Let $q=(a_1,a_2,a_3)\in\mathbb{R}^3$ and denote $f(x)$ as the squared distance of $x$ from $q$. Compute $$\int_{S^2} f(x) dS $$
attempt:
Since $f \in C^1 $ we know $\nabla f $ is a smooth vector field. So apply the divergence theorem to get: $$\int_{S^2} \langle \nabla f , N \rangle dS = \int_{\mathbb{B}^3} \triangle fdx$$
Now:
- $\nabla f(x) = 2(x-q)$ hence $\lvert \nabla f \rvert = 2 \lVert x-q\rVert$
- $N = outward\ normal = x$
- $\triangle f = 6$
$\langle \nabla f , N \rangle = \langle \nabla f , x-q+q \rangle = 2f + \langle \nabla f , q \rangle$
I don't see how to proceed from here.