\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)
| (%i2) |
load
(
basic
)
$
if get ( ' cartan , ' version ) = false then load ( cartan ) $ |
| (%i3) | init_cartan ( [ x1 , x2 , x3 ] ) ; |
\[\operatorname{ }\left[ \ensuremath{\mathrm{dx1}}\operatorname{,}\ensuremath{\mathrm{dx2}}\operatorname{,}\ensuremath{\mathrm{dx3}}\right] \]
| (%i6) |
ω
:
ω1
·
dx1
+
ω2
·
dx2
+
ω3
·
dx3
;
η : η1 · dx1 + η2 · dx2 + η3 · dx3 ; σ : σ1 · dx1 + σ2 · dx2 + σ3 · dx3 ; |
\[\operatorname{(\omega ) }\ensuremath{\mathrm{dx3}}\, \ensuremath{\mathrm{\omega 3}}+\ensuremath{\mathrm{dx2}}\, \ensuremath{\mathrm{\omega 2}}+\ensuremath{\mathrm{dx1}}\, \ensuremath{\mathrm{\omega 1}}\]
\[\operatorname{(\eta ) }\ensuremath{\mathrm{dx3}}\, \ensuremath{\mathrm{\eta 3}}+\ensuremath{\mathrm{dx2}}\, \ensuremath{\mathrm{\eta 2}}+\ensuremath{\mathrm{dx1}}\, \ensuremath{\mathrm{\eta 1}}\]
\[\operatorname{(\sigma ) }\ensuremath{\mathrm{dx3}}\, \ensuremath{\mathrm{\sigma 3}}+\ensuremath{\mathrm{dx2}}\, \ensuremath{\mathrm{\sigma 2}}+\ensuremath{\mathrm{dx1}}\, \ensuremath{\mathrm{\sigma 1}}\]
| (%i7) | facsum ( ω ~ η , dx1 , dx2 , dx3 ) ; |
\[\operatorname{ }-\ensuremath{\mathrm{dx2}}\, \ensuremath{\mathrm{dx3}}\, \left( \ensuremath{\mathrm{\eta 2}}\, \ensuremath{\mathrm{\omega 3}}-\ensuremath{\mathrm{\eta 3}}\, \ensuremath{\mathrm{\omega 2}}\right) -\ensuremath{\mathrm{dx1}}\, \ensuremath{\mathrm{dx3}}\, \left( \ensuremath{\mathrm{\eta 1}}\, \ensuremath{\mathrm{\omega 3}}-\ensuremath{\mathrm{\eta 3}}\, \ensuremath{\mathrm{\omega 1}}\right) -\ensuremath{\mathrm{dx1}}\, \ensuremath{\mathrm{dx2}}\, \left( \ensuremath{\mathrm{\eta 1}}\, \ensuremath{\mathrm{\omega 2}}-\ensuremath{\mathrm{\eta 2}}\, \ensuremath{\mathrm{\omega 1}}\right) \]
| (%i8) | factor ( ω ~ η ~ σ ) ; |
\[\operatorname{ }\ensuremath{\mathrm{dx1}}\, \ensuremath{\mathrm{dx2}}\, \ensuremath{\mathrm{dx3}}\, \left( \ensuremath{\mathrm{\eta 1}}\, \ensuremath{\mathrm{\sigma 2}}\, \ensuremath{\mathrm{\omega 3}}-\ensuremath{\mathrm{\eta 2}}\, \ensuremath{\mathrm{\sigma 1}}\, \ensuremath{\mathrm{\omega 3}}-\ensuremath{\mathrm{\eta 1}}\, \ensuremath{\mathrm{\sigma 3}}\, \ensuremath{\mathrm{\omega 2}}+\ensuremath{\mathrm{\eta 3}}\, \ensuremath{\mathrm{\sigma 1}}\, \ensuremath{\mathrm{\omega 2}}+\ensuremath{\mathrm{\eta 2}}\, \ensuremath{\mathrm{\sigma 3}}\, \ensuremath{\mathrm{\omega 1}}-\ensuremath{\mathrm{\eta 3}}\, \ensuremath{\mathrm{\sigma 2}}\, \ensuremath{\mathrm{\omega 1}}\right) \]
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