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).

$\mathrm{with}\left(\mathrm{liesymm}\right)\:$

$\mathrm{setup}\left(x\,y\,z\right)$

$\left[x\,y\,z\right]$

$\mathrm{Lie}\left(d\left(x\right)\,V\right)$

$\left(\frac{\partial }{\partial x}{V}_1\left(x\,y\,z\right)\right)d\left(x\right)+\left(\frac{\partial }{\partial y}{V}_1\left(x\,y\,z\right)\right)d\left(y\right)+\left(\frac{\partial }{\partial z}{V}_1\left(x\,y\,z\right)\right)d\left(z\right)$

$\mathrm{Lie}\left(f\left(x\,y\,z\right)d\left(x\right)\,V\right)$

$\left(V_1\left(x\,y\,z\right)\left(\frac{\partial }{\partial x}f\left(x\,y\,z\right)\right)+V_2\left(x\,y\,z\right)\left(\frac{\partial }{\partial y}f\left(x\,y\,z\right)\right)+V_3\left(x\,y\,z\right)\left(\frac{\partial }{\partial z}f\left(x\,y\,z\right)\right)\right)d\left(x\right)+f\left(x\,y\,z\right)\left(\left(\frac{\partial }{\partial x}{V}_1\left(x\,y\,z\right)\right)d\left(x\right)+\left(\frac{\partial }{\partial y}{V}_1\left(x\,y\,z\right)\right)d\left(y\right)+\left(\frac{\partial }{\partial z}{V}_1\left(x\,y\,z\right)\right)d\left(z\right)\right)$

$\mathrm{Lie}\left(f\left(x\,y\,z\right)d\left(x\right)\&ˆd\left(y\right)\,V\right)$

$\left(V_1\left(x\,y\,z\right)\left(\frac{\partial }{\partial x}f\left(x\,y\,z\right)\right)+V_2\left(x\,y\,z\right)\left(\frac{\partial }{\partial y}f\left(x\,y\,z\right)\right)+V_3\left(x\,y\,z\right)\left(\frac{\partial }{\partial z}f\left(x\,y\,z\right)\right)\right)d\left(x\right)\&ˆd\left(y\right)+f\left(x\,y\,z\right)\left(\left(\frac{\partial }{\partial x}{V}_1\left(x\,y\,z\right)\right)d\left(x\right)\&ˆd\left(y\right)-\left(\frac{\partial }{\partial z}{V}_1\left(x\,y\,z\right)\right)d\left(y\right)\&ˆd\left(z\right)+\left(\frac{\partial }{\partial y}{V}_2\left(x\,y\,z\right)\right)d\left(x\right)\&ˆd\left(y\right)+\left(\frac{\partial }{\partial z}{V}_2\left(x\,y\,z\right)\right)d\left(x\right)\&ˆd\left(z\right)\right)$