derivative of polar coordinates

Question

I am trying to undersatnd a proof and I am stuck on the derivative part in this proof. I can't understand why is there still a Fx and Fy in the polar coordinate.

Answer 1

The author is describing how partial derivatives work when they say "fixing $\theta$ and letting $r$ vary." The rest is indeed the chain rule $$ \frac{\partial f}{\partial r}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial r}\\ =f_x\cos\theta+f_y\sin \theta $$