$\nabla \cdot \hat n$ where $\hat n$ is a unit vector normal to a cylinder of radius $R$ and with a length $L=\infty$

Question

I'd like to calculate $\nabla \cdot \hat n$ where $\hat n$ is a unit vector normal to a cylinder of radius $R$ and with a length $L=\infty$.

What I've thought of is:

$\hat n= \hat R $ and using:

$\nabla \cdot \vec v = \frac{1}{s} \frac{\partial }{\partial s}(s v_s)+\frac{1}{s} \frac{\partial }{\partial \phi }( v_\phi ) + \frac{\partial }{\partial z}v_z$ giving:

I would get: $\nabla \cdot \hat n=\frac{1}{R} \frac{\partial }{\partial R}(R \cdot 1)=1/R$

Is this a correct way or how should I do it differently?

Answer 1

You've done it correctly. Remember: the divergence is specified only for vector fields, not for single vectors.

I answered a similar question here concerning the divergence in spherical coordinates just a few moments ago. That answer is more detailed, and you may find it of use.