These commands provide, within the DifferentialGeometry environment, the basic arithmetical operations for Matrices or Vectors of: vectors, differential forms, or tensors.  They are particularly useful for curvature calculations for connections on principle bundles of matrix groups.

These commands are part of the DifferentialGeometry:-Tools package, and so can be used in the form described above only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.

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

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

$A≔\mathrm{Vector}\left(\mathrm{evalDG}\left(\left[\mathrm{dx}-\mathrm{dy}\,\mathrm{dy}+\mathrm{dx}\right]\right)\right)$

$A≔\left[\begin{array}{c}\mathrm{dx}-\mathrm{dy}\\ \mathrm{dx}+\mathrm{dy}\end{array}\right]$

$B≔\mathrm{Vector}\left(\mathrm{evalDG}\left(\left[\mathrm{dx}+2\mathrm{dy}\,\mathrm{dx}+3\mathrm{dy}\right]\right)\right)$

$B≔\left[\begin{array}{c}\mathrm{dx}+2\mathrm{dy}\\ \mathrm{dx}+3\mathrm{dy}\end{array}\right]$

$C≔\mathrm{Matrix}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)$

$C≔\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

$E≔\mathrm{LinearAlgebra}:-\mathrm{Transpose}\left(A\right)$

$E≔\left[\begin{array}{cc}\mathrm{dx}-\mathrm{dy}& \mathrm{dx}+\mathrm{dy}\end{array}\right]$

$F≔\mathrm{Matrix}\left(\mathrm{evalDG}\left(\left[\left[\mathrm{dx}-\mathrm{dz}\,\mathrm{dy}\right]\,\left[\mathrm{dz}\,\mathrm{dx}+\mathrm{dy}+3\mathrm{dz}\right]\right]\right)\right)$

$F≔\left[\begin{array}{cc}\mathrm{dx}-\mathrm{dz}& \mathrm{dy}\\ \mathrm{dz}& \mathrm{dx}+\mathrm{dy}+3\mathrm{dz}\end{array}\right]$

$A\&MatrixPlusB$

$\left[\begin{array}{c}2\mathrm{dx}+\mathrm{dy}\\ 2\mathrm{dx}+4\mathrm{dy}\end{array}\right]$

$A\&MatrixMinusB$

$\left[\begin{array}{c}-3\mathrm{dy}\\ -2\mathrm{dy}\end{array}\right]$

$a\&MatrixMultA$

$\left[\begin{array}{c}a\mathrm{dx}-a\mathrm{dy}\\ a\mathrm{dx}+a\mathrm{dy}\end{array}\right]$

$C\&MatrixMultA$

$\left[\begin{array}{c}3\mathrm{dx}+\mathrm{dy}\\ 7\mathrm{dx}+\mathrm{dy}\end{array}\right]$

$E\&MatrixMultC$

$\left[\begin{array}{cc}4\mathrm{dx}+2\mathrm{dy}& 6\mathrm{dx}+2\mathrm{dy}\end{array}\right]$

$E\&MatrixWedgeB$

$\left[\begin{array}{c}5\mathrm{dx}\bigwedge \mathrm{dy}\end{array}\right]$

$C\&MatrixMultF$

$\left[\begin{array}{cc}\mathrm{dx}+\mathrm{dz}& 2\mathrm{dx}+3\mathrm{dy}+6\mathrm{dz}\\ 3\mathrm{dx}+\mathrm{dz}& 4\mathrm{dx}+7\mathrm{dy}+12\mathrm{dz}\end{array}\right]$

$F\&MatrixWedgeA$

$\left[\begin{array}{c}-2\mathrm{dx}\bigwedge \mathrm{dy}+\mathrm{dx}\bigwedge \mathrm{dz}-\mathrm{dy}\bigwedge \mathrm{dz}\\ -4\mathrm{dx}\bigwedge \mathrm{dz}-2\mathrm{dy}\bigwedge \mathrm{dz}\end{array}\right]$

$F\&MatrixWedgeF$

$\left[\begin{array}{cc}\mathrm{dy}\bigwedge \mathrm{dz}& 4\mathrm{dy}\bigwedge \mathrm{dz}\\ \mathrm{dy}\bigwedge \mathrm{dz}& -\mathrm{dy}\bigwedge \mathrm{dz}\end{array}\right]$