v=z\,t=1\right]\right)$

$\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E4}\,0\right]\,\left[\mathrm{E2}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{rrrr}0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right]\right]\right]\,\left[\left[1\,t\right]\,\left[x\,u\right]\,\left[z\,v\right]\right]\right]\right)$

$\mathrm{Y4}≔\mathrm{D\_u}$

$\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E2}\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)$

$\mathrm{X4}≔\mathrm{PullbackVector}\left(\mathrm{\Φ2}\,\mathrm{Y4}\,\mathrm{freevariable}='s'\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E4}\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,s_1\right]\,\left[\left[4\right]\,s_2\right]\right]\right]\right)$

$\mathrm{X5}≔\mathrm{PullbackVector}\left(\mathrm{\Φ2}\,\mathrm{D\_t}\,\mathrm{freevariable}='s'\right)$

$\mathrm{X5}≔\left(\right)$

$\mathrm{X6}≔\mathrm{PullbackVector}\left(\mathrm{\Φ2}\,\mathrm{Y4}\,\left[\mathrm{D\_x}\,\mathrm{D\_y}\right]\,\mathrm{freevariable}='s'\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E4}\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,s_1\right]\right]\right]\right)$

$\mathrm{X7}≔\mathrm{PullbackVector}\left(\mathrm{\Φ2}\,\mathrm{Y4}\,\left[\mathrm{D\_x}\right]\,\mathrm{freevariable}='s'\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E4}\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)$