Transformation from cartesian to polar Coordinates of Vector Field

Question

This is fairly low-level, still I would like to get a verification:

I vector field $$\mathbf{F}=F_x \hat{e_x} + F_y \hat{e_y} = F_r \hat{e_r} + F_{\phi} \hat{e_\phi}$$ given in cartesian coordinates, and want to transform it to polar coordinates.

I was reading Vector fields in cylindrical and spherical coordinates and Del in cylindrical and spherical coordinates at wikipedia aswell as entries on that topic here, still I am not sure whether this is correct:

$$F_r=\frac{x\cdot F_x + y\cdot F_y}{r}\\$$ $$F_{\phi}=\frac{-y\cdot F_x + x\cdot F_y}{r}\\$$

1) Could you please verify/falsify,

2) Provide me a reference for how to derive this.

Thanks!

Answer 1

All right ! See example 1, p. 3 in this pdf.