certain surface integral on the surface of the solid bounded by the sphere $x^2+y^2+z^2=10$ and the paraboloid $x^2+y^2=z-2$

Question

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 $\vec F=(4xz,-y^2,4yz)$. How to evaluate $\iint_S\vec F\cdot\vec n dS$ where $\vec n$ is the outward unit normal to the surface $S$? Should I use Gauss divergence theorem? Please help. Thanks in advance.

Answer 1

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 $$ \Phi=\int_0^{2\pi} \int_0^{3}\int_{r^2+2}^{\sqrt{10-r^2}}(4z+2r\sin\theta)\;rdzdrd\theta =-783 \pi $$