Please consider the following example; it calculates the commutator of two vector fields acting on a scalar field, that is to say it calculates [u,v]f.
from sage.all import *
%display latex
M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')
X.<t,x,y,z> = M.chart()
u0 = function(r'u_0')(t,x,y,z)
u1 = function(r'u_1')(t,x,y,z)
u2 = function(r'u_2')(t,x,y,z)
u3 = function(r'u_3')(t,x,y,z)
u = M.vector_field(u0,u1,u2,u3, latex_name=r'\mathbf{u}')
v0 = function(r'v_0')(t,x,y,z)
v1 = function(r'v_1')(t,x,y,z)
v2 = function(r'v_2')(t,x,y,z)
v3 = function(r'v_3')(t,x,y,z)
v = M.vector_field(v0,v1,v2,v3, latex_name=r'\mathbf{v}')
f = M.scalar_field(function('f')(t,x,y,z), latex_name='f')
commutator_f = u(v(f)) - v(u(f))
commutator_f
This works. Now I can look at the expression for [u,v]f using
commutator_f.expr()
This also works but has many terms that I would like to collect - I would like all terms that are derivatives of f to "go to the right". I tried
commutator_f.expr().collect(f)
but that does not work: TypeError: no canonical coercion from Algebra of differentiable scalar fields on the 4-dimensional Lorentzian manifold M to Symbolic Ring.
How can I do this please?
Using SageMath version 9.5, Release Date: 2022-01-30, on Ubuntu 22.04.
Thank you
GPN
