Chapter 6.7, Problem 327E

For the following exercises, without using Stokes’ theorem, calculate directly both the flux of curl $\text{F}\cdot \text{N}$ over the given surface and the circulation integral around its boundary, assuming all boundaries are oriented clockwise as viewed from above.

327. $\text{F}(x,y,z)=z\text{i}+x\text{j}+y\text{k}$ ; $S$ is hemisphere $z={(a^2-x^2-y^2)}^{\frac{1}{2}}$ .