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)\:$

$\mathrm{DGsetup}\left('\left[f=\mathrm{dgform}\left(0\right)\,\mathrm{\alpha }=\mathrm{dgform}\left(1\right)\,\mathrm{\beta }=\mathrm{dgform}\left(1\right)\,\mathrm{\sigma }=\mathrm{dgform}\left(2\right)\right]'\,\left[\right]\,M\right)$

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

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

$\left[f\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\sigma }\right]$

$\mathrm{\omega }≔\mathrm{evalDG}\left(2\mathrm{\alpha }\&wedge\mathrm{\beta }+4\mathrm{\sigma }\right)$

$\mathrm{\omega }≔2\mathrm{\alpha }\bigwedge \mathrm{\beta }+4\mathrm{\sigma }$

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

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

$\mathrm{DGinfo}\left(\mathrm{\omega }\,''CoefficientList''\,\left[\mathrm{\sigma }\right]\right)$

$\left[4\right]$

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

$\left[f\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\sigma }\right]$

$\mathrm{\Omega }≔\mathrm{evalDG}\left(\left[\mathrm{\alpha }\&w\mathrm{\beta }\,\mathrm{\sigma }\right]\right)$

$\mathrm{\Omega }≔\left[\mathrm{\alpha }\bigwedge \mathrm{\beta }\,\mathrm{\sigma }\right]$

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

$\mathrm{\zeta }≔3\mathrm{\alpha }\bigwedge \mathrm{\beta }+5\mathrm{\sigma }$

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

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

$\mathrm{\rho }≔\mathrm{ExteriorDerivative}\left(\mathrm{\alpha }\right)$

$\mathrm{\rho }≔d\mathrm{\alpha }$

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

$\left[f\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\sigma }\,d\mathrm{\alpha }\right]$

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

$0\mathrm{\alpha }\bigwedge \mathrm{\sigma }$

$\mathrm{DGsetup}\left(M\,\left[\right]\,\left[d\left(f\right)=f\mathrm{\alpha }\,d\left(\mathrm{\beta }\right)=4\mathrm{\sigma }+5\mathrm{\alpha }\&wedge\mathrm{\beta }\right]\right)$

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

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

$6\mathrm{\alpha }f\bigwedge \mathrm{\beta }+4\mathrm{\sigma }f$

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

$0$

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

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

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

$\left[f\,g\,\mathrm{\ω1}\,\mathrm{\ω2}\,\mathrm{\ω3}\,\mathrm{\alpha }\,\mathrm{\beta }\right]$

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

$\left[\mathrm{D\_}\mathrm{\ω1}\,\mathrm{D\_}\mathrm{\ω2}\,\mathrm{D\_}\mathrm{\ω3}\right]$

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

$\left[\mathrm{\ω1}\,\mathrm{\ω2}\,\mathrm{\ω3}\right]$

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

$X≔a\left(\mathrm{D\_}\mathrm{\ω1}\right)+b\left(\mathrm{D\_}\mathrm{\ω2}\right)+c\left(\mathrm{D\_}\mathrm{\ω3}\right)$

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

$\mathrm{\ω3}a-\mathrm{\ω1}c$

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

${\mathrm{\iota }}_1\mathrm{\beta }$

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

${\mathrm{\iota }}_2\mathrm{\beta }$

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

${\mathrm{\iota }}_3\mathrm{\beta }$

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

$\left[f\,g\,\mathrm{\ω1}\,\mathrm{\ω2}\,\mathrm{\ω3}\,\mathrm{\alpha }\,\mathrm{\beta }\,{\mathrm{\iota }}_1\mathrm{\beta }\,{\mathrm{\iota }}_2\mathrm{\beta }\,{\mathrm{\iota }}_3\mathrm{\beta }\right]$

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

${\mathrm{\iota }}_1\mathrm{\alpha }a+{\mathrm{\iota }}_2\mathrm{\alpha }b+{\mathrm{\iota }}_3\mathrm{\alpha }c$

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

$0\mathrm{\ω1}$

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

$-{\mathrm{\iota }}_{1,2}\mathrm{\beta }$

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

${\mathrm{\iota }}_{2,3}\mathrm{\beta }$

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

$\left[-\mathrm{\ω1}+\mathrm{\ω3}\,\mathrm{\ω2}\right]$

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

${\mathrm{\iota }}_{2}d\mathrm{\beta }+d{\mathrm{\iota }}_2\mathrm{\beta }$

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

$\left[f\,g\,\mathrm{\ω1}\,\mathrm{\ω2}\,\mathrm{\ω3}\,\mathrm{\alpha }\,\mathrm{\beta }\,{\mathrm{\iota }}_1\mathrm{\beta }\,{\mathrm{\iota }}_2\mathrm{\beta }\,{\mathrm{\iota }}_3\mathrm{\beta }\,{\mathrm{\iota }}_1\mathrm{\alpha }\,{\mathrm{\iota }}_2\mathrm{\alpha }\,{\mathrm{\iota }}_3\mathrm{\alpha }\,{\mathrm{\iota }}_{1,2}\mathrm{\beta }\,{\mathrm{\iota }}_{2,3}\mathrm{\beta }\,d\mathrm{\beta }\,{\mathrm{\iota }}_{2}d\mathrm{\beta }\,d{\mathrm{\iota }}_2\mathrm{\beta }\right]$

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

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

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

$2\mathrm{\alpha }\bigwedge {\mathrm{\iota }}_2\mathrm{\alpha }+d\mathrm{\alpha }$

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

$-\mathrm{D\_}\mathrm{\ω1}-\mathrm{i\_2\_3domega2}\left(\mathrm{D\_}\mathrm{\ω2}\right)$

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

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

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

$-\mathrm{D\_}\mathrm{\ω1}-q\left(\mathrm{D\_}\mathrm{\ω2}\right)$