How to define a vector field over a chart?

edit

Hello, I have the following code in sage:

E.<x,y,z> = EuclideanSpace()
CA = E.cartesian_coordinates(); CA
BP.<r, θ, ϕ> = E.chart()
BP_to_CA = BP.transition_map(CA, [r * sin(θ) * cos(ϕ), r * sin(θ) * sin(ϕ), r * cos(θ)])
g = E.metric()

E.set_default_chart(BP)
E.set_default_frame(BP.frame())
show( g.display() )
show( g.display_comp())

f = E.scalar_field({BP: function('F')(r, θ, ϕ)}, name='f')
f.display(BP)
f.laplacian().expr(BP).factor()

The transition map here is for spherical coordinates, I want to use it for other coordiante systems, so I don't want to use the spherical coordinates in sagemath. So I can define a scalar field and use gradient / laplacian-functions. Works fine. In order to calculate divergence and curl, I need to define a vector field. How can I do this? Thank's for your effort