Question: Let \(S\) be the part of the spherical surface \(x^{2}+y^{2}+z^{2}=4\), lying in \(x^{2}+y^{2}>1\), which is to say outside the cylinder of radius one with axis the \(z\) -axis.

1. (5 points) Compute the flux outward through \(S\) of the vector field \(\mathbf{F}=y \mathbf{i}-x \mathbf{j}+z \mathbf{k}\).
2. (5 points) Show that the flux of this vector field through any part of the cylindrical surface is zero.
3. (5 points) Using the divergence theorem applied to \(\mathrm{F}\), compute the volume of the region between \(S\) and the cylinder.

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