- #1
jameson2
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Homework Statement
Use the divergence theorem to find the outward flux of a vector field [tex]
F=\sqrt{x^2+y^2+z^2}(x\hat{i}+y\hat{j}+z\hat{k})[/tex] across the boundary of the region [tex]1\leq x^2+y^2+z^2 \leq4
[/tex]
Homework Equations
The Gauss Divergence Theorem states [tex]\int_D dV \nabla \bullet F=\int_S F\bullet dA [/tex] where D is a 3d region and S is it's boundary.
The Attempt at a Solution
First, I sketched out the boundary, which I think is a sphere of radius 2 with a cavity f radius 1 at the centre. The formula requires that S is oriented outwards.
I basically need to know if the way to do this is first to treat it first as a sphere of radius 2 without a cavity, and work out the outward flux throught this. Then treat the cavity as a sphere of radius 1 and work out the flux going into this, and add the two.
If this is the right approach, I'm not sure how to treat an inward pointing area element. The formula seems to heavily stress that the outward normal is taken, and I don't know if taking an inward normal is allowed.
Alternatively, I think I might be able to take the region as a whole straightaway, and then when integrating over the volume just take the limits of the radius as 2 and 1.
This way, which is the only way can actually get an answer at the moment, gives me an answer for the flux as[tex]\int_S F\bullet dA=48\pi [/tex]
Any help would be much appreciated.