This derivation is the application of the Divergence Theorem to a vector-valued surface integral by treating it component-wise. By defining the integrand as a product of a coordinate $x_i$ and a vector field $G =x r^{-5}$, we utilize the product rule for divergence to show that the spatial variation of the magnitude exactly cancels out a portion of the field's divergence. In the region excluding the origin, the divergence of the radial part $x r^{-5}$ simplifies to $-2 r^{-5}$, which, when combined with the gradient of the coordinate term, yields a remarkably simple scalar field. Ultimately, the transformation demonstrates that the outward flux of this specific weighted vector field is equivalent to a volume-distributed source characterized by the scalar function $\phi(x)=-r^{-5}$.
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