I have been asked to compute the flux through a hemisphere radius 2 centred at the origin oriented downward with $$ \overrightarrow{F}=(y,-x,2z) $$ I have worked out that $$ \hat{n}=-\frac{(x,y,z)}{2} $$ and in spherical co-ordinates, $$ dS=4\sin(\gamma)d\gamma d\theta. $$ So the flux is given by $$ \int_0^{2\pi}\int_0^{\frac{\pi}{2}}-2\cos(\gamma)4\sin(\gamma)d\gamma d\theta. $$

My question is, do I also need to calculate the flux through the bottom of the hemisphere, namely the disk $x^{2}+y^2=4$ on the $xy$ plane, or is this included in the integral above?

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    $\begingroup$ It's included neither in the question nor in the integral you wrote. $\endgroup$ – Ted Shifrin Mar 5 '19 at 18:29
  • $\begingroup$ @TedShifrin how do you mean? Surely as the bottom part of the hemisphere is part of the surface it should be included in some part of the calculation? $\endgroup$ – Anon Mar 5 '19 at 20:02
  • $\begingroup$ Only if the instructions say to include the disk at the bottom of the hemisphere. $\endgroup$ – Ted Shifrin Mar 5 '19 at 20:10
  • $\begingroup$ Okay, thanks for clarifying $\endgroup$ – Anon Mar 5 '19 at 20:16

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