I'd like to compute an exterior derivative in spherical coordinates. So far I have the following:
E.<x,y,z> = EuclideanSpace(3)
Ec.<r,theta,phi> = E.spherical_coordinates()
EcM = Ec.manifold()
EcM.set_default_frame(EcM.spherical_frame())
EcM.set_default_chart(EcM.spherical_coordinates())
F = function('F')
show(F)
F_1 = function('F_r')(r, theta, phi)
F_2 = function('F_theta')(r, theta, phi)
F_3 = function('F_phi')(r, theta, phi)
psi = EcM.diff_form(2, 'psi')
psi[2, 3] = F_1
psi[1, 3] = -F_2
psi[1, 2] = F_3
show(psi.display())
res = psi.exterior_derivative()
show(res.display())
which works for the spherical coordinate transformation that is the default for Euclidean space. But, I see that Sagemath has a different spherical coordinate map than I do:
> print(E.coord_change(E.spherical_coordinates(), E.cartesian_coordinates()).display())
x = r*cos(phi)*sin(theta)
y = r*sin(phi)*sin(theta)
z = r*cos(theta)
Instead, I'd like:
x = r * cos(theta) * cos(phi)
y = r * sin(theta) * sin(phi)
z = r * sin(phi)
How can I supply a different change of coordinates function for this case?
