Hello,
the lines below would suggest that Jacobi identity for vector fields [1] is
violated in diffeom moodule: am I missing something fundamental?
ric
[1] https://en.wikipedia.org/wiki/Lie_bracket_of_vector_fields#Properties
----------------------
Python console for SymPy 1.4 (Python 2.7.12)
These commands were executed:
from __future__ import division
from sympy import *
x, y, z, t = symbols('x y z t')
k, m, n = symbols('k m n', integer=True)
f, g, h = symbols('f g h', cls=Function)
Warning: this shell runs with SymPy 1.4 and so examples pulled from
other documentation may provide unexpected results.
Documentation can be found at http://docs.sympy.org/1.4.
from sympy.diffgeom import *
...
... LieD = LieDerivative
... Com = Commutator
...
... m = Manifold('AManifold',2)
... p = Patch('APatch',m)
... spherical = CoordSystem('Spherical',p,['r','theta'])
...
... xs = r, theta = spherical.coord_functions()
... es = e_r, e_theta = spherical.base_vectors()
...
... v1 = exp(r)*e_r + exp(r**2)*e_theta
... v2 = sin(theta)*e_r
... v3 = e_theta+e_r
...
... expand(LieD(LieD(v1, v2), v3) + LieD(LieD(v2, v3), v1) + LieD(LieD(v3, v1),
v2))
...
⎛ 2⎞ ⎛ 2⎞
⎝r ⎠ r r ⎝r ⎠
2⋅r⋅ℯ ⋅cos(θ)⋅∂ᵣ - ℯ ⋅sin(θ)⋅∂ᵣ - ℯ ⋅cos(θ)⋅∂ᵣ - ℯ ⋅sin(θ)⋅∂ᵣ
expand(Com(Com(v1, v2), v3) + Com(Com(v2, v3), v1) + Com(Com(v3, v1), v2))
⎛ 2⎞ ⎛ 2⎞
⎝r ⎠ r r ⎝r ⎠
2⋅r⋅ℯ ⋅cos(θ)⋅∂ᵣ - ℯ ⋅sin(θ)⋅∂ᵣ - ℯ ⋅cos(θ)⋅∂ᵣ - ℯ ⋅sin(θ)⋅∂ᵣ
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