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Using SageMath 8.7, I confirm that the command grad(Phi) runs for ever... Actually, this is not the computation of the gradient by itself that takes time but its simplification. The quantity phi_ges from which the scalar field Phi is formed is a quite lengthy symbolic expression. Sage evaluates

diff(phi_ges, z)

quite fast but then the command

diff(phi_ges, z).simplify_full()

runs for ages... It turns out that grad(Phi) is using a simplification sequence pretty similar to simplify_full() (the details are here) to simplify its output. Hence the observed behavior. Maybe this is an issue of the simplification of long symbolic expressions involving Heaviside functions in Sage...

Using SageMath 8.7, I confirm that the command grad(Phi) runs for ever... Actually, this is not the computation of the gradient by itself that takes time but its simplification. The quantity phi_ges from which the scalar field Phi is formed is a quite lengthy symbolic expression. Sage evaluates

diff(phi_ges, z)

quite fast but then the command

diff(phi_ges, z).simplify_full()

runs for ages... It turns out that grad(Phi) is using a simplification sequence pretty similar to simplify_full() (the details are here) to simplify its output. Hence the observed behavior. Maybe this is an issue of the simplification of long symbolic expressions involving Heaviside functions in Sage...

S.default_chart()._calc_method._simplify_dict['SR'] = simplify

Using SageMath 8.7, I confirm that the command grad(Phi) runs for ever... Actually, this is not the computation of the gradient by itself that takes time but its simplification. The quantity phi_ges from which the scalar field Phi is formed is a quite lengthy symbolic expression. Sage evaluates

diff(phi_ges, z)

quite fast but then the command

diff(phi_ges, z).simplify_full()

runs for ages... It turns out that grad(Phi) is using a simplification sequence pretty similar to simplify_full() (the details are here) to simplify its output. Hence the observed behavior. Maybe this is an issue of the simplification of long symbolic expressions involving Heaviside functions in Sage...

EDIT: a possible workaround is to replace simplify_chain_real by the mere simplify function as the simplifying function associated to the default chart (x,z) of the Euclidean space S; to do this, it suffices to insert the line

S.default_chart()._calc_method._simplify_dict['SR'] = simplify

just after the definition S.<x,z> = EuclideanSpace(2).

Using SageMath 8.7, I confirm that the command grad(Phi) runs for ever... Actually, this is not the computation of the gradient by itself that takes time but its simplification. The quantity phi_ges from which the scalar field Phi is formed is a quite lengthy symbolic expression. Sage evaluates

diff(phi_ges, z)

quite fast but then the command

diff(phi_ges, z).simplify_full()

runs for ages... It turns out that grad(Phi) is using a simplification sequence pretty similar to simplify_full() (the details are here) to simplify its output. Hence the observed behavior. Maybe this is an issue of the simplification of long symbolic expressions involving Heaviside functions in Sage...

EDIT: a possible workaround is to replace simplify_chain_real by the mere simplify function as the simplifying function associated to the default chart (x,z) of the Euclidean space S; to do this, it suffices to insert the line

S.default_chart()._calc_method._simplify_dict['SR'] = simplify

just after the definition S.<x,z> = EuclideanSpace(2).

Explanations:

• S.default_chart() is the chart of Cartesian coordinates $(x,z)$ on the Euclidean plane $S$
• _calc_method is the calculus method, i.e. the tool managing the symbolic backend that is used when perfoming calculations with the chart $(x,z)$: either the symbolic ring SR (SageMath's default) or SymPy
• _simplify_dict is the dictionary of simplifying functions associated to each symbolic backend: by default, _simplify_dict['SR'] = simplify_chain_real and _simplify_dict['sympy'] = simplify_chain_real_sympy
• the function simplify in the right-hand side of the suggested command is SageMath's minimal simplifying function: it is fast, but it does not do much. You can replace it by any function that maps a symbolic expression to another symbolic expression.

%time print((-grad(Phi)).display())

CPU times: user 498 ms, sys: 18.6 ms, total: 517 ms
Wall time: 396 ms

Using SageMath 8.7, I confirm that the command grad(Phi) runs for ever... Actually, this is not the computation of the gradient by itself that takes time but its simplification. The quantity phi_ges from which the scalar field Phi is formed is a quite lengthy symbolic expression. Sage evaluates

diff(phi_ges, z)

quite fast but then the command

diff(phi_ges, z).simplify_full()

runs for ages... It turns out that grad(Phi) is using a simplification sequence pretty similar to simplify_full() (the details are here) to simplify its output. Hence the observed behavior. Maybe this is an issue of the simplification of long symbolic expressions involving Heaviside functions in Sage...

EDIT: a possible workaround is to replace simplify_chain_real by the mere simplify function as the simplifying function associated to the default chart (x,z) of the Euclidean space S; to do this, it suffices to insert the line

S.default_chart()._calc_method._simplify_dict['SR'] = simplify

just after the definition S.<x,z> = EuclideanSpace(2).

Explanations:

• S.default_chart() is the chart of Cartesian coordinates $(x,z)$ on the Euclidean plane $S$
• _calc_method is the calculus method, i.e. the tool managing the symbolic backend that is used when perfoming calculations with the chart $(x,z)$: either the symbolic ring SR (SageMath's default) or SymPy
• _simplify_dict is the dictionary of simplifying functions associated to each symbolic backend: by default, _simplify_dict['SR'] = simplify_chain_real and _simplify_dict['sympy'] = simplify_chain_real_sympy
• the function simplify in the right-hand side of the suggested command is SageMath's minimal simplifying function: it is fast, but it does not do much. You can replace it by any function that maps a symbolic expression to another symbolic expression.

With the above command,

%time print((-grad(Phi)).display())

returns

CPU times: user 498 ms, sys: 18.6 ms, total: 517 ms
Wall time: 396 ms