This worksheet provides additional information for working with abstract differential forms, that is, differential forms which are defined abstractly without reference to any underlying system of coordinates. This new functionality of DifferentialGeometry is intended to supercede the difforms package.

There are 2 different ways of using DifferentialGeometry to calculate with abstract differential forms.  The first way mirrors the scenario current provided by difforms -- one defines a list of forms (using DGsetup) by simply specifying their degrees. Then one can calculate wedge products and exterior derivatives of these forms. Equations for exterior derivatives can be specified. For the second method one indicates which of the 1-forms being defined constitute a co-frame for the underlying manifold.  In this setting, the vector fields dual to the given 1-forms are automatically created by DGsetup.  Wedge products, interior products, exterior derivatives and Lie derivatives can all be computed.

The functionalities provided by DGzip, GetComponents, Annihilator, DGbasis and DGinfo are available as appropriate.

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

$\mathrm{DGsetup}\left(\'\left[f\=\mathrm{dgform}\left(0\right)\,\mathrm{\α}\=\mathrm{dgform}\left(1\right)\,\mathrm{\β}\=\mathrm{dgform}\left(1\right)\,\mathrm{\σ}\=\mathrm{dgform}\left(2\right)\right]\'\,\left[\right]\,M\right)$

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

$\mathrm{DGinfo}\left(''AbstractForms''\right)$

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

$\mathrm{\ω}≔\mathrm{evalDG}\left(2\mathrm{\α}\&wedge\mathrm{\β}\+4\mathrm{\σ}\right)$

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

$\mathrm{DGinfo}\left(\mathrm{\ω}\,''CoefficientSet''\right)$

$\left\{2\,4\right\}$

$\mathrm{DGinfo}\left(\mathrm{\ω}\,''CoefficientList''\,\left[\mathrm{\σ}\right]\right)$

$\left[4\right]$

$\mathrm{DGinfo}\left(''AbstractForms''\right)$

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

$\mathrm{\Ω}≔\mathrm{evalDG}\left(\left[\mathrm{\α}\&w\mathrm{\β}\,\mathrm{\σ}\right]\right)$

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

$\mathrm{zeta}≔\mathrm{DGzip}\left(\left[3\,5\right]\,\mathrm{\Ω}\,''plus''\right)$

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

$\mathrm{GetComponents}\left(\mathrm{zeta}\,\mathrm{\Ω}\right)$

$\left[3\,5\right]$

$\mathrm{\ρ}≔\mathrm{ExteriorDerivative}\left(\mathrm{\α}\right)$

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

$\mathrm{DGinfo}\left(''AbstractForms''\right)$

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

$\mathrm{ExteriorDerivative}\left(\mathrm{\ρ}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,3\right]\,\left[\left[\left[1\,3\right]\,0\right]\right]\right]\right)$

$\mathrm{DGsetup}\left(M\,\left[\right]\,\left[d\left(f\right)\=f\mathrm{\α}\,d\left(\mathrm{\β}\right)\=4\mathrm{\σ}\+5\mathrm{\α}\&wedge\mathrm{\β}\right]\right)$

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

$\mathrm{ExteriorDerivative}\left(f\mathrm{\β}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[3\right]\,4f\right]\,\left[\left[1\,2\right]\,6f\right]\right]\right]\right)$

$\mathrm{RemoveFrame}\left(M\right)$

$0$

$\mathrm{DGsetup}\left(\left[f\=\mathrm{dgform}\left(0\right)\,g\=\mathrm{dgform}\left(0\right)\,\left[\mathrm{\ω1}\,\mathrm{\ω2}\,\mathrm{\ω3}\right]\,\'\mathrm{\α}\'\=\mathrm{dgform}\left(2\right)\,\'\mathrm{\β}\'\=\mathrm{dgform}\left(3\right)\right]\,\left[d\left(\mathrm{\ω1}\right)\=\mathrm{\ω2}\&w\mathrm{\ω3}\right]\,N\right)$

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

$\mathrm{DGinfo}\left(''AbstractForms''\right)$

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

$\mathrm{DGinfo}\left(''FrameBaseVectors''\right)$

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

$\mathrm{DGinfo}\left(''FrameBaseForms''\right)$

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

$X≔\mathrm{evalDG}\left(a\mathrm{D\_omega1}\+b\mathrm{D\_omega2}\+c\mathrm{D\_omega3}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,N\,\left[\right]\right]\,\left[\left[\left[1\right]\,a\right]\,\left[\left[2\right]\,b\right]\,\left[\left[3\right]\,c\right]\right]\right]\right)$

$\mathrm{Hook}\left(X\,\mathrm{\ω1}\&w\mathrm{\ω3}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\right]\,\left[\left[\left[1\right]\,-c\right]\,\left[\left[3\right]\,a\right]\right]\right]\right)$

