The procedure ComplementaryBasis(S, T) returns a list C of vectors, differential p-forms or tensors such that the span of [S, C] equals the span of the vectors, differential p-forms or tensors  defined by T.

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

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

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

$\mathrm{S1}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\right]$

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

$\mathrm{T1}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

$\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{C1}≔\mathrm{ComplementaryBasis}\left(\mathrm{S1}\,\mathrm{T1}\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{S2}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_x}\+\mathrm{D\_y}\right]$

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

$\mathrm{T2}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_x}\+\mathrm{D\_z}\,\mathrm{D\_x}-\mathrm{D\_y}\,\mathrm{D\_z}\,\mathrm{D\_u}\right]\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,-1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{C2}≔\mathrm{ComplementaryBasis}\left(\mathrm{S2}\,\mathrm{T2}\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{S3}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\right]$

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

$\mathrm{T3}≔\left[\mathrm{D\_x}\,\mathrm{D\_u}\,\mathrm{D\_v}\right]$

$\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[5\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{C3}≔\mathrm{ComplementaryBasis}\left(\mathrm{S3}\,\mathrm{T3}\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[5\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{S4}≔\mathrm{evalDG}\left(\left[\mathrm{dx}\&w\mathrm{dy}\,\mathrm{dx}\&w\mathrm{dz}\right]\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,3\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{T4}≔\mathrm{evalDG}\left(\left[\mathrm{dx}\&w\mathrm{dy}\,\mathrm{dx}\&w\mathrm{dz}\,\mathrm{dx}\&w\mathrm{du}\,\mathrm{dy}\&w\mathrm{dv}\right]\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[2\,5\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{C4}≔\mathrm{ComplementaryBasis}\left(\mathrm{S4}\,\mathrm{T4}\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[2\,5\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{S5}≔\mathrm{evalDG}\left(\left[\mathrm{dx}\&t\mathrm{D\_y}\,\mathrm{dx}\&t\mathrm{D\_z}\right]\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,3\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{T5}≔\mathrm{evalDG}\left(\left[\mathrm{dx}\&t\mathrm{D\_y}\,\mathrm{dx}\&t\mathrm{D\_z}\,\mathrm{dx}\&t\mathrm{D\_u}\,\mathrm{dy}\&t\mathrm{D\_v}\right]\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,5\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{C5}≔\mathrm{ComplementaryBasis}\left(\mathrm{S5}\,\mathrm{T5}\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,5\right]\,1\right]\right]\right]\right)\right]$