= Solution
From the preceding gradient, $\partial_x^2(r^{-1})=-r^{-3}+3x^2r^{-5}$. Summing the three Cartesian second <partial derivatives> gives
$$
\Delta(1/r)=-3r^{-3}+3(x^2+y^2+z^2)r^{-5}=0,\qquad r>0.
$$
Thus the reciprocal radius is <harmonic> away from the origin. Its negative gradient is
$$
\boxed{E=\frac{r}{r^3},\qquad\nabla\times E=0\quad(r>0).}
$$
The <curl> vanishes because $E$ is a gradient field and mixed second derivatives commute. The exclusion of the origin is essential: the potential and field are singular there.
Back to article page
