I'm trying to figure out how to calculate curl ($\nabla \times \vec{V}^{\,}$) when the velocity vector is represented in cylindrical coordinates. The way I thought I would do it is by calculating this determinant:
$$\left|\begin{matrix} e_r & e_{\theta} & e_{z} \\ \frac{\partial }{\partial r} & \frac{1}{r} \frac{\partial }{\partial \theta} & \frac{\partial }{\partial z} \\ v_r & v_\theta & v_z \end{matrix}\right|$$
Which gives:
$$\left\lbrack \frac{1}{r} \frac{\partial v_z}{\partial \theta} - \frac{\partial v_\theta}{\partial z}, \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r}, \frac{\partial v_\theta}{\partial r} - \frac{1}{r} \frac{\partial v_r}{\partial \theta}\right\rbrack$$
But I think the correct curl is:
$$\left\lbrack \frac{1}{r} \frac{\partial v_z}{\partial \theta} - \frac{\partial v_\theta}{\partial z}, \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r}, \frac{1}{r} \frac{\partial rv_\theta}{\partial r} - \frac{1}{r} \frac{\partial v_r}{\partial \theta}\right\rbrack$$
Can anyone explain why this is? It seems sort of like the way to calculate it is with:
$$\left|\begin{matrix} e_r & e_{\theta} & e_{z} \\ \frac{1}{r}\frac{\partial }{\partial r} & \frac{1}{r} \frac{\partial }{\partial \theta} & \frac{\partial }{\partial z} \\ v_r & rv_\theta & v_z \end{matrix} \right|$$
Is that correct?