You can also use the Euclidean space.
E.<x,y> = EuclideanSpace()
cartesian = E.default_chart()
polar.<r,ph> = E.polar_coordinates()
Coframes
[(E2,(dx,dy)),(E2,(dr,dϕ)),(E2,(er,eϕ))].
Thus
dph = E.coframes()[1][2]
dph.display(polar)
dϕ=dϕ,
and
dph.display(cartesian)
dϕ=(−yx2+y2)dx+(xx2+y2)dy.
Or
dph.apply_map(lambda h:h.subs({x^2+y^2:r^2}))
dph.display(cartesian)
dϕ=−yr2dx+xr2dy.