I was given the above question. I am totally stuck. I have been spending quite a bit of time trying to solve it. I can't really give my steps because I haven't gotten anywhere with it yet. I'm looking for some help. No, I'm not trying to get you to do my homework for me - I need to know how to solve this kind of problem - and I simply can't figure it out. I don't want just the answer to the question - I'm trying to develop an understanding of how to work problems like this myself in the future.
Any help is much appreciated! Thank you!
a) ${\rm div}\ F = 0+x^2+y^2 $ so that \begin{align*} \int\int_{\partial Q}\ F\cdot n \ dS &= \int_Q \ {\rm div}\ F \ d{\rm vol}_Q\\&= \int_0^1\int_0^{2\pi} \int_{r=0}^{\sqrt{4-z}}\ (x^2+y^2)\ rdrd\theta dz \\&= \int \int\int \ r^3 drd\theta dz \\&= \int\ \frac{1}{4} (4-z)^2 2\pi\ dz \\&= \frac{\pi}{6} (4^3-3^3) \end{align*}
b) Here $\partial Q$ is a union of top, bottom and side faces : For the side, we have a parametrization $T(x,y)=(x,y,4-x^2-y^2)$ so that $$ T_x = (1,0,-2x),\ T_y = (0,1,-2y) $$
Hence $ n = \frac{(2x,2y,1)}{\sqrt{4r^2+1}} $ so that $ n\cdot F =0 $
$$ \int_{{\rm top\ face}} \ F\cdot n = \int_{x^2+y^2\leq 3} \ -2xe^{r^2+1} = 0 $$ by symmetry
$$ \int_{{\rm bottom \ face}}\ F\cdot n = \int_{ x^2+y^2\leq 4}\ 0 =0 $$
Hence $\int\int_{\partial Q}\ F\cdot n \ dS =0$.