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liesymm

 hasclosure
 verify closure with respect to d()

 Calling Sequence hasclosure(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 tested for closure with respect to the exterior derivative d().

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)
 > $\mathrm{a2}≔\left(\mathrm{w2}+{u}^{2}\right)\left(d\left(x\right)\right)&^\left(d\left(t\right)\right)-\left(d\left(\mathrm{w2}\right)\right)&^\left(d\left(x\right)\right)$
 ${\mathrm{a2}}{:=}{-}\left({{u}}^{{2}}{+}{\mathrm{w2}}\right){}\left({d}{}\left({t}\right)\right){&^}\left({d}{}\left({x}\right)\right){+}\left({d}{}\left({x}\right)\right){&^}\left({d}{}\left({\mathrm{w2}}\right)\right)$ (3)
 > $\mathrm{hasclosure}\left(\left[\mathrm{a1},\mathrm{a2}\right]\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{close}\left(\left[\mathrm{a1},\mathrm{a2}\right]\right)$
 $\left[{d}{}\left({u}\right){-}{\mathrm{w1}}{}{d}{}\left({t}\right){-}{\mathrm{w2}}{}{d}{}\left({x}\right){,}{-}\left({{u}}^{{2}}{+}{\mathrm{w2}}\right){}\left({d}{}\left({t}\right)\right){&^}\left({d}{}\left({x}\right)\right){+}\left({d}{}\left({x}\right)\right){&^}\left({d}{}\left({\mathrm{w2}}\right)\right){,}\left({d}{}\left({t}\right)\right){&^}\left({d}{}\left({\mathrm{w1}}\right)\right){+}\left({d}{}\left({x}\right)\right){&^}\left({d}{}\left({\mathrm{w2}}\right)\right)\right]$ (5)
 > $\mathrm{hasclosure}\left(\right)$
 ${\mathrm{true}}$ (6)