The command DGmap is similar to the command map.  DGmap will apply the procedure f to the coefficients of the object X.  The integer n indicates the position of the coefficients of X in the argument list of  f.  Thus DGmap(1, f, X, arg1, arg2, ..., argN) will replace the coefficient C of X with  f(C, arg1, arg2, ..., argN); DGmap(2, f, X, arg1, arg2, ..., argN) will replace the coefficient C of X with f(arg1, C, arg2, ..., argN); and so on.

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

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

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

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

$X≔\mathrm{evalDG}\left(\left(\mathrm{C1}x+\mathrm{C2}\right)\mathrm{D\_x}+\mathrm{C1}y\mathrm{D\_y}\right)$

$X≔\left(\mathrm{C1}x+\mathrm{C2}\right)\mathrm{D\_x}+\mathrm{C1}y\mathrm{D\_y}$

$\mathrm{DGmap}\left(1\,\mathrm{diff}\,X\,\mathrm{C1}\right)$

$x\mathrm{D\_x}+y\mathrm{D\_y}$

$\mathrm{DGmap}\left(1\,\mathrm{diff}\,X\,\mathrm{C2}\right)$

$\mathrm{D\_x}$

$\mathrm{\omega }≔\mathrm{evalDG}\left(t^2{x}^2\mathrm{dx}-t^{3}x{y}^2\mathrm{dy}\right)$

$\mathrm{\omega }≔x^2{t}^2\mathrm{dx}-y^{2}x{t}^3\mathrm{dy}$

$\mathrm{DGmap}\left(1\,\mathrm{int}\,\mathrm{\omega }\,t=0..1\right)$

$\frac{x^2\mathrm{dx}}{3}-\frac{y^{2}x\mathrm{dy}}{4}$

$T≔\mathrm{evalDG}\left(\frac{\exp \left(x\right)-1}{x}\mathrm{dx}\&t\mathrm{dx}+\frac{\sin \left(2x\right)}{x}\mathrm{dy}\&t\mathrm{dy}\right)$

$T≔\frac{\left({\ⅇ}^x-1\right)\mathrm{dx}}{x}\mathrm{dx}+\frac{\sin \left(2x\right)\mathrm{dy}}{x}\mathrm{dy}$

$\mathrm{DGmap}\left(1\,\mathrm{limit}\,T\,x=0\right)$

$\mathrm{dx}\mathrm{dx}+2\mathrm{dy}\mathrm{dy}$

$\mathrm{\Phi }≔\mathrm{Transformation}\left(M\,M\,\left[x=sx+\left(s-1\right)y\,y=sy\right]\right)$

$\mathrm{\Phi }≔\left[x=sx+\left(s-1\right)y\,y=sy\right]$

$\mathrm{DGmap}\left(2\,\mathrm{subs}\,\mathrm{\Phi }\,\left\{s=1\right\}\right)$

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