Given the vector field $\vec{v}=wr\vec{i}_{\theta}-v_0\vec{i}_r$ (cylindrical-coordinates)
I'm supposed to find the local and convectiv acceleration/derivation
$$\frac{Df}{Dt}=\frac{\partial f}{\partial t}+v\cdot\nabla f$$
For the local derivation I should get $\frac{\partial \vec{v}}{\partial t}=0$?
I'm not sure on how I do the convectiv derivation, It gets messy really fast and I get confused. Do I first take $v\cdot\nabla$ or $\nabla f$? If you could show me with an example, it would be really appriciated!
$$D\vec{v}=\partial_{t}\vec{v}+(\vec{v}\cdot\nabla)\vec{v}=$$ $$=\frac{\partial\vec{v}}{\partial{t}}+v_{r}\frac{\partial\vec{v}}{\partial{r}}+v_{\varphi}\frac{1}{r}\frac{\partial\vec{v}}{\partial\varphi}+v_{z}\frac{\partial\vec{v}}{\partial{z}}$$