(%i2) load ( basic ) $
if get ( ' cartan , ' version ) = false then load ( cartan ) $
(%i4) x : [ x1 , x2 , x3 ] ;
dx : init_cartan ( x ) ;

\[\operatorname{(x) }\left[ \ensuremath{\mathrm{x1}}\operatorname{,}\ensuremath{\mathrm{x2}}\operatorname{,}\ensuremath{\mathrm{x3}}\right] \]

\[\operatorname{(dx) }\left[ \ensuremath{\mathrm{dx1}}\operatorname{,}\ensuremath{\mathrm{dx2}}\operatorname{,}\ensuremath{\mathrm{dx3}}\right] \]

1-Form ω
(%i7) ω : [ ω1 , ω2 , ω3 ] ;
depends ( ω , x ) ;
ω : ω . dx ;

\[\operatorname{(\omega ) }\left[ \ensuremath{\mathrm{\omega 1}}\operatorname{,}\ensuremath{\mathrm{\omega 2}}\operatorname{,}\ensuremath{\mathrm{\omega 3}}\right] \]

\[\operatorname{ }\left[ \operatorname{\omega 1}\left( \ensuremath{\mathrm{x1}}\operatorname{,}\ensuremath{\mathrm{x2}}\operatorname{,}\ensuremath{\mathrm{x3}}\right) \operatorname{,}\operatorname{\omega 2}\left( \ensuremath{\mathrm{x1}}\operatorname{,}\ensuremath{\mathrm{x2}}\operatorname{,}\ensuremath{\mathrm{x3}}\right) \operatorname{,}\operatorname{\omega 3}\left( \ensuremath{\mathrm{x1}}\operatorname{,}\ensuremath{\mathrm{x2}}\operatorname{,}\ensuremath{\mathrm{x3}}\right) \right] \]

\[\operatorname{(\omega ) }\ensuremath{\mathrm{dx3}}\, \ensuremath{\mathrm{\omega 3}}+\ensuremath{\mathrm{dx2}}\, \ensuremath{\mathrm{\omega 2}}+\ensuremath{\mathrm{dx1}}\, \ensuremath{\mathrm{\omega 1}}\]

Äußere Ableitung dω
(%i9) dω : ext_diff ( ω ) $
facsum ( % , dx1 , dx2 , dx3 ) ;

\[\operatorname{ }\ensuremath{\mathrm{dx2}}\, \ensuremath{\mathrm{dx3}}\, \left( \frac{d}{d \ensuremath{\mathrm{x2}}} \ensuremath{\mathrm{\omega 3}}-\frac{d}{d \ensuremath{\mathrm{x3}}} \ensuremath{\mathrm{\omega 2}}\right) +\ensuremath{\mathrm{dx1}}\, \ensuremath{\mathrm{dx3}}\, \left( \frac{d}{d \ensuremath{\mathrm{x1}}} \ensuremath{\mathrm{\omega 3}}-\frac{d}{d \ensuremath{\mathrm{x3}}} \ensuremath{\mathrm{\omega 1}}\right) +\ensuremath{\mathrm{dx1}}\, \ensuremath{\mathrm{dx2}}\, \left( \frac{d}{d \ensuremath{\mathrm{x1}}} \ensuremath{\mathrm{\omega 2}}-\frac{d}{d \ensuremath{\mathrm{x2}}} \ensuremath{\mathrm{\omega 1}}\right) \]

2-Form η
(%i11) depends ( [ η12 , η13 , η23 ] , x ) ;
η : η12 · dx1 ~ dx2 + η13 · dx1 ~ dx3 + η23 · dx2 ~ dx3 ;

\[\operatorname{ }\left[ \operatorname{\eta 12}\left( \ensuremath{\mathrm{x1}}\operatorname{,}\ensuremath{\mathrm{x2}}\operatorname{,}\ensuremath{\mathrm{x3}}\right) \operatorname{,}\operatorname{\eta 13}\left( \ensuremath{\mathrm{x1}}\operatorname{,}\ensuremath{\mathrm{x2}}\operatorname{,}\ensuremath{\mathrm{x3}}\right) \operatorname{,}\operatorname{\eta 23}\left( \ensuremath{\mathrm{x1}}\operatorname{,}\ensuremath{\mathrm{x2}}\operatorname{,}\ensuremath{\mathrm{x3}}\right) \right] \]

\[\operatorname{(\eta ) }\ensuremath{\mathrm{dx2}}\, \ensuremath{\mathrm{dx3}}\, \ensuremath{\mathrm{\eta 23}}+\ensuremath{\mathrm{dx1}}\, \ensuremath{\mathrm{dx3}}\, \ensuremath{\mathrm{\eta 13}}+\ensuremath{\mathrm{dx1}}\, \ensuremath{\mathrm{dx2}}\, \ensuremath{\mathrm{\eta 12}}\]

Äußere Ableitung dη
(%i13) dη : ext_diff ( η ) $
factor ( % ) ;

\[\operatorname{ }\ensuremath{\mathrm{dx1}}\, \ensuremath{\mathrm{dx2}}\, \ensuremath{\mathrm{dx3}}\, \left( \frac{d}{d \ensuremath{\mathrm{x1}}} \ensuremath{\mathrm{\eta 23}}-\frac{d}{d \ensuremath{\mathrm{x2}}} \ensuremath{\mathrm{\eta 13}}+\frac{d}{d \ensuremath{\mathrm{x3}}} \ensuremath{\mathrm{\eta 12}}\right) \]

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