I'm asked to determine the flow of the field $$\left(\frac{x}{x^2+y^2+z^2},\frac{y}{x^2+y^2+z^2},\frac{z}{x^2+y^2+z^2}\right)$$ out from $$K : 2 \leq x^2 + y^2 + z^2 \leq 3.$$
My attempt:
What is relevant to the question is how much flow is produced $x^2 + y^2 + z^2 \leq 3.$
I performed the substitution: $$\begin{cases}x = r\sin{\theta}\cos{\phi}\\ y = r\sin{\theta}\sin{\phi}\\ z = r\cos{\theta}\end{cases}$$
where E : $0\leq r \leq \sqrt{3}, \quad 0 \leq \theta \leq \pi, \quad 0 \leq \phi \leq 2\pi.$
The divergence of the field is equal to
$$\frac{1}{x^2+y^2+z^2}=\frac{1}{r^2}$$
I use Gauss's theorem:
$$\iint_{\partial K}(F\cdot n)dS = \iiint_K(divF)dxdydz$$ $$= \iiint_E \left(\frac{1}{r^2}\cdot r^2\sin{\theta}\right)drd\phi d\theta = \sqrt{3}\cdot 2\pi \cdot \int_0^\pi \sin{\theta}d\theta = \sqrt{3}\cdot 2\pi \cdot (-1-(-1)) = 0$$
The answer is supposed to be $4\pi(\sqrt{3} - \sqrt{2})$.
What am I doing wrong and how would I solve this?