Considering the following identity transformation in cylindrical coordinate:
$$\mathbf{v}(R,\theta,z)=R\;\mathbf{e}_{R}+\theta\;\mathbf{e}_{\theta}+z\;\mathbf{e}_{z} $$
Taking its derivative according to the formula for the derivative of a vector field in cylindrical coordinates:
I expect to obtain the identity but the formula above gives also some non-diagonal terms. Could you explain me why or give me a good reference?