I am reading the following preprint: https://arxiv.org/pdf/1010.5821.pdf and my goal is to deduce (A.3) from (A.2) [see page 9].
Given $f\in L^{2^*}(\mathbb{S}^n)$ we define $F\in L^{2^*}(\mathbb{R}^n)$ such that $$F(x) = J_S^{1/2^*}(x) f(S(x))$$ where $S:\mathbb{R}^n\to \mathbb{S}^n$ such that $S(x)=\left(\frac{2x}{1+|x|^2},\frac{1-|x|^2}{1+|x|^2}\right)$ and the jacobian of the map $J_{S}(x) = \left(\frac{2}{1+|x|^2}\right)^{n}.$ Then I want to show that, $$\int_{\mathbb{R}^n}|\nabla F|^2 dx = \int_{\mathbb{S}^n} |\nabla f|^2 + \frac{n(n-2)}{4}|f|^2 d\omega.$$
I tried the compute the derivatives using the representation of $F$ but the computations do not seem to simplify.