$\mathrm{Hook}\left(\mathrm{D\_omega1}\,\mathrm{\β}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,N\,2\right]\,\left[\left[\left[6\right]\,1\right]\right]\right]\right)$

$\mathrm{Hook}\left(\mathrm{D\_omega2}\,\mathrm{\β}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,N\,2\right]\,\left[\left[\left[7\right]\,1\right]\right]\right]\right)$

$\mathrm{Hook}\left(\mathrm{D\_omega3}\,\mathrm{\β}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,N\,2\right]\,\left[\left[\left[8\right]\,1\right]\right]\right]\right)$

$\mathrm{DGinfo}\left(''AbstractForms''\right)$

$\left[f\,g\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,2\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,3\right]\,\left[\left[\left[5\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,2\,''hook''\,\left[\left[1\right]\,5\right]\right]\,\left[\left[\left[6\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,2\,''hook''\,\left[\left[2\right]\,5\right]\right]\,\left[\left[\left[7\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,2\,''hook''\,\left[\left[3\right]\,5\right]\right]\,\left[\left[\left[8\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{Hook}\left(X\,\mathrm{\α}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\right]\,\left[\left[\left[9\right]\,a\right]\,\left[\left[10\right]\,b\right]\,\left[\left[11\right]\,c\right]\right]\right]\right)$

$\mathrm{Hook}\left(\mathrm{D\_omega1}\,\mathrm{i\_1beta}\right)$

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

$\mathrm{Hook}\left(\mathrm{D\_omega1}\,\mathrm{i\_2beta}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\right]\,\left[\left[\left[12\right]\,-1\right]\right]\right]\right)$

$\mathrm{Hook}\left(\mathrm{D\_omega3}\,\mathrm{i\_2beta}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\right]\,\left[\left[\left[13\right]\,1\right]\right]\right]\right)$

$\mathrm{Annihilator}\left(\left[\mathrm{D\_omega1}\+\mathrm{D\_omega3}\right]\right)$

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

$\mathrm{LieDerivative}\left(\mathrm{D\_omega2}\,\mathrm{\β}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,N\,3\right]\,\left[\left[\left[15\right]\,1\right]\,\left[\left[16\right]\,1\right]\right]\right]\right)$

$\mathrm{DGinfo}\left(''AbstractForms''\right)$

$\left[f\,g\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,2\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,3\right]\,\left[\left[\left[5\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,2\,''hook''\,\left[\left[1\right]\,5\right]\right]\,\left[\left[\left[6\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,2\,''hook''\,\left[\left[2\right]\,5\right]\right]\,\left[\left[\left[7\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,2\,''hook''\,\left[\left[3\right]\,5\right]\right]\,\left[\left[\left[8\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\,''hook''\,\left[\left[1\right]\,4\right]\right]\,\left[\left[\left[9\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\,''hook''\,\left[\left[2\right]\,4\right]\right]\,\left[\left[\left[10\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\,''hook''\,\left[\left[3\right]\,4\right]\right]\,\left[\left[\left[11\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\,''hook''\,\left[\left[1\,2\right]\,5\right]\right]\,\left[\left[\left[12\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\,''hook''\,\left[\left[2\,3\right]\,5\right]\right]\,\left[\left[\left[13\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,4\,''d''\,5\right]\,\left[\left[\left[14\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,3\,''hook''\,\left[\left[2\right]\,14\right]\right]\,\left[\left[\left[15\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,N\,3\,''d''\,7\right]\,\left[\left[\left[16\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{DGsetup}\left(N\,\left[\right]\,\left[d\left(\mathrm{\ω3}\right)\=0\,d\left(\mathrm{\β}\right)\=\mathrm{\α}\&w\mathrm{\α}\,\mathrm{hook}\left(\mathrm{D\_omega2}\,\mathrm{\β}\right)\=\mathrm{\α}\right]\right)$

$\mathrm{updated\; frame:\; N}$

$\mathrm{LieDerivative}\left(\mathrm{D\_omega2}\,\mathrm{\β}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,N\,3\right]\,\left[\left[\left[17\right]\,1\right]\,\left[\left[4\,10\right]\,2\right]\right]\right]\right)$

$\mathrm{LieBracket}\left(\mathrm{D\_omega2}\,\mathrm{D\_omega3}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,N\,\left[\right]\right]\,\left[\left[\left[1\right]\,-1\right]\,\left[\left[2\right]\,-\mathrm{i\_2\_3domega2}\right]\right]\right]\right)$

$\mathrm{DGsetup}\left(N\,\left[q\=\mathrm{dgform}\left(0\right)\right]\,\left[\mathrm{hook}\left(\left[\mathrm{D\_omega2}\,\mathrm{D\_omega3}\right]\,\mathrm{domega2}\right)\=q\right]\right)$

$\mathrm{updated\; frame:\; N}$

$\mathrm{LieBracket}\left(\mathrm{D\_omega2}\,\mathrm{D\_omega3}\right)$

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