= Solution
For a continuously differentiable <vector field> on a bounded region with suitably piecewise smooth boundary, the <divergence theorem> gives
$$
\oint_S\mathbf F\cdot\hat{\mathbf n}\,dS=\int_V\nabla\cdot\mathbf F\,dV,
$$
where the <normal vector> points outward. In three dimensions, the <position vector> is $\mathbf r=(x,y,z)$ and $\nabla\cdot\mathbf r=1+1+1=3$. Therefore
$$
\boxed{\oint_S\mathbf r\cdot d\mathbf S=3V.}
$$
The result depends only on the enclosed <volume>, not on the shape or its location relative to the origin. Reversing the surface orientation reverses the sign.
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