ApplyTransformation(Phi, pt) returns the coordinates of the point Phi(pt) in N.

The second argument is of the form [a1, a2, ...] or [x1 = a1, x2 = a2, ...], where x1, x2, ... are the coordinates on M.

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

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

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

$\mathrm{\Phi }≔\mathrm{Transformation}\left(M\,N\,\left[u=x^2+y^2+z^2\,v=xy+yz+xz\right]\right)$

$\mathrm{\Phi }≔\left[u=x^2+y^2+z^2\,v=xy+xz+yz\right]$

$\mathrm{p1}≔\left[1\,2\,3\right]$

$\mathrm{p1}≔\left[1\,2\,3\right]$

$\mathrm{ApplyTransformation}\left(\mathrm{\Phi }\,\mathrm{p1}\right)$

$\left[14\,11\right]$

$\mathrm{p2}≔\left[x=5\,y=0\,z=1\right]$

$\mathrm{p2}≔\left[x=5\,y=0\,z=1\right]$

$\mathrm{ApplyTransformation}\left(\mathrm{\Phi }\,\mathrm{p2}\right)$

$\left[u=26\,v=5\right]$