Divergence Theorem — Question 9

Question 9

On $\mathbb R^3\setminus\{\mathbf 0\}$, consider $$\mathbf F(x,y,z)=\frac{\langle x,y,z\rangle}{(x^2+y^2+z^2)^{3/2}}.$$ Let $S$ be the outward-oriented unit sphere.

Tasks

1. Compute the flux through $S$ directly.

2. Explain why zero divergence away from the origin does not permit applying the Divergence Theorem to the full unit ball.

3. Reconcile the result using a punctured ball.

Question 9 – Solution

Strategy. The nonzero flux is caused by a singularity inside the sphere; remove a small ball so the theorem’s hypotheses hold.

Step 1: Direct flux On the unit sphere, $\mathbf F=\mathbf n$, so $$\boxed{\iint_S\mathbf F\cdot\mathbf n\,dS =\iint_S1\,dS=4\pi}.$$ Although $\nabla\cdot\mathbf F=0$ for $r>0$, the field is undefined at the origin and is not smooth throughout the full ball.

Step 2: Puncture the ball Remove the ball $r<\varepsilon$. On the shell $\varepsilon\le r\le 1$, the divergence is zero, so the total outward flux is zero.

Step 3: Track both boundaries The outer flux is $4\pi$. On $r=\varepsilon$, the shell’s outward normal is $-\mathbf e_r$, while $\mathbf F=\mathbf e_r/\varepsilon^2$. Thus $$\Phi_{r=\varepsilon}=-\frac 1{\varepsilon^2}(4\pi\varepsilon^2)=-4\pi.$$ Therefore $4\pi-4\pi=0$, consistent with the theorem on the punctured region.

Verification The inner flux remains $-4\pi$ as $\varepsilon\to 0^+$, recording the enclosed point source rather than disappearing.