&^ - Maple Help

liesymm

 &^
 the wedge product

 Calling Sequence a &^ b &^(a, b, c)

Parameters

 a, b, c - expression involving differential forms relative to specific coordinates

Description

 • This routine is part of the liesymm package and is loaded via with(liesymm) .
 • It computes the wedge product of differential forms relative to the coordinates defined by setup().
 • All 1-forms are generated by applying d() to the coordinates.
 • All wedge products are automatically simplified to a wedge product of n 1-forms by extracting coefficients of wedge degree 0.
 • All results of a wedge product are reported using an address ordering of the 1-forms to facilitate simplifications.  Thus $d\left(y\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(x\right)$ may simplify to $-d\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(y\right)$ and if so will do so consistently within a given session.
 • The ordering used for simplifications of the products of 1-forms is available as $\left\{\mathrm{wedgeset}\left(1\right)\right\}$.

Examples

 > $\mathrm{with}\left(\mathrm{liesymm}\right):$
 > $\mathrm{setup}\left(x,y,z\right)$
 $\left[{x}{,}{y}{,}{z}\right]$ (1)
 > $d\left(t\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(x\right)$
 ${0}$ (2)
 > $d\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(y\right)$
 ${-}{d}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({x}\right)$ (3)
 > $\left(\left(5d\left(x\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(y\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left(3d\left(z\right)\right)$
 ${-}{15}{}{\mathrm{&^}}{}\left({d}{}\left({z}\right){,}{d}{}\left({y}\right){,}{d}{}\left({x}\right)\right)$ (4)
 > $\mathrm{&^}\left(ad\left(x\right),bd\left(y\right),cd\left(z\right)\right)$
 ${-}{a}{}{b}{}{c}{}{\mathrm{&^}}{}\left({d}{}\left({z}\right){,}{d}{}\left({y}\right){,}{d}{}\left({x}\right)\right)$ (5)