An Expression for Nabla in Cylindrical Coordinates

Question

I'm looking for an expression for $$\big(\,u\cdot\nabla\,\big)\,\psi$$ in cylindrical coordinates where $\psi$ is a scalar field and $u$ is a vector field. The Wikipedia page given here https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates seems only to have an expression for this when $\psi$ is also a vector field which is quite frustrating. Does anybody have a source for an expression for this or alternatively a way of deducing it? Thanks :)

Answer 1

See you can analyse it in this way.
Let $u=u_\rho \hat e_\rho + u_\phi \hat e_\phi + u_z \hat e_z$,
$\nabla=\frac{\hat e_\rho}{h_\rho} \frac{\partial}{\partial u_\rho} + \frac{\hat e_\phi}{h_\phi} \frac{\partial}{\partial u_\phi} + \frac{\hat e_z}{h_z} \frac{\partial}{\partial u_z}$, and
$\psi=\psi'(x,y,z)=\psi(\rho,\phi,z)$
where SYMBOLS have their usual meanings.

Hence we have $$(u\cdot\nabla)\psi=\frac{u_\rho}{h_\rho} \frac{\partial \psi}{\partial u_\rho} + \frac{u_\phi}{h_\phi} \frac{\partial \psi}{\partial u_\phi} + \frac{u_z}{h_z} \frac{\partial \psi}{\partial u_z}$$

Hope my answer is clear enough.