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Let $\vec{F}:U ⊆ R ^3 → R ^3$ be the vector field

$(x/ (x ^2 + y ^2 + z ^2 ) ^{3/2} , y/ (x ^2 + y^ 2 + z^ 2 ) ^{3/2} , z/ (x^ 2 + y^ 2 + z^ 2 ) ^{3/2}) $

where U is $R ^3$ \ (0, 0, 0). Compute the flux of the vector field $\vec{F}$ through the surface S given by taking the sphere of radius 12345 centered at the origin, chopping off a small section from the top of the sphere, and replacing the chopped-off section with a flat disk, so that S remains closed.

Assume S is oriented with an outward-pointing normal.

I understand how to compute the flux of the vector field, but I am confused on how I can compute it through the surface S. Any help would be appreciated, thanks!

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Use Gauss’s divergence theorem to do this problem. Note that the vector field in spherical coordinates is $F=\frac{1}{r^2}\hat{r}$. It’s divergence is zero in the region (Call that volume V) between the flat portion (disk D) and spherical cap C you just cut off. Let the remaining portion of the spherical surface be R.

Since the volume integral of the divergence over V is zero, by the divergence theorem, the flux outward through the flat portion is the same as the flux outward through the spherical cap.

Divergence theorem says $$\int_V \operatorname{div} F = 0 = \int_C F \cdot \hat{n} - \int_D F \cdot \hat{n}$$

Here I have chosen the unit normal vectors on both surfaces to be the ones pointing away from the origin. Hence the negative sign. So

$$ \int_C F \cdot \hat{n} = \int_D F \cdot \hat{n}$$ $$ So the flux through S is the same as the flux through the whole sphere.

$$ \int_S F \cdot \hat{n} = \int_{R+D} F \cdot \hat{n} = \int_{R+C} F \cdot \hat{n} = \int_{\mbox{sphere}} F \cdot \hat{n} = \frac{1}{R^2} A$$ where A is the area of the sphere of the given radius R.

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  • $\begingroup$ Okay I understand your explanation, but I don't get how to solve it. Can you please show the computation? Thanks $\endgroup$ – Connor Brown Dec 9 '17 at 3:38
  • $\begingroup$ @ConnorBrown which computation? The flux through the whole sphere or the application of Guass’s theorem over V? $\endgroup$ – Mathemagical Dec 9 '17 at 3:39
  • $\begingroup$ Both, if possible! $\endgroup$ – Connor Brown Dec 9 '17 at 3:40
  • $\begingroup$ Okay thanks. So now I would just plug in 12345 for R and pi$12345^2$ for A and solve? $\endgroup$ – Connor Brown Dec 9 '17 at 4:05
  • $\begingroup$ @ConnorBrown right, except...the surface area of a sphere is not πr ²... there is a factor of 4, no? $\endgroup$ – Mathemagical Dec 9 '17 at 4:06

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