Let $\Sigma$ be the portion of the paraboloid $z=1-x^{2}-y^{2}$ where $x\geq0$ and $z\geq0$. Consider the vector fields $\mathbf{u}=(xy,xz^{2},x^{2}y)$ and $\mathbf{v}=\mathrm{curl}\,{\mathbf{u}}$. Compute the flux of $\mathbf{v}$ through $\Sigma$.
The solution manual solves the problem by applying Stokes theorem as follows-
$$\iint_\Sigma \nabla \times \mathbf{u} \cdot d\boldsymbol{\Sigma} = \oint_{\partial\Sigma_1} \mathbf{u} \cdot d\mathbf{r}\ +\oint_{\partial\Sigma_2} \mathbf{u} \cdot d\mathbf{r}\,$$
where $\partial\Sigma_1$ and $\partial\Sigma_2$ are the boundary curves of $\Sigma$. $\partial\Sigma_1$ in the xy-plane and $\partial\Sigma_2$ in the yz-plane.
The problem for me is while I understand that $\mathbf{v}=\mathrm{curl}\,{\mathbf{u}}$ I feel that to apply Stokes correctly one should instead calculate
$$\iint_\Sigma \nabla \times (\nabla \times \mathbf{u}) \cdot d\boldsymbol{\Sigma} = \oint_{\partial\Sigma_1} \mathbf{v} \cdot d\mathbf{r}\ +\oint_{\partial\Sigma_2} \mathbf{v} \cdot d\mathbf{r}\,.$$
Why am I wrong?
Any help appreciated.
Both formulas you wrote are correct, but the first one is the flux of $\mathbf{v}$ through $\Sigma$: $$\text{flux}=\int \int_\Sigma\mathbf{v} \cdot d\mathbf{\Sigma} =\int \int_\Sigma \nabla \times \mathbf{u} \cdot d\mathbf{\Sigma}$$ while the second one is the flux through $\Sigma$ of $\nabla \times \mathbf{v}$