I'm having a hard time trying to derive the divergence in cylindrical coordinates from its expression in cartesian coordinates $\dfrac {\partial F_{x}} {\partial x}+\dfrac {\partial F_{y}} {\partial y}+\dfrac {\partial F_{z}} {\partial z}$. I'm trying to proceed as follows: from the cartesian coordinate system $(x,y,z)$, defining
\begin{align} r&=\sqrt{x^2+y^2} \\ \theta&=\arctan \frac y x \end{align}
With the chain rule, we have:
\begin{align} \frac \partial {\partial x}&=\left( \frac {\partial r} {\partial x} \frac {\partial} {\partial r} + \frac {\partial \theta} {\partial x} \frac {\partial } {\partial \theta} \right) = \left( \cos \theta \frac {\partial} {\partial r} - \frac {\sin \theta} r \frac {\partial } {\partial \theta} \right) \\ \frac \partial {\partial y}&=\left( \frac {\partial r} {\partial y} \frac {\partial} {\partial r} + \frac {\partial \theta} {\partial y} \frac {\partial } {\partial \theta} \right) = \left( \sin \theta \frac {\partial} {\partial r} + \frac {\cos \theta} r \frac {\partial } {\partial \theta} \right) \end{align}
As $F_{x}=F_{r} \cos F_{\theta}$ and $F_{y}=F_{r} \sin F_{\theta}$:
\begin{align} \frac {\partial F_{x}} {\partial x} &= \cos \theta \cos F_{\theta} \frac {\partial F_{r}} {\partial r} - F_{r} \sin F_{\theta} \cos \theta \frac {\partial F_{\theta}} {\partial r} - \frac {\cos F_{\theta} \sin \theta} r \frac {\partial F_{r}} {\partial \theta} + \frac {F_{r}} r \sin \theta \sin F_{\theta} \frac {\partial F_{\theta}} {\partial \theta} \\ \frac {\partial F_{y}} {\partial y} &= \sin \theta \sin F_{\theta} \frac {\partial F_{r}} {\partial r} + F_{r} \cos F_{\theta} \sin \theta \frac {\partial F_{\theta}} {\partial r} + \frac {\sin F_{\theta} \cos \theta} r \frac {\partial F_{r}} {\partial \theta} + \frac {F_{r}} r \cos \theta \cos F_{\theta} \frac {\partial F_{\theta}} {\partial \theta} \end{align}
However, this sum leads to a weird combination of trigonometric sums and differences of $F_{\theta}$ and $\theta$. It seems to me that if I suppose $F_{\theta}=\theta$. I'll get the usual formula, but I can't see how this is true.
edit: I've used a wrong definition for a vector field on cylindrical coordinates. The correct definition is $\vec F=F_{r}\hat r+F_{\theta} \hat \theta + F_{z} \hat e_{z}$, where $\hat r=\cos \theta \hat i + \sin \theta \hat j$ and $\hat \theta=-\sin \theta \hat i + \cos \theta \hat j$. With this definition, the deduction above works.
