How to find the del of spherical vector field?

Question

Given $\vec{v}=wr\vec{i}_{\theta}-v_0\vec{i}_r$

How do I solve $\nabla\cdot \vec{v}$ and $\nabla\times\vec{v}?$

Answer 1

Assuming $\nabla = (\partial/\partial\theta, \partial/\partial r)$ and that $r$ and $\theta$ are independent of each other, you note that $$ \begin{split} \nabla \cdot \vec{v} &= \left(\frac{\partial}{\partial\theta}, \frac{\partial}{\partial r}\right) \cdot (wr, -v_0) \\ &= \frac{\partial [wr]}{\partial\theta} + \frac{\partial[-v_0]}{\partial r} \\ &= r \frac{\partial w}{\partial\theta} - \frac{\partial v_0}{\partial r} \end{split} $$ and everything else will depend on how $w$ and $v_0$ depend on $r$ and $\theta$.

If you are using polar coordinates, note that $$ \nabla = \left(\frac{\partial}{\partial r}, \frac{1}{r} \frac{\partial}{\partial \theta}\right) $$

Can you complete this and find the curl in a similar way?

Answer 2

You seem to be using polar coordinates, so you could have a look at the following wiki page:

https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates#Del_formula