liesymm - Maple Programming 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)
 > $\mathrm{Lie}\left(d\left(x\right),V\right)$
 $\left(\frac{{\partial }}{{\partial }{x}}{}{V}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right)\right){}{d}{}\left({x}\right){+}\left(\frac{{\partial }}{{\partial }{y}}{}{V}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right)\right){}{d}{}\left({y}\right){+}\left(\frac{{\partial }}{{\partial }{z}}{}{V}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right)\right){}{d}{}\left({z}\right)$ (2)
 > $\mathrm{Lie}\left(f\left(x,y,z\right)d\left(x\right),V\right)$
 $\left({V}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right){}\left(\frac{{\partial }}{{\partial }{x}}{}{f}{}\left({x}{,}{y}{,}{z}\right)\right){+}{V}{[}{2}{]}{}\left({x}{,}{y}{,}{z}\right){}\left(\frac{{\partial }}{{\partial }{y}}{}{f}{}\left({x}{,}{y}{,}{z}\right)\right){+}{V}{[}{3}{]}{}\left({x}{,}{y}{,}{z}\right){}\left(\frac{{\partial }}{{\partial }{z}}{}{f}{}\left({x}{,}{y}{,}{z}\right)\right)\right){}{d}{}\left({x}\right){+}{f}{}\left({x}{,}{y}{,}{z}\right){}\left(\left(\frac{{\partial }}{{\partial }{x}}{}{V}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right)\right){}{d}{}\left({x}\right){+}\left(\frac{{\partial }}{{\partial }{y}}{}{V}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right)\right){}{d}{}\left({y}\right){+}\left(\frac{{\partial }}{{\partial }{z}}{}{V}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right)\right){}{d}{}\left({z}\right)\right)$ (3)
 > $\mathrm{Lie}\left(f\left(x,y,z\right)\left(d\left(x\right)\right)&^\left(d\left(y\right)\right),V\right)$
 $\left({V}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right){}\left(\frac{{\partial }}{{\partial }{x}}{}{f}{}\left({x}{,}{y}{,}{z}\right)\right){+}{V}{[}{2}{]}{}\left({x}{,}{y}{,}{z}\right){}\left(\frac{{\partial }}{{\partial }{y}}{}{f}{}\left({x}{,}{y}{,}{z}\right)\right){+}{V}{[}{3}{]}{}\left({x}{,}{y}{,}{z}\right){}\left(\frac{{\partial }}{{\partial }{z}}{}{f}{}\left({x}{,}{y}{,}{z}\right)\right)\right){}\left({d}{}\left({x}\right)\right){&^}\left({d}{}\left({y}\right)\right){+}{f}{}\left({x}{,}{y}{,}{z}\right){}\left(\left(\frac{{\partial }}{{\partial }{x}}{}{V}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right)\right){}\left({d}{}\left({x}\right)\right){&^}\left({d}{}\left({y}\right)\right){-}\left(\frac{{\partial }}{{\partial }{z}}{}{V}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right)\right){}\left({d}{}\left({y}\right)\right){&^}\left({d}{}\left({z}\right)\right){+}\left(\frac{{\partial }}{{\partial }{y}}{}{V}{[}{2}{]}{}\left({x}{,}{y}{,}{z}\right)\right){}\left({d}{}\left({x}\right)\right){&^}\left({d}{}\left({y}\right)\right){+}\left(\frac{{\partial }}{{\partial }{z}}{}{V}{[}{2}{]}{}\left({x}{,}{y}{,}{z}\right)\right){}\left({d}{}\left({x}\right)\right){&^}\left({d}{}\left({z}\right)\right)\right)$ (4)