= Closure of outward face area vectors
{title2=$\sum_F A_F\mathbf n_F=\mathbf0$}
For a closed <polyhedron>, let $A_F$ be a face area and $\mathbf n_F$ its outward <unit normal vector>. The sum of the face area <vectors> is zero. To prove it, fix any constant <vector> $\mathbf q$ and apply the <divergence theorem> to the constant <vector field> $\mathbf q$: its <divergence> vanishes, while its outward flux is $\mathbf q\cdot\sum_F A_F\mathbf n_F$. This is zero for every $\mathbf q$, so the sum vanishes. For a <tetrahedron>, the same identity follows directly by expanding the four oriented face <cross products>; multiplying every face <vector> by two leaves the identity intact.
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