Cylindrical coordinates in axis symmetric flow

Question

I am getting stuck in a detail in a paper. It's about the axi symmetric Navier Stokes equations $$u_t - \nu\Delta u + u\cdot \nabla u + \nabla p=0$$ We consider in cylindrical coordinates $u=(u^{r}, u^{\theta},u^{z})$. And we have the following vorticity equation in cylindrical form. $$\omega^r = \frac{1}{r}\frac{\partial u^z}{\partial \theta} - \frac{\partial u^\theta}{\partial z} = - \frac{\partial u^\theta}{\partial z}, \\ \omega^\theta = \frac{\partial u^r}{\partial z} - \frac{\partial u^z}{\partial r}, \\ \omega^z = \frac{1}{r}\frac{\partial}{\partial r}(r u^\theta) - \frac{1}{r} \frac{\partial u^r}{\partial \theta} = \frac{1}{r}\frac{\partial}{\partial r}(r u^\theta).$$ My question goes as follows: Suppose $J=\frac{w^{r}}{r}$, then how to derive the following steps? $$\int J(\omega^r \partial_{r}+\omega^z \partial_{z})\frac{u^r}{r}rdrdz=\int [\nabla\times (u^{\theta}e_{\theta})](J\nabla \frac{u^{\theta}}{r}) rdrdz=\int (u^{\theta}e_{\theta})(\nabla J\times\nabla \frac{u^{\theta}}{r}) rdrdz$$ I'm assuming some integration by parts is involved but couldn't derive them; besides, where does the cross product come from? The equality above that bugs me is from the bottom of page 11 of this paper.