There's a typo in the last two expressions you write: $u^{\theta } /r$ should be $u^r /r$.

Then, it seems to check out (for any well-behaved $J$): In the first expression, just insert the expressions for the $\omega $ in terms of the $u$; in the second expression, note $$ \nabla \times (u^{\theta } e_{\theta } ) = e_z \partial_{r} u^{\theta } + e_z u^{\theta } /r - e_r \partial_{z} u^{\theta } $$ and $$ \nabla (u^r /r) = e_r \left( \frac{1}{r} \partial_{r} u^r -\frac{1}{r^2 } u^r \right) + e_{\theta } \frac{1}{r^2 }\partial_{\theta } u^r +e_z \frac{1}{r} \partial_{z} u^r $$ and multiply that out - no integration needed, already the integrands are identical. To get to the third expression, consider the volume integral of $$ \nabla \cdot \left[ (u^{\theta } e_{\theta } ) \times (J\nabla(u^r /r)) \right] $$ You don't say anything about the boundary conditions, but I suppose the field in the square brackets doesn't have net flux out of the volume - then this vanishes by Gauss' theorem. Also, the $u$ have to regular enough so that different components of $\nabla $ commute on them ...

There's a typo in the last two expressions you write: $u^{\theta } /r$ should be $u^r /r$.

Then, it seems to check out (for any well-behaved $J$): In the first expression, just insert the expressions for the $\omega $ in terms of the $u$; in the second expression, note $$ \nabla \times (u^{\theta } e_{\theta } ) = e_z \partial_{r} u^{\theta } + e_z u^{\theta } /r - e_r \partial_{z} u^{\theta } $$ and $$ \nabla (u^r /r) = e_r \left( \frac{1}{r} \partial_{r} u^r -\frac{1}{r^2 } u^r \right) + e_{\theta } \frac{1}{r^2 }\partial_{\theta } u^r +e_z \frac{1}{r} \partial_{z} u^r $$ and multiply that out - no integration needed, already the integrands are identical. To get to the third expression, consider the volume integral of $$ \nabla \cdot \left[ (u^{\theta } e_{\theta } ) \times (J\nabla(u^r /r)) \right] $$ You don't say anything about the boundary conditions, but I suppose the field in the square brackets doesn't have net flux out of the volume - then this vanishes by Gauss' theorem. Also, the $u$ have to be regular enough so that different components of $\nabla $ commute on them ...

There's a typo in last two expressions you write: $u^{\theta } /r$ should be $u^r /r$.

Then, it seems to check out (for any well-behaved $J$): In the first expression, just insert the expressions for the $\omega $ in terms of the $u$; in the second expression, note $$ \nabla \times (u^{\theta } e_{\theta } ) = e_z \partial_{r} u^{\theta } + e_z u^{\theta } /r - e_r \partial_{z} u^{\theta } $$ and $$ \nabla (u^r /r) = e_r \left( \frac{1}{r} \partial_{r} u^r -\frac{1}{r^2 } u^r \right) + e_{\theta } \frac{1}{r^2 }\partial_{\theta } u^r +e_z \frac{1}{r} \partial_{z} u^r $$ and multiply that out - no integration needed, already the integrands are identical. To get to the third expression, consider the volume integral of $$ \nabla \cdot \left[ (u^{\theta } e_{\theta } ) \times (J\nabla(u^r /r)) \right] $$ You don't say anything about the boundary conditions, but I suppose the field in the square brackets doesn't have net flux out of the volume - then this vanishes by Gauss' theorem. Also, the $u$ also have to regular enough so that different components of $\nabla $ commute on them ...

There's a typo in the last two expressions you write: $u^{\theta } /r$ should be $u^r /r$.

Then, it seems to check out (for any well-behaved $J$): In the first expression, just insert the expressions for the $\omega $ in terms of the $u$; in the second expression, note $$ \nabla \times (u^{\theta } e_{\theta } ) = e_z \partial_{r} u^{\theta } + e_z u^{\theta } /r - e_r \partial_{z} u^{\theta } $$ and $$ \nabla (u^r /r) = e_r \left( \frac{1}{r} \partial_{r} u^r -\frac{1}{r^2 } u^r \right) + e_{\theta } \frac{1}{r^2 }\partial_{\theta } u^r +e_z \frac{1}{r} \partial_{z} u^r $$ and multiply that out - no integration needed, already the integrands are identical. To get to the third expression, consider the volume integral of $$ \nabla \cdot \left[ (u^{\theta } e_{\theta } ) \times (J\nabla(u^r /r)) \right] $$ You don't say anything about the boundary conditions, but I suppose the field in the square brackets doesn't have net flux out of the volume - then this vanishes by Gauss' theorem. Also, the $u$ have to regular enough so that different components of $\nabla $ commute on them ...