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liesymm

 close
 compute the closure of a set of differential forms

 Calling Sequence close(forms)

Parameters

 forms - list or set of differential forms

Description

 • This routine is part of the liesymm package and is loaded via with(liesymm).
 • A set of differential forms is closed with respect to the exterior derivative d() by adding additional forms to the original set.

Examples

 > $\mathrm{with}\left(\mathrm{liesymm}\right):$
 > $\mathrm{setup}\left(t,x,u,\mathrm{w1},\mathrm{w2}\right)$
 $\left[{t}{,}{x}{,}{u}{,}{\mathrm{w1}}{,}{\mathrm{w2}}\right]$ (1)
 > $\mathrm{a1}≔d\left(u\right)-\mathrm{w1}d\left(t\right)-\mathrm{w2}d\left(x\right)$
 ${\mathrm{a1}}{≔}{d}{}\left({u}\right){-}{\mathrm{w1}}{}{d}{}\left({t}\right){-}{\mathrm{w2}}{}{d}{}\left({x}\right)$ (2)
 > $\mathrm{a2}≔\left(\mathrm{w2}+{u}^{2}\right)d\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(t\right)-d\left(\mathrm{w2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(x\right)$
 ${\mathrm{a2}}{≔}{-}\left({{u}}^{{2}}{+}{\mathrm{w2}}\right){}{d}{}\left({t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({x}\right){+}{d}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({\mathrm{w2}}\right)$ (3)
 > $\mathrm{hasclosure}\left(\left[\mathrm{a1},\mathrm{a2}\right]\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{close}\left(\left[\mathrm{a1},\mathrm{a2}\right]\right)$
 $\left[{d}{}\left({u}\right){-}{\mathrm{w1}}{}{d}{}\left({t}\right){-}{\mathrm{w2}}{}{d}{}\left({x}\right){,}{-}\left({{u}}^{{2}}{+}{\mathrm{w2}}\right){}{d}{}\left({t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({x}\right){+}{d}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({\mathrm{w2}}\right){,}{d}{}\left({t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({\mathrm{w1}}\right){+}{d}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({\mathrm{w2}}\right)\right]$ (5)
 > $\mathrm{hasclosure}\left(\right)$
 ${\mathrm{true}}$ (6)