Lie - Maple Help

liesymm

 Lie
 the Lie derivative

 Calling Sequence Lie(form, V)

Parameters

 form - expression involving differential forms relative to specific coordinates V - name or an explicit isovector [V1,V2,...Vn]

Description

 • The Lie derivative of the differential form form is constructed with respect to $\mathrm{V1},...,\mathrm{Vn}$ where n is the number of coordinates.
 • This routine is part of the liesymm package and is loaded via with(liesymm).

Examples

 > $\mathrm{with}\left(\mathrm{liesymm}\right):$
 > $\mathrm{setup}\left(x,y,z\right)$
 $\left[{x}{,}{y}{,}{z}\right]$ (1)
 > $\mathrm{Lie}\left(d\left(x\right),V\right)$
 $\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{V}}_{{1}}{}\left({x}{,}{y}{,}{z}\right){}{d}{}\left({x}\right){+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{V}}_{{1}}{}\left({x}{,}{y}{,}{z}\right){}{d}{}\left({y}\right){+}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{V}}_{{1}}{}\left({x}{,}{y}{,}{z}\right){}{d}{}\left({z}\right)$ (2)
 > $\mathrm{Lie}\left(f\left(x,y,z\right)d\left(x\right),V\right)$
 $\left({{V}}_{{1}}{}\left({x}{,}{y}{,}{z}\right){}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}{,}{z}\right){+}{{V}}_{{2}}{}\left({x}{,}{y}{,}{z}\right){}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}{,}{z}\right){+}{{V}}_{{3}}{}\left({x}{,}{y}{,}{z}\right){}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}{,}{z}\right)\right){}{d}{}\left({x}\right){+}{f}{}\left({x}{,}{y}{,}{z}\right){}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{V}}_{{1}}{}\left({x}{,}{y}{,}{z}\right){}{d}{}\left({x}\right){+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{V}}_{{1}}{}\left({x}{,}{y}{,}{z}\right){}{d}{}\left({y}\right){+}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{V}}_{{1}}{}\left({x}{,}{y}{,}{z}\right){}{d}{}\left({z}\right)\right)$ (3)
 > $\mathrm{Lie}\left(f\left(x,y,z\right)d\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}d\left(y\right),V\right)$
 ${-}\left({{V}}_{{1}}{}\left({x}{,}{y}{,}{z}\right){}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}{,}{z}\right){+}{{V}}_{{2}}{}\left({x}{,}{y}{,}{z}\right){}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}{,}{z}\right){+}{{V}}_{{3}}{}\left({x}{,}{y}{,}{z}\right){}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}{,}{z}\right)\right){}{d}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({x}\right){-}{f}{}\left({x}{,}{y}{,}{z}\right){}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{V}}_{{2}}{}\left({x}{,}{y}{,}{z}\right){}{d}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({x}\right){+}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{V}}_{{2}}{}\left({x}{,}{y}{,}{z}\right){}{d}{}\left({z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({x}\right){+}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{V}}_{{1}}{}\left({x}{,}{y}{,}{z}\right){}{d}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({x}\right){+}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{V}}_{{1}}{}\left({x}{,}{y}{,}{z}\right){}{d}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({z}\right)\right)$ (4)