Maple: How to use partial differential operators?

Question

I am trying to calculate the commutator $[v,w]=vw-wv$ for given infinitesimals

$$v=\dfrac{\partial}{\partial x}$$ and

$$w=x\dfrac{\partial}{\partial t}$$

I know how to calculate the commutator by hand, but I would like to learn how I can do these kind of calculations with maple.

Here is the result of the calculations:

$$[v,w]=\dfrac{\partial}{\partial x}\left( x\dfrac{\partial}{\partial t}\right)-x\dfrac{\partial}{\partial t}\dfrac{\partial}{\partial x}$$

Answer 1

You can easily do this using Lie algebra package in Maple 18, for sake of example I give you following code:

with(DifferentialGeometry): with(LieAlgebras):

Gamma := evalDG([D_x, D_y, x*D_x, y*D_x, x*D_y, y*D_y]);

L := LieAlgebraData(Gamma, Alg3);

DGsetup(L)

MultiplicationTable("LieTable")

this last command will return you Lie table for commutator.

For more details see tutorial on Lie algebra in Maple help.

You can also use command SymmetryCommutator command in PDEtools package of Maple.

For further detail see Mapleprime