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The vectorfield $F$ is given by $F(x,y,z) = [x^2y+z^2, xcoz(z)-xy^2, x^3+3z]$.

Let $S$ be the cylindric surface defined by $x^2+4y^2=1$ where $0 \leq z \leq 8$.

Calculate $$\iint_S F \cdot \hat{N} dS$$ where the unitnormal $\hat{N}$ points away from the $z$ axis.

Once can easily find div $F = 3$. From there I simply used the divergence theorem and the fact that we're dealing with an ellipic cylinder to find the flux. We simply get $3*volume(T) = 3*\pi*1*\frac{1}{2}*8 = 12\pi$.

I thought I was done, but in the solution they find the flux across the surface like I did, and then they treat the surface where $z=0$ and where $z=8$. Doing this we get that the flux out of the surface at $z=0$ is equal to $0$ whereas the flux out of the surface at $z=8$ is equal to $12\pi$, and thus $$\iint_S F \cdot \hat{N} dS = 0$$

What they have found here is obviously the flux across the "cylindric" portion of the surface, excluding the ellipse at the bottom and the top. Shouldn't this be specified in the task itself? I thought since the inequalities are not strict, we are simply finding the flux across the entire surface, including at $z=0$ and $z=8$. Shouldn't the inequalities be strict (as in $0<z<8$ ), if we're only going to talk about the flux out of the "cylindric" part of the surface?

I hope the question is clear even tho my use of "cylindric part" is probably not the right terminology.

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The question seems asking to find the flux across the lateral surface and the method used is the following

  • determine the overall flux of $12\pi$ by divergence theorem
  • calculate directly the flux for the bottom ($=0$) and the top ($=12\pi$)
  • determine by difference the flux across the lateral surface ($=12\pi-12\pi=0$)
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  • $\begingroup$ Yes, but is that obvious from the way the task was asked? I thought the task asked about the flux across the entire surface, how am I supposed to know that they were infact asking for the flux across the lateral surface, when this was not specified? $\endgroup$
    – novo
    May 21, 2018 at 16:08
  • $\begingroup$ @novo The problem states "Let $S$ be the cylindric surface defined by $x^2+4y^2=1$ where $0 \leq z \leq 8$" and thus I think that top and bottom are excluded. $\endgroup$
    – user
    May 21, 2018 at 16:13
  • $\begingroup$ Ofc, I forgot that the equation defines an elliptic cylinder with no height and bottom initially, and to use the divergence theorem we need to close it at z=0 and z=8. Sorry, that was a bad misinterpretation from my side. Thank you for the clarification, sir! $\endgroup$
    – novo
    May 21, 2018 at 17:13

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