Evaluation of  $\int {\int }_S\left(\nabla \times \mathbf{F}\right)·\mathbf{N} \mathrm{d\σ}$  requires the curl of F:

$\nabla \times \mathbf{F}\=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ {\partial }_x& {\partial }_y& {\partial }_z\\ z& -x& -y\end{array}\right|$ = $\left[\begin{array}{c}-1\\ 1\\ -1\end{array}\right]$ 

Then, it requires a unit normal field, here taken from the position-vector parametrization

$\mathbf{R}\=\left[\begin{array}{c}\sin \(\mathrm{\φ}\)\cos \(\mathrm{\θ}\)\\ \sin \(\mathrm{\φ}\)\sin \(\mathrm{\θ}\)\\ \cos \(\mathrm{\φ}\)\end{array}\right]$ 

as $\mathbf{N}\={\mathbf{R}}_{\mathrm{\φ}}\times {\mathbf{R}}_{\mathrm{\θ}}$ = $\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \cos \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)& \cos \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)& -\sin \left(\mathrm{\φ}\right)\\ -\sin \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)& \sin \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)& 0\end{array}\right|$ = $\left[\begin{array}{c}{\sin \left(\mathrm{\phi }\right)}^2\cos \left(\mathrm{\theta }\right)\\ {\sin \left(\mathrm{\phi }\right)}^2\sin \left(\mathrm{\theta }\right)\\ \cos \left(\mathrm{\phi }\right)\sin \left(\mathrm{\phi }\right)\end{array}\right]$. For $\mathrm{d\σ}$, take

$\mathrm{d\σ}\=∥{\mathbf{R}}_{\mathrm{\phi }}\times {\mathbf{R}}_{\mathrm{\θ}}∥ \mathrm{dA}$ = $\sin \left(\mathrm{\φ}\right) \mathrm{dA}$ 

Hence,

 $\left(\nabla \times \mathbf{F}\right)·\mathbf{N}$ = $\left[\begin{array}{c}-1\\ 1\\ -1\end{array}\right]·\left[\begin{array}{c}{\sin \left(\mathrm{\phi }\right)}^2\cos \left(\mathrm{\theta }\right)\\ {\sin \left(\mathrm{\phi }\right)}^2\sin \left(\mathrm{\theta }\right)\\ \cos \left(\mathrm{\phi }\right)\sin \left(\mathrm{\phi }\right)\end{array}\right]$ = $$$\sin \left(\mathrm{\phi }\right) \left(\sin \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)-\sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)-\cos \left(\mathrm{\phi }\right)\right)$

and

${\int }_0^{2 \mathrm{\π}}{\int }_0^{\mathrm{\π}}{\sin }^2\left(\mathrm{\phi }\right) \left(\sin \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)-\sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)-\cos \left(\mathrm{\phi }\right)\right) \mathit{\ⅆ}\mathrm{\φ} \mathit{\ⅆ}\mathrm{\θ}$ = $0$ 

Should the "Clear All and Reset" button in the Task Template be pressed, all the data that has been input to the template will be lost. In that event, the reader should simply re-launch the example to recover the appropriate inputs to the template.