I'm a little puzzled with the work of a force considering rectangular and polar coordinates.
In rectangular coordinates we have:
(1) [F]=(Fx [i] + Fy [j]) is the force vector.
(2) [dr]=(dx [i] + dy [j]) is small displacement.
Then the work is:
(3) W = [F] [dr] = (Fx dx + Fy dy)
In...