Question: Let \(\mathbf{F}=-2 x z \mathbf{i}+y^{2} \mathbf{k}\)

1. ( 3 points) Calculate curl \(\mathrm{F}\).
2. (5 points) Show that \(\iint_{\mathcal{R}} \operatorname{curl} \mathbf{F} \cdot \mathbf{n} d S=0\) for any finite portion \(\mathcal{R}\) of the unit sphere \(S\) given by \(x^{2}+y^{2}+z^{2}=1\) with outward facing normal vector \(\mathbf{n}\).
3. (5 points) Show that \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}=0\) for any simple closed curve \(C\) on the unit sphere \(x^{2}+y^{2}+z^{2}=1\)

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