# Using Gauss' Theorem to evaluate a flux integral?

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• Sep 17th 2010, 10:50 AM
chocaholic
Using Gauss' Theorem to evaluate a flux integral?
Use Gauss' Theorem to evaluate the flux integral:
SS(curlF).ndS
where SS represents the double integral with respect to the surface S and where
F(x,y,z)=(x,y,z) and surface S is given by:
S={(x,y,z) such that z=1-x^2-y^2 and z>=0}.

I know how Gauss' theorem relates a flux integral to a triple integral of the div of the vector field but when I try to make curlF the vector field so that I can use Gauss' theorem to evaluate the triple integal of div(curlF) over the region bounded by S I can't because I get (curlF)=0 and I don't know where to go from there.Any help would be greatly appreciated.Thanks in advance.
• Sep 18th 2010, 05:44 AM
HallsofIvy
Rather a silly problem! Yes, the flow F(x,y,z)= (x, y, z), which is a flow directly away from the origin is "curl free" or "irrotational": $\displaystyle \nabla\time F= curl F= 0$. You don't need Gauss's theorem, the flux integral is 0. Are you sure you have the right F and are you sure you want $\displaystyle \int\int curl F\cdot dS$ rather than just $\displaystyle \int\int F\cdot dS$? If I remember correctly, the latter is the "flux" of F through the surface.
• Sep 19th 2010, 01:51 AM
chocaholic
Hi hallsofivy yeah I couldn't believe they'd ask something like this for 10 marks either but that's exactly what the question says. I figured that for 10 marks the answer couldn't just be 0 and therefore I must be doing something wrong but now I think they made an error in the question either with F or in what integral should be evaluated.