= Volume integral of a homogeneous function
{title2=$\int_Vf\,dV=\frac1{n+3}\int_{\partial V}f\,x\cdot n\,dS$}
For a sufficiently regular <homogeneous function> $f$ of degree $n$ on a three-dimensional region, <Euler theorem for homogeneous functions> gives $x\cdot\nabla f=nf$. Hence $\nabla\cdot(fx)=(n+3)f$, and the <divergence theorem> gives the displayed reduction when $n\ne-3$. At degree $-3$ only the zero-flux identity follows, so division by $n+3$ is unavailable. Singular functions additionally require an excision or limiting argument; the surface contribution at a removed singularity must not be silently dropped. On the sloping side of a cone with its apex at zero, $x$ is tangent to the side and $x\cdot n=0$, so only the cap contributes to this formula.
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