spherical coordinate simplify

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This is because when you declare a symbolic expression like f(th) = abs(sin(th)) it has no knowledge of the range of spherical coordinates on E. There are two solutions to your issue:

1/ Stick to Sage's standard symbolic expression but use a better simplifying function, which takes into account the range of spherical coordinates in Euclidean space:

sage: E.<xi,th,ph> = EuclideanSpace(coordinates='spherical')
sage: f(th) = abs(sin(th))
sage: from sage.manifolds.utilities import simplify_chain_real
sage: simplify_chain_real(f(th))
sin(th)

See the documentation of simplify_chain_real for details.

2/ Use chart functions instead of mere symbolic expressions:

sage: E.<xi,th,ph> = EuclideanSpace(coordinates='spherical')
sage: X = E.spherical_coordinates(); X
Chart (E^3, (xi, th, ph))
sage: f = X.function(abs(sin(th)))
sage: f
abs(sin(th))
sage: f.simplify()
sin(th)