<s>[INST] Laplacian of a Smooth Function $f(\vec x) = 1/\|\vec x \|$ for $\| \vec x \| \geq 1$ This is a multivariable calculus problem from a past prelim exam. I have an answer for this written up (posted below), but it seemed rather time-intensive. If there is a slicker way to approach this problem, I'd appreciate seeing it. Thanks!

Recall that for a smooth function $f: \mathbb{R}^3 \to \mathbb{R}$, the Laplacian of $f$ is defined by
  $$ \Delta f = \nabla \cdot ( \nabla f). $$
Suppose that $f: \mathbb{R}^3 \to \mathbb{R}$ is a smooth function satisfying $f(\vec{x}) = 1/\|\vec{x}\|$ for $\|\vec{x}\| \geq 1$.
  
  
*
  
*Verify that $\Delta f(\vec{x}) = 0$ for $\|\vec{x}\| \geq 1$.
  
*Compute $\int_{\mathbb{R}^3} \Delta f \, dV$.

 [/INST] *

*In rectangular coordinates, for $\|\vec{x}\| \geq 1$,
$
\begin{align*}
f(\vec{x}) &= \frac{1}{\sqrt{x^2+y^2+z^2}} \\
\nabla f &= \left\langle \frac{-x}{(x^2+y^2+z^2)^{3/2}}, \frac{-y}{(x^2+y^2+z^2)^{3/2}}, \frac{-z}{(x^2+y^2+z^2)^{3/2}} \right\rangle \\
&= \frac{-1}{(x^2+y^2+z^2)^{3/2}} \left\langle x,y,z \right\rangle \\
\nabla \cdot \nabla f &= \frac{ (-1) (x^2+y^2+z^2)^{3/2} - (-x)\frac{3}{2}(x^2+y^2+z^2)^{1/2}(2x) }{(x^2+y^2+z^2)^3} \\
&+ \frac{ (-1) (x^2+y^2+z^2)^{3/2} - (-y)\frac{3}{2}(x^2+y^2+z^2)^{1/2}(2y) }{(x^2+y^2+z^2)^3} \\
&+ \frac{ (-1) (x^2+y^2+z^2)^{3/2} - (-z)\frac{3}{2}(x^2+y^2+z^2)^{1/2}(2z) }{(x^2+y^2+z^2)^3} \\
&= \frac{-3(x^2+y^2+z^2) + 3x^2+3y^2+3z^2}{(x^2+y^2+z^2)^{5/2}} \\
&= 0
\end{align*}
$

*Since $\Delta f = 0$ for $\|\vec x\| \geq 1$, then $\int_{\mathbb{R}^3} \Delta f \, dV = \int_{\| \vec x \| \leq 1} \Delta f \, dV$. Then by the divergence theorem,
$\begin{align*}
\int_{\| \vec x \| \leq 1}   \nabla \cdot ( \nabla f) dV &= \iint_{\|\vec{x}\|=1} (\nabla f) \cdot \vec n \, dS \\
&= \iint_{\|\vec{x}\|=1} \frac{-1}{(x^2+y^2+z^2)^{3/2}} \left\langle x,y,z \right\rangle \cdot \frac{ \left\langle x,y,z \right\rangle }{(x^2+y^2+z^2)^{1/2}}dS \\
&= \iint_{\|\vec{x}\|=1} \frac{-(x^2+y^2+z^2)}{(x^2+y^2+z^2)^2} dS \\
&= \iint_{\|\vec{x}\|=1}-1 dS \\
&= -4 \pi
\end{align*}$
</s>