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Let $S$ be the surface of the solid bounded by the sphere $x^2+y^2+z^2=10$ and the paraboloid $x^2+y^2=z-2$ , let $\vec F=(4xz,-y^2,4yz)$ , then how to evaluate $\iint_S\vec F.\vec n dS$ , where $\vec n$ is the outward unit normal to the surface $S$ ? Should I use Gauss divergence theorem ? Please help . Thanks in advance

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Yes, using the Gauss divergence theorem is a good idea: $$ \Phi=\iint_S \vec{F}\cdot d\vec{S} = \iiint_E \nabla \cdot \vec{F}\; dV, $$ where $E$ is the region bounded by the sphere $x^2+y^2+x^2=10$ and the paraboloid $x^2+y^2=z-2$, that is: $$ E=\{(r,\theta,z)\;|\; 0 \le \theta \le 2\pi ,0\le r \le3,r^2+2\le z \le \sqrt{10-r^2}\}, $$ and $\nabla \cdot \vec{F}$ is the divergence of your field: $$ \nabla \cdot \vec{F} =4z-2y+4y=4z+2y $$ If follows that $$ \Phi=\int_0^{2\pi} \int_0^{3}\int_{r^2+2}^{\sqrt{10-r^2}}(4z+2r\sin\theta)\;rdzdrd\theta =-783 \pi $$

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