Vorticity of a velocity field in cylindrical coordinates [closed]

I'm currently working on a obtaining the vorticity of my velocity field $u_r, u_\theta, u_x$. I know that this is equal to the curl of the velocity field $\nabla \times u$:

$$\nabla \times u = \frac{1}{r} \begin{bmatrix} \boldsymbol{e_r} & r\boldsymbol{e_\theta} & \boldsymbol{e_x} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial\theta} & \frac{\partial}{\partial x} \\ u_r & r u_\theta & u_x \end{bmatrix}$$

Am I right in saying that this is the same as:

$$\nabla \times u = \frac{1}{r}\left[ \left(\frac{\partial u_x}{\partial\theta} - \frac{\partial (r u_\theta)}{\partial x}\right)\boldsymbol{e_r} + \left(r\left(\frac{\partial u_x}{\partial r} - \frac{\partial u_r}{\partial x}\right)\right)\boldsymbol{e_\theta} + \left(\frac{\partial(r u_\theta)}{\partial r} - \frac{\partial u_r}{\partial \theta}\right)\boldsymbol{e_x} \right]$$

I'm currently thinking that I'm interpreting the $r\boldsymbol{e_\theta}$ incorrectly and that the second term should actually have a $\frac{1}{r}$ and not a $r$ in front.

closed as off-topic by Kyle Kanos, tpg2114♦, stafusa, Chris♦, Emilio PisantyFeb 12 '18 at 15:04

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The correct curl in cylindrical coordinates is $$\left(\frac{1}{r}\frac{\partial u_x}{\partial \theta}- \frac{\partial u_\theta}{\partial x}\right)\mathbf{e_r}+ \left(\frac{\partial u_r}{\partial x}-\frac{\partial u_x}{\partial r}\right)\mathbf{e_\theta}+ \frac {1}{r}\left(\frac{\partial (r u_\theta)}{\partial r}-\frac{\partial u_r}{\partial \theta}\right)\mathbf{e_x},$$ as you can see in Wikipedia. Your only mistake was the sign of the term in the direction $\mathbf{e_\theta}$.