I'm trying to derive divergence in cylindrical coordinates.
Suppose we have a vector function expressed in cylindrical coordinates: $$\textbf{F}=F_R\textbf{e}_R+F_\theta\textbf{e}_\theta+F_z\textbf{k}$$
I know that grad in cylindrical coordinates is:
$$\nabla=\textbf{e}_R\frac{\partial}{\partial R}+\textbf{e}_\theta\frac{1}{R}\frac{\partial}{\partial \theta}+\textbf{k}\frac{\partial}{\partial z}$$
So:
$$\nabla \cdot \textbf{F} = (\textbf{e}_R\frac{\partial}{\partial R}+\textbf{e}_\theta\frac{1}{R}\frac{\partial}{\partial \theta}+\textbf{k}\frac{\partial}{\partial z}) \cdot \textbf{F} $$
My question is, when you expand this dot product, why do you get:
$$\nabla \cdot \textbf{F} = (\textbf{e}_R\cdot\frac{\partial \textbf{F}}{\partial R}+\frac{1}{R}\textbf{e}_\theta \cdot \frac{\partial \textbf{F}}{\partial \theta}+\textbf{k} \cdot \frac{\partial \textbf{F}}{\partial z})$$
and not:
$$\nabla \cdot \textbf{F} = (\textbf{e}_R \cdot \textbf{F} \frac{\partial}{\partial R}+\textbf{e}_\theta \cdot \textbf{F} \frac{1}{R}\frac{\partial}{\partial \theta}+\textbf{k} \cdot \textbf{F} \frac{\partial}{\partial z})$$
Which mathematical laws prescribe the former instead of the latter?
Thanks for any help,
Jack