026ca6…: “Apart from the breezy butterfly”

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Consider the octahedron with vertices: $(1,0,0) \quad (0,1,0)\quad (0,0,1)$ $(-1,0,0) \quad (0,-1,0)\quad (0,0,-1)$ Let this octahedron be oriented with outward pointing normal vectors. Compute the flux of the field $\vec {F}=\vector {x,y,z}$ across the surface of the octahedron. $\text {Flux} = \answer {4}$

You will need to compute: $\iint _O \vec {F}\dotp \uvec {n}\d S$ where $O$ is the octahedron above.

One way to proceed is to break the integral above into $8$ different integrals, one for each face of the octahedron.

Start by considering the face of the octahedron in the first octant, call it $O_{I}$ . In this case $\uvec {n} = \vector {1/\sqrt {3},1/\sqrt {3},1/\sqrt {3}}$ .

Here $\vec {F}\dotp \uvec {n} = x/\sqrt {3}+y/\sqrt {3}+z/\sqrt {3}$ .

Moreover on this side, $x + y + z = 1$ .

$\begin{align*} \iint_{O_I} \vec{F}\dotp\uvec{n}\d S &= \iint_{O_I} 1/\sqrt{3} \d S\\ &= 1/\sqrt{3} \iint_{O_I} \d S\\ &= 1/\sqrt{3} \cdot \sqrt{3}/2\\ &= 1/2. \end{align*}$

The calculations for the other five sides are similar.