Jacobian of a Change of Variable I am studying this article for my PhD, where every thing was quite clear except the Jacobian claimed in the proof of proposition 5 (which is not related to the topic of the article).
In particular, I am confused about the Jacobian of the change of variable
$$S^2\times\mathbb{R}^3\mapsto\mathbb{R}^3\times\{{p}\}^\perp$$
$$(w,V_*)\mapsto(p,q),$$
where $p=(V_*.w)w$ and $q=V_*-p$, which turned out to be
$$dV_*dw=\frac{2}{p^2\sin(p,p+q)}dpdq.$$
I found an answer on stack exchange which was to view the Jacobian the infinitesimal area of the paralellogram of sides $|p|dV_*$ and $|p|dw$. I didn't understand what does the side $|p|dV_*$ have to do here!I hope someone could give me a formal explanation, and I'm very grateful.
 A: Here is the begining of an answer. I did not make all the computations (especially in the end) but hope it will help you answer your own question.
Let $f$ be the differentiable function
\begin{align}
f : \mathbb{R}^3 \times \mathbb{R}^3 & \longrightarrow \mathbb{R}^3 \times \mathbb{R}^3 \\
(x,v) & \longmapsto \left(\langle x,v\rangle x, v-\langle x,v\rangle x \right)
\end{align}
Where $\langle \cdot,\cdot\rangle$ stands for the euclidean structure. It is differentiable as for $(h_1,h_2)\in \mathbb{R}^3\times\mathbb{R}^3$, one has
\begin{align}
f(x+h_1,v+h_2) = f(x,v)+ L_{x,v}(h_1,h_2) + o\left(\|h_1,h_2\|\right)
\end{align}
where
\begin{align}
L_{x,v}(h_1,h_2) = \left(\left(\langle h_1,v\rangle + \langle x,h_2 \rangle \right) x + \langle x,v\rangle h_1, h_2 -\left(\langle h_1,v\rangle + \langle x,h_2 \rangle \right) x - \langle x,v\rangle h_1  \right)
\end{align}
By restriction, $f$ is differentiable on $\mathbb{S}^2\times \mathbb{R}^3 \subset \mathbb{R}^3\times\mathbb{R}^3$ and the differentiable is just the restriction of $L$ on the tangent spaces.
I do not show that $f$ is injective and that $L_{x,v}$ is invertible on $T_{x,v}\left(\mathbb{S}^2\times \mathbb{R}^3 \right)$ as this is due to easy but messy computations. This shows that $f$ is a diffeomorphism from $\mathbb{S}^2\times \mathbb{R}^3$ to its image. Moreover, the image of $f$ is
\begin{align}
f\left(\mathbb{S}^2\times \mathbb{R}^3\right) = \bigcup_{x \in \mathbb{S}^2} \mathbb{R}^3\times \{x\}^{\perp}
\end{align}
We now have a clean writing of $f$, and the jacobian of $f$ is the determinant of $L_{x,v}$ in orthonormal bases.
Moreover, in the article they are using $f^{-1}$. Maybe the inverse function $f^{-1}$ has a nicer form and can help you. For the $\sin(p,p+q)$ part, I feel like, as $p+q = v$ and $\langle x,v\rangle = \langle p,v\rangle$ (easy computations), we can hope that using $f$ above show a formula like $f^*\mathrm{dVol}_{RHS} = \langle x,v\rangle^2 \frac{\sin\left(x,v\right)}{2}\mathrm{dVol}_{LHS}$ (for volume on the left hand side and right hand side) and thus just use $f$.
